Graded multiplicities of the nullcone for the

algebraic symmetric pair of type G

by

Anthony Paul van Groningen

A Dissertation Submitted inPartial Fulfillment of the

Requirements for the Degree of

Doctor of Philosophyin

Mathematics

at

The University of Wisconsin-MilwaukeeMay 2007

Graded multiplicities of the nullcone for the

algebraic symmetric pair of type G

by

Anthony Paul van Groningen

A Dissertation Submitted inPartial Fulfillment of the

Requirements for the Degree of

Doctor of Philosophyin

Mathematics

at

The University of Wisconsin-MilwaukeeMay 2007

Major Professor Date

Major Professor Date

Graduate School Approval Date

ii

Abstract

Graded multiplicities of the nullcone for the

algebraic symmetric pair of type G

by

Anthony Paul van Groningen

The University of Wisconsin-Milwaukee, 2007Under the Supervision of Professor Allen D. Bell and

Professor Jeb W. Willenbring

The regular functions on the nullcone associated to the algebraic symmetric pair

(G2, so(4, C)) of Type G is decomposed as a graded representation of K = SO(4, C).

This is done in two complementary ways. First a formula for the graded multiplicities

in terms of the principal branching rule for sp(2, C) is derived using an application

of Howe duality. Second, an explicit formula for the q-multiplicity of each K-type is

given. As a consequence, a closed rational form for the Hilbert series of C [ G2 ]K is

determined.

Branching from a simple Lie algebra to a principal three-dimensional subalgebra is

discussed in the context of Kostant’s multiplicity formula. The support of the relevant

partition function is described in terms of the Bruhat order. Using the methods of

Heckman, asymptotic estimates for the principal branching multiplicities of a simple

Lie algebra g are shown to be related to the exponents g.

iii

The case of g = sp(2, C) is studied in further detail and these calculations are

used to describe the asymptotic distribution of the graded multiplicities in Type G.

The symmetric pair (G2, so(4, C)) leads to the study of binary (3, 1)-forms for

which a complete system of covariant generators is known. From this the Brion

polytope for the nullcone is determined.

Major Professor Date

Major Professor Date

iv

c© Copyright by Anthony Paul van Groningen, 2007All Rights Reserved

v

To Mum, Dad, and goddaughter Sophia.

In memory of Tony.

vi

Table of Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Lie theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Partitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Representations of SL(2, C) . . . . . . . . . . . . . . . . . . . 5

Representations of GL(2, C) . . . . . . . . . . . . . . . . . . . 7

Representations of SL(2, C)× SL(2, C) . . . . . . . . . . . . . 8

Representations of Sp(l, C) . . . . . . . . . . . . . . . . . . . . 8

1.1.2 The principal TDS . . . . . . . . . . . . . . . . . . . . . . . . 9

Kostant’s branching formula for uprin . . . . . . . . . . . . . . 11

1.1.3 Algebraic symmetric pairs . . . . . . . . . . . . . . . . . . . . 13

Functions on the nullcone . . . . . . . . . . . . . . . . . . . . 15

Graded multiplicities . . . . . . . . . . . . . . . . . . . . . . . 17

2 Principal branching multiplicity 20

2.1 Bruhat order and weight restriction . . . . . . . . . . . . . . . . . . . 21

2.1.1 Bruhat order and principal branching . . . . . . . . . . . . . . 24

2.2 µ-Partition functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Asymptotics for principal branching . . . . . . . . . . . . . . . 32

vii

2.3 Principal branching for sp(2, C) . . . . . . . . . . . . . . . . . . . . . 35

3 Graded multiplicities in Type G 43

3.1 The symmetric pair (G2, so4) . . . . . . . . . . . . . . . . . . . . . . . 43

3.2 An alternating formula for graded multiplicities . . . . . . . . . . . . 52

3.3 a priori formulae for the q-multiplicities . . . . . . . . . . . . . . . . 56

3.3.1 Step 1. The generating function . . . . . . . . . . . . . . . . . 58

3.3.2 Step 2: Averaging over the Weyl group . . . . . . . . . . . . . 63

3.4 Hilbert series for C [ G2 ]SO(4,C) . . . . . . . . . . . . . . . . . . . . . . 64

4 Invariant theory 68

4.1 Covariants of double binary forms . . . . . . . . . . . . . . . . . . . . 68

4.2 Syzygies for the binary (3, 1)-forms . . . . . . . . . . . . . . . . . . . 73

5 Asymptotic methods for graded multiplicities 77

5.1 The Brion polytope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.1.1 The Brion polytope for Type G . . . . . . . . . . . . . . . . . 79

5.2 Density of the graded multiplicities in Type G . . . . . . . . . . . . . 81

A Computer proof of Lemmas 3.10, 3.12 91

B Syzygies for (3, 1)-covariants 93

B.1 Program for computing syzygies . . . . . . . . . . . . . . . . . . . . . 93

B.2 Computing the Hilbert series in Macaulay2 . . . . . . . . . . . . . . . 100

viii

List of Figures

2.1 L−1µ (10) ∩ E≥0 with µ = (2, 1). . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Hasse diagram for the Weyl group of sp(2, C) with nodes labeled by

Lw(k, l, m) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.1 Vogan diagram for G2. . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.1 The Brion polytope for the nullcone of (G2, so(4, C)). . . . . . . . . . 80

5.2 Partitioning of the Brion polytope. . . . . . . . . . . . . . . . . . . . 87

5.3 Graph of the density function z = g(x, y). . . . . . . . . . . . . . . . 90

ix

List of Tables

3.1 Structure constants N(i, j) for G2. . . . . . . . . . . . . . . . . . . . 45

3.2 Structure constants αj(hi) for G2. . . . . . . . . . . . . . . . . . . . . 46

4.1 Todd’s generators for Cov(F 3,1). . . . . . . . . . . . . . . . . . . . . . 72

5.1 Piecewise linear approximation for the graded multiplicities. . . . . . 84

5.2 `w(x, y) for w ∈ W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3 Density of the graded multiplicities. . . . . . . . . . . . . . . . . . . . 88

x

Acknowledgements

This work would not of been possible without the continuing support of my advisors

Allen Bell and Jeb Willenbring. I am incredibly grateful for their time, patience,

and insight. In addition, I wish to thank committee members Craig Guilbault, Ian

Musson, and Yi Ming Zou for their input into this work. Let me also assert my

gratitude to Mark Teply who sadly passed away last year.

I have had so many wonderful professors at UWM who have influenced me both

as a mathematician and as an educator—Ric Ancel, Eric Key, Ming Lee, James

Arnold, Jay Beder, Hans Volkmer, Boris Okun, and the aforementioned members

of my committee—thank you all. I am grateful to the departmental staff who have

always been eager to assist in the day-to-day activities of a graduate student. Fellow

students at UWM have helped make my time here a pleasure. In particular, I must

thank Mirek Pryszczepko and Joseph Shomberg for their friendship these past years.

I am grateful to live in a time and place where intellectual pursuits are accessible

to all and not merely the folly of the elite. This would be impossible without the

efforts of those in my family who came before—be they butcher or bullockie, engineer

or florist, soldier or pirate.

Thank you Mum, Dad, Monique, and Gerard for your support through this ar-

duous process and thanks to my friends Jonathan, Jason, Brad, and Andy for your

unwavering mateship.

While preparing this dissertation I was a Research Assistant for Dr. Willenbring

with funding provided by the Graduate School Research Committee Award.

xi

The whole of arithmetic now appearedwithin the grasp of mechanism.

Charles Babbage

I rest not, ’tis best not, the world is a wide one–And, caged for an hour, I pace to and fro;I see things and dree things and plan while I’m sleeping,I wander for ever and dream as I go.

I have stood by Table Mountain,On the Lion at Capetown,

And I watched the sunset fadingFrom the roads that I marked down;

And I looked out with my brothersFrom the heights behind Bombay,

Gazing north and west and eastward,Over roads I’ll tread some day.

For my ways are strange ways and new ways and old ways,And deep ways and steep ways and high ways and low;I’m at home and at ease on a track that I know not,And restless and lost on a road that I know.

The Wander-lightHenry Lawson

xii

1

Chapter 1

Introduction

The theory of algebraic symmetric pairs plays a crucial role both in the classification

of real forms of simple complex Lie algebras and in the classification of compact

symmetric spaces. The central object of study in this dissertation is the symmetric

pair (g, k) where g is the exceptional complex Lie algebra of type G2 and k = so(4, C).

This pair corresponds to the split real form of G2. Let K = SO(4, C). There is a

Cartan decomposition g = k⊕ p where p is an irreducible representation of K. By a

theorem of Kostant and Rallis, the algebra C [ p ] of regular functions on p is a free

module over the algebra C [ p ]K of K-invariant functions on p. Precisely,

C [ p ] ∼= C [ p ]K ⊗H

where H is the space of harmonic functions on p. Moreover, H = ⊕d≥0Hd inherits a

gradation from C [ p ] with each homogeneous component Hd invariant under K. This

leads to the question of how Hd decomposes as a representation of K.

This problem has a geometric interpretation. Let J be the ideal of C [ p ] generated

by the u ∈ C [ p ]K satisfying u(0) = 0. Then the vanishing set of J determines an

affine variety N ⊂ p called the nullcone. Restricting the harmonic functions to the

nullcone gives an isomorphism of graded K-representations H ∼= C [N ]. Understand-

ing the graded decomposition of H under the group K is equivalent to understanding

2

the graded decomposition of C [N ].

In Chapter 3 these questions are settled. First we give a formula for the graded

multiplicities in terms of a branching rule from sp(2, C) to the so-called principal TDS.

While elegant, this formula involves an alternating sum which at first glance obscures

matters. A second approach describes the so-called q-multiplicity of each irreducible

representation of K. For a given K-type λ, the q-multiplicity is a polynomial in q

whose coefficient of qd records the multiplicity of λ in Hd. By computing a generating

function for the q-multiplicities, various combinatorial information is deduced. In

particular, a closed form for the Hilbert series of the algebra C [ G2 ]SO(4,C) is computed.

The K-structure of H is closely related to a certain subgroup M of K. The lift of M

to the simply-connected covering group Spin(4, C) of K is described.

In Chapter 2, principal branching rules analogous to that used in computing the

graded multiplicities are discussed. Kostant’s multiplicity formula is used to com-

pute the branching multiplicities in terms of a partition function. The support of

these partition functions are described in terms of the Bruhat order. Employing the

methods of Heckman, the asymptotic behavior of the branching rule for a simple Lie

algebra g to its principal TDS is shown to be related to the exponents of g. Additional

analysis is carried out in the case of sp(2, C). This is put to use in Chapter 5 where

an asymptotic description of the graded multiplicities for the pair (G2, so(4, C)) is

provided. The distribution of graded multiplicities is shown to behave in a piecewise

linear fashion. The Brion polytope for the nullcone is determined, thus providing an

asymptotic picture of the support of the graded multiplicities.

In Chapter 4, the symmetric pair (G2, so(4, C)) is shown to be related to a problem

in classical invariant theory. In 1946, J. A. Todd found a complete set of generators

for the algebra of covariants of the double binary (3, 1)-forms. Exploiting the above

results and using a computer, polynomial identities among the covariant generators

for the (3, 1)-forms are determined.

3

1.1 Background

1.1.1 Conventions

Lie theory

Let g be a complex semisimple Lie algebra of rank l > 0 and let G be its adjoint

group. Thus, G is the connected complex Lie group with Lie algebra g. Each choice

of Cartan subalgebra h of g determines a root system Φ := Φ(h) ⊂ h∗. Choose a set

of positive roots Φ+ for Φ and let ∆ ⊂ Φ+ be a base of simple roots for Φ. Let β ∈ Φ+

and write β =∑

α∈∆ nα α, where the nα are non-negative integers. The height of β

is the number ht(β) defined by

ht(β) =∑α∈∆

nα. (1.1)

Let E be the real span of Φ in h∗ and let ( , ) denote the restriction of the Killing

form to E. As ( , ) is non-degenerate on h∗ we can associate to each λ ∈ E an element

Hλ ∈ h satisfying (λ, γ) = γ(Hλ) for all γ ∈ h∗. Let

hλ :=2Hλ

(λ, λ).

The weight lattice of g is the set

P (g) = λ ∈ h∗ | (λ, hα) ∈ Z for all α ∈ ∆ .

The dominant integral weights of g is the set

P+(g) = λ ∈ P (g) | (λ, hα) ≥ 0 for all α ∈ ∆ .

4

The set P+(g) parametrizes the finite-dimensional regular representations of g by

associating to each λ ∈ P+(g) the representation L(λ) with highest weight λ.

Write ∆ = α1, . . . , αl . The fundamental weights for g are the $1, . . . , $l ∈ h∗

satisfying

〈$i, hαj〉 = δij

for 1 ≤ i, j ≤ l. Since g is semisimple,

P+(g) =l⊕

i=1

Z≥0 $i.

Let T ⊂ G be a maximal torus in G. Then G ∼= (C×)l as an algebraic group where

the number l is the rank of G (and of g). A rational character of G is a homomorphism

of algebraic groups χ : T → C×. Every rational character is determined by an l-tuple

λ = (λ1, λ2, . . . , λl) of integers. The corresponding character is denoted eλ whose

value at (s1, . . . , sl) ∈ T is

eλ = eλ(s1, . . . , sl) = sλ11 · · · sλl

l

where the character eλ is identified with the corresponding rational expression.

For a finite-dimensional regular representation V of G, the character of V is the

rational character ch(V ) : T → C×

ch(V ) =∑

λ∈P (g)

dimC V (λ) eλ

where

V (λ) =

v ∈ V | s.v = eλ(s)v for all s ∈ T

.

Suppose W = ⊕d≥0Wd is a graded vector space such that each Wd is G-invariant. By

5

the q-character of W we mean the formal expression

chq (W ) =∑d≥0

ch (Wd) qd

where q is an indeterminate. The Hilbert series of W is the formal expression

Hilb(W ) =∑d≥0

(dimC Wd) qd.

Partitions

A partition λ is a finite decreasing sequence λ = (λ1 ≥ λ2 ≥ . . . ≥ λn ≥ 0) of non-

negative integers. We write |λ | :=∑

i λi for the size of λ. If N = |λ | then we say

λ is a partition of N . The depth of λ is the number `(λ) := sup i | λi 6= 0 . The

conjugate partition of λ, denote λ] is the partition whose ith entry is

(λ])

i= # j | i ≤ λj .

Define λ! by λ! =∏`(λ)

i λi! We need the following simple observation.

Lemma 1.1. Let λ = (λ1 ≥ λ2 ≥ . . . ≥ λn ≥ 0) be a partition. Then

λ]! = 1λ12λ2 · · ·nλn .

Representations of SL(2, C)

Let K = SL(2, C). For k ∈ Z≥0, let F k = SkC2 denote the kth symmetric power of

the standard representation C2 of K. By convention, let F k = 0 for k < 0. Write

C2 = spanCx0, x1 . Then F k may be taken as the space of homogeneous k-forms

6

in the variables x0 and x1. A typical f ∈ F k can be written as

f =k∑

i=0

(k

i

)ai x

k−i0 xi

1. (1.2)

For each k we have a representation ρk : K → GL(F k) given by linear change of

coordinates. Explicitly, for g = [ a bc d ] ∈ K

ρk(g)(f) =k∑

i=0

(k

i

)(ax0 + bx1)

k−i (cx0 + dx1)i (1.3)

where f is given by (1.2).

Let T denote the maximal torus of K consisting of the diagonal matrices in K.

The character of the representation F k is denoted χk and its value at diag(s, s−1) ∈ T

is

χk =sk+1 − s−k−1

s− s−1.

Let k = Lie(K). Then

X =

0 1

0 0

, H =

1 0

0 −1

, Y =

0 0

1 0

is a suitable basis for k. The triple (H, X, Y ) satisfy the commutation relations

[ H, X ] = 2X, [ H, Y ] = −2Y, [ X, Y ] = H. (1.4)

Every finite-dimensional irreducible representation of k occurs as a differential dρk :

k → gl(F k). Explicitly, k acts on F k by the differential operators

dρk(X) = x0∂

∂x1

, dρk(H) = x0∂

∂x0

− x1∂

∂x1

, dρk(Y ) = x1∂

∂x0

. (1.5)

7

Theorem 1.2. For each k ≥ 0, there exists a unique (up to scalar) K-invariant

non-degenerate bilinear form Γk on F k. If k is even then Γk is symmetric; if k is odd

then Γk is skew-symmetric.

Proof. As K-representations, F k ⊗ F k ∼= S2F k ⊕ ∧2F k with

S2F k ∼=⊕i≥0

F 2k−4i and ∧2 F k ∼=⊕j≥0

F 2k−4j−2. (1.6)

The trivial representation F 0 = C occurs in the symmetric square when k is even and

in the exterior square when k is odd. Projection onto the trivial term determines (up

to scalar) a K-equivariant map Γk : S2F k → C or Γk : ∧2F k → C as the case may

be.

If k ≥ 0 is even (resp. odd) set G(k) = SO(F k, Γk) (resp. G(k) = Sp(F k, Γk)).

Then Theorem 1.2 provides an embedding ρk : K → G(k). The differential dρk : k →

Lie(G(k)) determines an embedding of Lie algebras.

Lemma 1.3. Let g = Lie(G(k)) and l = rank(g). As a representation of k,

g ∼=l⊕

i=1

F 2(2i−1).

Proof. When k is even (resp. odd) the adjoint representation is ∧2F k (resp. S2F k)

and l = k/2 (resp. l = (k + 1)/2). The K-decomposition of g is then provided by

(1.6).

Representations of GL(2, C)

Let G = GL(2, C). The finite dimensional polynomial representations of G are indexed

by partitions λ = (λ1 ≥ λ2 ≥ 0). We denote the representation corresponding to such

a λ by F λ2 . The reason for this notation is motivated by the following fact.

8

Proposition 1.4. [Ful97, Section 8.2] The restriction of the GL(2, C)-representation

F λ2 to SL(2, C) is F λ1−λ2.

Representations of SL(2, C)× SL(2, C)

The group K := SL(2 C)× SL(2, C) plays a crucial role in this dissertation. For each

k, l ≥ 0, let F k,l = SkC2⊗SlC2 and let

ρk,l = ρk⊗ρl : SL(2, C)× SL(2, C) → GL(F k,l).

We have written ⊗ to emphasize that this is an outer tensor product with each factor

of SL(2, C) acting independently on either SkC2 or SlC2. The space F k,l may be

viewed as the binary (k, l)-forms. A typical element f ∈ F k,l is of the form

f =k∑

i=0

l∑j=0

(k

i

)(l

j

)ai,j xk−i

0 xi1y

l−j0 yj

1.

The action of K is given by the left (resp. right) factor of SL(2, C) changing co-

ordinates in the x0, x1 (resp. y0, y1 ) as in (1.3). Let k = Lie(K). Then

k = sl(2, C) ⊕ sl(2, C) ∼= so(4, C). The finite-dimensional irreducible representations

of k are the differentials dρk,l : k → gl(F k,l). The action k is given by the differential

operators (1.5) in both sets of variables.

Representations of Sp(l, C)

Let G = Sp(l, C) and let g = sp(l, C). Let $1, . . . , $l be the fundamental weights for

g corresponding to a choice of Cartan subalgebra of g and compatible positive roots.

Since g is simple, each λ ∈ P+(g) is of the form

λ =l∑

i=1

ai $i

9

where each ai ∈ Z≥0. Alternatively, λ may be expressed as the partition of depth at

most l

λ = (λ1 ≥ λ2 ≥ . . . ≥ λl ≥ 0)

where λi − λi+1 = ai, i = 1, . . . , l. We denote the irreducible representation corre-

sponding to λ by V λl or just V λ when the rank of G is clear.

1.1.2 The principal TDS

Let g be a simple complex Lie algebra of rank l > 0. The adjoint representation ad :

g → gl(g) of g is given by ad(X)(Y ) = [ X, Y ], for all X, Y ∈ g. An element X ∈ g is

called semisimple (resp. nilpotent) if the operator ad(X) : g → g is semisimple (resp.

nilpotent). Let G be the adjoint group of g. Then G acts on g via the adjoint action

Ad : G → GL(g) which we refer to simply as conjugation.

Definition 1.5. 1. A triple (H, X, Y ) of non-zero linear independent elements of

g is called a standard triple of g if H, X, Y satisfy the commutation relations

(1.4).

2. A Lie subalgebra u of g is called a TDS (three-dimensional subalgebra) of g if

u ∼= sl(2, C) as Lie algebras.

Clearly, any standard triple of g spans a TDS of g. If u is a TDS of g then

u contains a standard triple (H, X, Y ). In this case, H is necessarily a semisimple

element of u and X, Y are nilpotent in u. These properties also hold as elements of g

by the preservation of Jordan form [Hum78, Corollary 6.4].

Lemma 1.6. Suppose (H, X, Y ) is a standard triple in g. Then H is a semisimple

element of g and X, Y are nilpotent elements of g.

If X ∈ g and u is any subset of g then the centralizer of X in u is defined as

uX = Y ∈ u | [ X, Y ] = 0 .

10

Definition 1.7. 1. X ∈ g is called regular in g if dimC gX ≤ dimC gY for all Y ∈ g.

2. A standard triple (H, X, Y ) in g is called a principal standard triple of g if X

is regular.

3. A TDS u of g is called a principal TDS g if it contains a regular nilpotent

element of g.

Clearly, a TDS u of g is a principal TDS if and only if it is spanned by a principal

standard triple.

Let h be a Cartan subalgebra for g and let Φ := Φ(h). Choose a positive root

system Φ+ for Φ and let ∆ ⊂ Φ+ be a base of simple roots. Let E denote the real

span of Φ and let ( , ) denote the restriction of the Killing form to E. Set N = |Φ+ |,

and h = 2Nl

. The number h is the Coxeter number of g.

Theorem 1.8. [CM93, Theorem 3.4.12,4.1.6] Let H0 be the unique element of h

satisfying α(H0) = 2 for all α ∈ ∆. Then there exists a principal standard triple

(H0, X, Y ) of g containing H0. Let uprin be the principal TDS spanned by H0, X, and

Y . Any other principal TDS of g is conjugate to uprin.

Remark 1.9. The algebra uprin is unique only up to choice of Cartan subalgebra h

and choice of positive root system for Φ(h). On the other hand, suppose u is a

principal TDS spanned by a principal standard triple (H, X, Y ). Then there is a

Cartan subalgebra h of g such that H ∈ h and the set ∆ := α ∈ Φ(h) | α(H) = 2

is a base for Φ(h). Relative to this data, u is the same principal TDS defined in 1.8.

Consider g as a representation of uprin via the restriction of the adjoint represen-

tation ad : uprin → gl(g). The decomposition of g into irreducible representations of

uprin is a beautiful result connecting many important invariants of g.

Theorem 1.10. [Kos59, Theorem 5.2] Let u be a TDS of g. Then g decomposes into

l odd-dimensional irreducible representations of u if and only if u is a principal TDS.

11

In this case,

g =l⊕

i=1

F 2ei

where e1, . . . , el are non-negative integers.

By Lemma 1.3 we obtain the following description of the principal TDS for the

simple groups sp(l, C) = sp2lC and so(2l + 1, C).

Corollary 1.11. Let u = sl(2, C), k ≥ 1, and let g = spkC (when k is even) or

so(k, C) (when k is odd). Then the image dρk(u) of the map dρk : u → g is a

principal TDS.

The integers e1, . . . , el of 1.10 are invariants called the exponents of g. In fact,

the numbers 2ei + 1, i = 1, . . . , l are the Betti numbers of the complex manifold G

[CM93, Section 4.4]. We need the following connection between the exponents of g

and root heights.

Theorem 1.12. [Hum90, Theorem 3.20] Write the exponents of g in decreasing order

λ := (e1 ≥ e2 ≥ . . . ≥ el ≥ 0) as a partition. Then

1. |λ | =∑l

i=1 ei = |Φ+ |.

2. e1 = h− 1 and el = 1. In particular, the conjugate partition λ] has depth h− 1.

3. Write λ] = (k1 ≥ k2 ≥ . . . ≥ kh−1 ≥ 0). Then

ki = #

α ∈ Φ+ | ht(α) = i

.

In particular, k1 = l, k2 = l − 1, and kh−1 = 1.

Kostant’s branching formula for uprin

For λ ∈ P+(g) and k ≥ 0 define

Bg(k, λ) = dimC HomSL(2,C)

(F k, L(λ)

). (1.7)

12

Since all principal three-dimensional subalgebras are conjugate, the multiplicities (1.7)

are independent of the choice of Cartan subalgebra and positive root system used to

define uprin.

Suppose uprin is the span of the principal standard triple (H0, X, Y ) with X regular

in g and

α(H0) = 2, for all α ∈ ∆. (1.8)

Then hprin := C H0 is a Cartan subalgebra for uprin contained in h. For γ ∈ h∗, write

γ ∈ h∗prin for the restriction of γ to hprin. Let δ be the unique element of E satisfying

(δ, α) = 2 for all α ∈ ∆. Then δ(H0) = 2 and h∗prin = C δ. We may identify h∗prin with

C by identifying δ with 2 ∈ C. By (1.4), Φprin := 2,−2 is the root system of uprin

determined by hprin and Φ+prin := 2 is a set of positive roots for Φprin.

Let λ ∈ E. Since ∆ is a basis for E we can write λ =∑

α∈∆ nα α. Then

λ =∑α∈∆

nα α = 2∑α∈∆

nα = (δ, λ). (1.9)

In particular, if β ∈ Φ+ is a positive root then by (1.1)

β = 2 ht(β). (1.10)

Consider the partition function ℘g : R → R defined by

∑k∈Z≥0

℘g (k) qk =1− q2∏

β∈Φ+ (1− q2 ht(β)). (1.11)

As in 1.12, let ki denote the number of positive roots of g of height i, i = 1, ..., h− 1.

Then, by (1.10)

ki = #

α ∈ Φ+ | α = 2i

.

13

By (1.11), ∑k∈Z≥0

℘g (k) qk =1− q2∏h−1

i=1 (1− q2i)ki. (1.12)

Let Wg denote the Weyl group of g and let

ρ =1

2

∑α∈Φ+

α.

For each w ∈ Wg, define Lw : R× E → R by

Lw(k, λ) = w(λ + ρ)− ρ− k. (1.13)

Theorem 1.13 (Principal branching formula). For λ ∈ P+(g) and k ≥ 0,

Bg(k, λ) =∑

w∈Wg

sgn(w) ℘g (Lw (k, λ)) .

Moreover, if Bg(k, λ) 6= 0 then λ− k ∈ 2Z.

Proof. Use [GW98, Theorem 8.2.1] or [Kna02, Theorem 9.20]. The partition function

used in Kostant’s formula is obtained by considering the multiset Φ+ \Φ+prin.

By (1.11), ℘g (n) = 0 for n odd. Since all simple roots restrict to 2, all λ in the

root lattice of g restrict to even values. The parity of each Lw(k, λ) agrees as w ∈ W

varies because the Weyl group preserves the root lattice of g. Thus the parity of each

Lw(k, λ) is determined by the parity of L1(k, λ) = λ− k.

1.1.3 Algebraic symmetric pairs

In this section we review the necessary facts from the seminal paper [KR71] of Kostant

and Rallis. Let g be a semisimple complex Lie algebra and suppose θ : g → g is an

involutional automorphism of g. Thus θ is an automorphism of Lie algebras satisfying

14

θ2 = 1. Let

k = X ∈ g | θ(X) = X

and

p = X ∈ g | θ(X) = −X

denote the +1-eigenspace of θ and the −1-eigenspace of θ, respectively. Then g = k⊕p

is called the Cartan decomposition of g. Moreover, k is a Lie subalgebra of g and p

is a representation of k via the adjoint action. The pair (g, k) is called an algebraic

symmetric pair.

Let G be the adjoint group of g. We let Kθ be the subgroup of G consisting of

those elements which commute with θ

Kθ = g ∈ G | Ad(g)θ = θ Ad(g) .

Alternatively, since G is connected, θ determines an automorphism Θ : G → G of Lie

groups and

Kθ = g ∈ G | Θ(g) = g .

In fact, k and p are both invariant under Kθ. Moreover, k is the Lie algebra of Kθ

and p is a representation of Kθ.

A Lie subalgebra u of g is called a toral subalgebra if it is abelian and each element

of u is semisimple in g. Any two maximal toral subalgebras of g which contained in p

are conjugate under the action of Kθ. Thus, the dimension r = dimC a of any maximal

toral subalgebra a of p is an invariant of the symmetric pair (g, k) and is called the

real rank. The following definition generalizes the notion of regularity defined earlier

for the a djoint case.

Definition 1.14. An element X ∈ p is called regular in p if dimC kX ≤ dimC kY , for

all Y ∈ p.

15

Let R denote the set of regular elements of p, S the set of semisimple elements of

p, and N the set of nilpotent elements p. Then N is called the nullcone of p.

Lemma 1.15. [KR71, Prop. 8] Let X ∈ p. Then X ∈ R if and only if dimC pX = r.

Maximal toral subalgebras arise as the centralizers of regular semisimple elements.

Lemma 1.16. [KR71, Lemma 20] Let X ∈ p and let a := pX . Then X ∈ R ∩ S if

and only if a is a maximal toral subalgebra of p. In this case, the centralizer of a in

Kθ is equal to the stabilizer of X in Kθ

g ∈ Kθ | Ad (g) a = a = g ∈ Kθ | Ad (g) X = X .

Functions on the nullcone

Let C [ p ] denote the space of regular functions on p. Then C [ p ] is a representation

of Kθ according to the rule

(k.f)(X) = f(k−1X),

for all k ∈ Kθ, f ∈ C [ p ], and X ∈ p. Let C [ p ]Kθ be the subring of C [ p ] of

Kθ-invariant functions. The Chevalley Restriction Theorem asserts that C [ p ]Kθ =

C [ u1, . . . , ur ] is a polynomial ring minimally generated by invariants u1, . . . , ur. Note

that the number of generators is the real rank r.

Lemma 1.17. The ideal of C [ p ] generated by the Kθ invariants u1, . . . , ur is a radical

ideal and the variety it defines is the nullcone

N = X ∈ p | ui(X) = 0 for i = 1, . . . , r . (1.14)

Moreover, if Kθ is a connected group then N is irreducible as an affine variety.

16

Proof. The fact that N is given by (1.14) is [KR71, Prop. 11]. By [KR71], R∩N is

a single Kθ-orbit which is dense in N . Hence, if Kθ is connected N is irreducible as

an affine algebraic variety.

Let S (p) be the symmetric algebra on p viewed as a representation of Kθ. Let

∂ : p → EndC (p) be the map associating to each X ∈ p its directional derivative ∂X

(∂X f) (Y ) = limt→0

f(Y + tX)− f(Y )

t, for all Y ∈ p.

This extends uniquely to a Kθ-map ∂ : S (p) → EndC (p). Moreover, C [ p ] ∼= S (p) as

algebras and as graded Kθ-representations. We may then view ∂ : C [ p ] → EndC (p).

Definition 1.18. A function f ∈ C [ p ] is called harmonic if ∂(u)f = 0 for all

u ∈ C [ p ]Kθ satisfying u(0) = 0.

Let H (p) ⊂ C [ p ] denote the space of harmonic functions on p. Then

H (p) = f ∈ C [ p ] | ∂(ui)f = 0, for i = 1, . . . , r .

For d ≥ 0, define

Hd (p) = H (p) ∩ C [ p ]d .

Then H (p) = ⊕d≥0Hd(p) is a graded vector space and each Hd (p) is Kθ-invariant.

Lemma 1.19. Restriction of functions from p to the nullcone N is an isomorphism

of graded Kθ-representations H (p) ∼= C [N ].

A central result in [KR71] is the following separation of variables for C [ p ].

Theorem 1.20. [GW98, Theorem 12.4.1] Let a be a maximal toral subalgebra of p

and let Mθ be the centralizer of a in Kθ. Then multiplication of functions gives an

17

isomorphism of Kθ-representations.

H (p)⊗ C [ p ]Kθ ∼= C [ p ] .

As a representation of Kθ,

H (p) ∼= IndKθMθ

(1) (1.15)

is induced from the trivial representation of Mθ.

The key step to proving Theorem 1.20 is to observe that if Z ∈ a is a regular

semisimple element of p then the restriction of functions from p to the Kθ-orbit OZ

of Z is an isomorphism of Kθ-representations H (p) ∼= C [OZ ]. Now, OZ∼= Kθ/Mθ

by Lemma 1.16. Thus, C [OZ ] ∼= C [ Kθ/Mθ ] proving that H (p) is indeed induced

from the trivial representation of Mθ.

Graded multiplicities

Suppose λ ∈ P+(k) and L(λ) is the corresponding finite-dimensional irreducible rep-

resentation of Kθ with highest weight λ. By (1.15) and Frobenius reciprocity,

dimC HomKθ(L(λ),H (p)) = dimC

(L(λ)Mθ

).

In particular, L(λ) occurs in the harmonics with finite multiplicity. This fact, however,

does not explain in which degrees L(λ) appears. One would like to know the graded

multiplicity of λ in H (p), by which we mean the numbers

fd(λ) := dimC HomKθ(L(λ),Hd(p)) , for λ ∈ P+(k) and d ∈ Z≥0. (1.16)

18

It is convenient to arrange these statistics as the coefficients of a polynomial. For

λ ∈ P+(k), the q-multiplicity of λ is defined as

pλ(q) :=∑d≥0

fd(λ) qd (1.17)

where q is an indeterminate. In particular,

pλ(1) = dimC HomKθ(L(λ),H(p)) = dimC

(L (λ)Mθ

)< ∞.

Lemma 1.21. Let di be the degree of the Kθ invariant ui, i = 1, . . . , r. Let wt(p∗) ⊂

P (k) be the set weights of p∗ as a Kθ-representation. Then

∑λ∈P+(k)

pλ(q) ch (L(λ)) =

∏ri=1(1− qdi)∏

γ∈wt(p∗)(1− qeγ),

Proof. We have

chq C [ p ] =1∏

γ∈wt(p∗) ( 1− q eγ ). (1.18)

By (1.17),

chq H(p) =∑

λ∈P+(k)

pλ(q) ch (L(λ)) .

By Theorem 1.20,

chq C [ p ] = chq C [ p ]Kθ · chq H(p)

=1∏r

i=1 (1− qdi)· chq H(p).

Solving for chq H(p) and combining with (1.18) gives the desired result.

For examples of symmetric pairs see Goodman and Wallach [GW98, Ch. 12.4.3]

or Helgason [Hel01, Ch. X, Section 6]. Hesselink has found an alternating formula

for the graded multiplicities in the adjoint case. Let k be a semisimple complex Lie

19

algebra and g = k⊕ k. Consider the pair (g, k) with k diagonally embedded in g.

Theorem 1.22 ([Hes80]). For λ ∈ P+(k) and d ≥ 0

fd(λ) =∑w∈W

sgn(w) ℘d (w (λ + ρ)− ρ) ,

where W is the Weyl group of K and ℘d (γ) is the number of ways to write

γ =∑

α∈Φ+k

nα α

with nα ∈ Z≥0 and∑

nα = d.

Other interesting cases where the graded multiplicity is understood are more spo-

radic. The pair (sl(4, C), so(4, C)) was treated in [WW00] while the pair (F4, Spin(9))

is multiplicity free [Joh76]. Wallach used an example of a symmetric pair to answer

a question related to the space of 4-qubits [Wal05].

20

Chapter 2

Principal branching multiplicity

There are two fundamental difficulties in applying a branching formula such as that

given by Theorem 1.13

Bg(k, λ) =∑w∈W

sgn(w) ℘g (Lw(k, λ)) .

Firstly, the alternation over the Weyl group W of g will result in including many

terms which contribute nothing to the total. When Lw(k, λ) < 0, ℘((Lw(k, λ)) =

0 and including such terms may be avoided. In Section 2.1 we make the simple

observation that w ∈ W is more likely to fall into this category as w gets larger in

the sense of the Bruhat order.

Secondly, it is difficult to obtain useful expressions for evaluating the partition

function ℘g. In Section 2.2 we apply G. J. Heckman’s asymptotic methods to show

that the principal branching multiplicity is asymptotically related to the exponents

of g. In Section 2.3 the principal branching rule for sp(2, C) is investigated in further

detail.

21

2.1 Bruhat order and weight restriction

Let E be a Euclidean space of dimension l > 0 with standard basis ε1, . . . , εl and

inner-product ( , ). Let Φ ⊂ E be an irreducible crystallographic root system of rank

l and make a choice Φ+ ⊂ Φ of positive roots. Let ∆ ⊂ Φ+ be a base of simple roots

for Φ. Each root β determines a reflection tβ ∈ GL(E) according to the formula

tβ(λ) = λ−Nβ(λ) β,

where λ ∈ E and

Nβ(λ) :=2(λ, β)

(β, β).

It is a part of the definition of a root system that Nβ(α) ∈ Z whenever α, β ∈ Φ.

If α ∈ ∆ then we write sα in place of tα and we let S = sα | α ∈ ∆ be the

corresponding set of simple reflections.

Let W be the Weyl group determined by the above data, i.e. W is the (finite)

subgroup of GL(E) generated by S. In fact, W acts orthogonally on E which may be

expressed as saying

(λ, wγ) = (w−1λ, γ),

for all w ∈ W and λ, γ ∈ E. For each w ∈ W the length of w is the number

`(w) = #

α ∈ Φ+ | wα /∈ Φ+

.

The Bruhat order of W is then the partial ordering < on W defined as the transitive

closure of the relations

w < w · tβ ⇐⇒ `(w) < `(w · tβ),

for w ∈ W and β ∈ Φ+. Note that < ultimately depends on the choice of simple

22

roots ∆. We need the following fact which connects the Bruhat order with the action

of W on E.

Lemma 2.1. [BB05, Prop. 4.4.6] For all w ∈ W and β ∈ Φ+

w(β) ∈ Φ− ⇐⇒ `(w · tβ) < `(w).

Let

D := λ ∈ E | (λ, α) ≥ 0 for all α ∈ ∆ .

We refer to elements of D as dominant vectors. Fix a dominant vector δ ∈ D and

let J = α ∈ ∆ | (δ, α) = 0 . The stabilizer of δ in the Weyl group is the so-called

parabolic subgroup WJ := 〈 sα | α ∈ J 〉. Define

JW = w ∈ W | sα · w > w for all α ∈ J .

Then JW forms the set of the unique minimal length right coset representatives for

the quotient WJ\W . Moreover, (JW, <) is a partially ordered set under the restricted

Bruhat order. 1

Definition 2.2. For λ ∈ D and k ∈ R, define

Tδ(k, λ) = w ∈ W | (δ, wλ) ≥ k .

Recall that a subset A of a partially ordered set (P, <) is called an order ideal of

P if for all a, b ∈ P

b < a and a ∈ A =⇒ b ∈ A.

Proposition 2.3. Let λ ∈ D, k ∈ R. Set A := Tδ(k, λ) and B := A ∩ JW . Then

1See [BB05] or [Hum90] for details regarding parabolic subgroups and quotients. Both texts,however, focus on the left coset representatives of the quotient W/WJ .

23

1. A is an order ideal in W ;

2. B is an order ideal in JW ;

3. A =⋃

JW · w where the union is taken over the maximal elements of B.

Proof. Let w ∈ W , β ∈ Φ+, and let w′ = w · tβ. For any λ ∈ E,

(δ, w′λ) = (w−1δ, tβλ) = (w−1δ, λ)−Nβ(λ)(w−1δ, β)

So,

(δ, wλ) = (δ, w′λ) + Nβ(λ)(δ, wβ).

If λ ∈ D is dominant then Nβ(λ) ≥ 0. If, in addition, w < w′ then `(w) < `(w′) and

wβ ∈ Φ+ by Lemma 2.1. Since δ ∈ D we also have (δ, wβ) ≥ 0.

By the definition of Bruhat order we’ve shown for all w,w′ ∈ W ,

w < w′ =⇒ (δ, w′λ) ≤ (δ, wλ) for all λ ∈ D.

In particular, if w < w′ ∈ W and w′ ∈ Tδ(k, λ) then (δ, wλ) ≥ (δ, w′λ) ≥ k. Hence,

w ∈ Tδ(k, λ) and Tδ(k, λ) is an order ideal.

Suppose now that w ∈ Tδ(k, λ) and u ∈ WJ is in the stabilizer of δ. Then

(δ, uwλ) = (u−1δ, wλ) = (δ, wλ) ≥ k.

This shows that if w ∈ A then the right coset WJ · w ⊂ A. The remaining claims

follow from this fact.

Let

ρ =1

2

∑α∈Φ+

α.

24

Definition 2.4. For λ ∈ D and k ∈ R, define

T ∗δ (k, λ) = w ∈ W | (δ, w(λ + ρ)− ρ) ≥ k .

Lemma 2.5. For any λ ∈ E and k ∈ R

T ∗δ (k, λ) = Tδ(k + (δ, ρ), λ + ρ).

In particular, if λ ∈ D then T ∗δ (k, λ) is an order ideal of W .

Proof. For λ ∈ E,

w ∈ T ∗δ (k, λ) ⇐⇒ (δ, w(λ + ρ)− ρ) ≥ k

⇐⇒ (δ, w(λ + ρ))− (δ, ρ) ≥ k

⇐⇒ (δ, w(λ + ρ)) ≥ k + (δ, ρ)

⇐⇒ w ∈ Tδ(k + (δ, ρ), λ + ρ).

If λ is dominant then so is λ + ρ. By 2.3, Tδ(k + (δ, ρ), λ + ρ) is an order ideal.

2.1.1 Bruhat order and principal branching

Suppose now that g is a simple complex Lie algebra with rank l ≥ 2 and fix a

Cartan subalgebra h of g. Let Φ := Φ(h) and choose a set Φ+ of positive roots

for Φ. Let ∆ ⊂ Φ+ be a base of simple roots for Φ. Enumerate the simple roots

∆ = α1, . . . , αl . Let E be the real span of Φ and W the Weyl group of g with

Bruhat order < determined by ∆. Let uprin be the principal TDS of g determined by

the above data as in Theorem 1.8. Let δ ∈ E satisfying (δ, α) = 2 for all α ∈ ∆. The

next proposition refines Theorem 1.13.

25

Proposition 2.6. For k ≥ 0, λ ∈ P+(g), λ− k ∈ 2Z

Bg(k, λ) =∑w∈S

sgn(w) ℘g (Lw(k, λ))

where S may be taken to be the order ideal T ∗δ (k, λ) of (W, <). In particular, if S = ∅,

then Bg is identically zero. If λ− k < 0 then Bg(k, λ) = 0.

Proof. By Section 1.1.2,

Lw(k, λ) = w(λ + ρ)− ρ− k = (δ, w(λ + ρ)− ρ)− k.

By Definition 2.4, Lw(k, λ) ≥ 0 if and only if w ∈ T ∗δ (k, λ).

Suppose λ − k < 0. Then (δ, λ) < k and it must be that the identity element

of W is not in T ∗δ (k, λ). Since T ∗

δ (k, λ) is an order ideal, T ∗δ (k, λ) = ∅. Therefore,

℘g (Lw(k, λ)) = 0 for all w ∈ W and the branching multiplicity vanishes.

2.2 µ-Partition functions

In this section we apply G. J. Heckman’s method [Hec82] for establishing asymptotic

estimates of partition functions for a special case relevant to the partition function

(1.11). We then use this to establish an asymptotic estimate for the principal branch-

ing multiplicities Bg(k, λ).

Fix an integer M > 0 and choose a partition µ = (µ1 ≥ µ2 ≥ . . . µn ≥ 0) of M .

We associate to µ a partition function ℘µ : R → R given by the generating function

∑k≥0

℘µ(k) qk =1∏n

i=1 (1− qi)µi. (2.1)

26

Thus, for a non-negative integer k, ℘µ(k) is the number of ways to write

k =n∑

i=1

i

(µi∑

j=1

ai,j

),

where each ai,j must be a non-negative integer. In particular, ℘µ(0) = 1. Note

that since each µi ≥ 0, ℘µ(k) is finite. If k ∈ R is not a non-negative integer then

℘µ(k) = 0.

Let E = RM be a Euclidean space of dimension M . The partition µ determines

an indexing of the standard basis

E = span e(i, j) | 1 ≤ i ≤ n, 1 ≤ j ≤ µi .

Define Lµ : E → R to be the linear function determined by

Lµ (e (i, j)) = i, for 1 ≤ i ≤ n, 1 ≤ j ≤ µi.

Set

E≥0 :=∑

1≤i≤n1≤j≤µi

R≥0 e(i, j).

Then

℘µ(k) = # of integral points in L−1µ (k) ∩ E≥0.

For example, Figure 2.1 shows the set L−1µ (10) ∩ E≥0 ⊂ R3 for µ = (2, 1).

Heckman’s approach to estimating ℘µ(k) is to compute the volume of L−1µ (k)∩E≥0.

normalized by the volume of a fundamental block formed by the integral points.

The integral solutions to the equation

n∑i=1

i

(µi∑

j=1

ai,j e(i, j)

)= 0 (2.2)

27

0.0

0.00

2.5

e(1,2)

1

5.0

2.5

7.5

2

10.0

5.0

e(2,1)

12.5

3

e(1,1)

4

7.5

5

10.0

6

12.5

Figure 2.1: L−1µ (10) ∩ E≥0 with µ = (2, 1).

in the unknowns ai,j form a lattice Kµ of rank M − 1 in ker Lµ. Define vectors

vi,j = −i · e(1, 1) + 1 · e(i, j), for 1 ≤ i ≤ n, 1 ≤ j ≤ µi, (i, j) 6= (1, 1). (2.3)

These M − 1 non-zero vectors are independent and determine a set of generators for

Kµ as an abelian group

Kµ =⊕

1≤i≤n1≤j≤µi

(i,j) 6=(1,1)

Z vi,j.

Let Nµ denote the volume of a fundamental domain in Kµ.

Lemma 2.7.

Nµ =

√√√√ n∑i=1

µi i2. (2.4)

Proof. We must compute the volume of the (M − 1)-parallelepiped P determined by

28

the M − 1 vectors vi,j

P :=

∑

1≤i≤n1≤j≤µi

(i,j) 6=(1,1)

ti,j vi,j | 0≤ ti,j ≤ 1 for all i, j

.

Then Nµ =√

det (AAt), were A is the matrix whose rows are the vi,j’s. Making

the natural choice of ordering of the vi,j’s we may take

A =

−1... Iµ1−1

−1−2

... Iµ2

−2−3

... Iµ3

−3

.... . .

−n... Iµn

−n

(M−1)×M

where Ik denotes the k × k identity matrix. By [Blo79, 5.7.4],

det(AAt

)=

M∑k=1

(det Bk)2

where Bk is the (M−1)×(M−1)-submatrix of A obtained by deleting the kth column

from A. By performing minor expansion along the kth-row of Bk the determinant is

easily computed: by deleting any of the first µ1 columns of A the resulting submatrix

has squared determinant 1; by deleting any of the next µ2 columns of A the resulting

29

submatrix has squared determinant 2; and so on. Hence,

det(AAt

)=

n∑i=1

µii2.

Recall that µ] is the conjugate partition of µ and µ! =∏n

i=1 µi!.

Lemma 2.8. For k ≥ 0,

vol(L−1

µ (k) ∩ E≥0

)=

Nµ

(M − 1)! µ]!kM−1.

Proof. Let ui,j = ki· e(i, j), for 1 ≤ i ≤ n, 1 ≤ j ≤ µi. Then L−1

µ (k) ∩ E≥0 is the

convex hull in E of the set ui,j | 1 ≤ i ≤ n, 1 ≤ j ≤ µi . This is an (M −1)-simplex

in RM whose volume is given by

1

(M − 1)!

√det (AAt)

where A is the matrix with rows wi,j := ui,j − u1,1, 1 ≤ i ≤ n, 1 ≤ j ≤ µi, (i, j) 6=

(1, 1). With the appropriate ordering of the wi,j’s, we may take

30

A =

−k... k · Iµ1−1

−k−k

...k

2· Iµ2

−k−k

...k

3· Iµ3

−k

.... . .

−k...

k

n· Iµn

−k

(M−1)×M

.

In a similar fashion as Lemma 2.7 we delete columns to obtain the determinant

of AAt

det(AAt

)=

n∑i=1

µi

[kµ1(

k2

)µ2 · · ·(

kn

)µn

ki

]2

=n∑

i−1

µi i2

[kM−1

1µ12µ2 · · ·nµn

]2

which by Lemma 1.1 is

=n∑

i−1

µi i2

[kM−1

µ]!

]2

= N2µ

[kM−1

µ]!

]2

.

Taking the square root and dividing by (M − 1)! obtains the result.

The volume of L−1µ (k) ∩ E≥0 normalized by Nµ provides the following first order

estimate.

Proposition 2.9. Let µ be a partition and set M = |µ |.

31

1. There exists a constant C > 0 such that for all non-negative integers k,

∣∣∣∣℘µ(k)− kM−1

(M − 1)! µ]!

∣∣∣∣ ≤ C (1 + k)M−2 .

2. Assume that `(µ) ≥ 2, i.e. µ1 ≥ µ2 > 0. Fix ξ ∈ Z. There exists a constant

Cξ > 0 such that for all k ∈ Z,

∣∣∣∣℘µ(k + ξ)− kM−1

(M − 1)! µ]!

∣∣∣∣ ≤ Cξ (1 + k)M−2 .

Proof. In the notation of [Hec82, Section 2] we have V = R, L = Z, A = i | µi > 0 ,

mi = µi. Then∑

i∈A mi = M and the Z-span of A is the rank one lattice Z in V .

Apply [Hec82, Lemma 2.3] for part (1).

For part (2), if i ∈ A then A\ i still determines a rank one lattice by assumption

on the depth of µ. We can then apply [Hec82, Lemma 2.4].

Example 2.10. Let µ = (1, 1, 1). The generating function for ℘µ is

1

(1− q)(1− q2)(1− q3).

We have M = 3, µ] = (3), and µ]! = 3! = 6. The estimate given by 2.9 is k2/12. In

fact, for integral k ≥ 0

℘µ(k) =

[(k + 3)2

12

]where [n] denotes the nearest integer to n [Slo].

Example 2.11. Let µ = (2, 1). The generating function for ℘µ is

1

(1− q)2(1− q2).

We have M = 3, µ] = (2, 1), µ]! = 2. The estimate in this example is k2/4. In fact,

32

for integral k ≥ 0

℘µ(k) =

⌊(k + 2)2

4

⌋where bnc denotes the floor of n [Slo].

2.2.1 Asymptotics for principal branching

Let g be a simple complex Lie algebra with rank l ≥ 2 and fix a Cartan subalgebra

h of g. Let Φ := Φ(h) and choose a set of positive roots Φ+ for Φ. Let ∆ ⊂ Φ+ be a

base of simple roots for Φ. Let E denote the real span of Φ and let W be the Weyl

group of g. Let N = |Φ+ | be the number of positive roots and let h = 2Nl

be the

Coxeter number of g. For i = 1, . . . , h− 1, let ki be the number of positive roots with

height i. Let

µ = (k1 − 1 ≥ k2 ≥ k3 ≥ . . . ≥ kh−1).

Then µ is indeed a partition by Theorem 1.12 (3) and |µ | = N − 1. Let

λ = (e1 ≥ e2 ≥ . . . ≥ el)

be the exponents of g written in decreasing order. Note that µ] 6= λ due to the

subtraction in the first entry of µ.

Lemma 2.12. µ]! = λ!

Proof. Use the fact that el = 1.

By (2.1),

∑k≥0

℘µ(k) qk =1

(1− q)k1−1∏n

i=2(1− qi)ki=

1− q∏h−1i=1 (1− qi)ki

33

Comparing with the partition function ℘g defined in (1.11),

∑k≥0

℘g(k) qk =1− q2∏h−1

i=1 (1− q2i)ki

=∑k≥0

℘µ(k) q2k.

Thus,

℘g =

℘µ(k/2), if k is even;

0, otherwise.

In this context, Proposition 2.9 is restated as follows.

Lemma 2.13. 1. There exists a constant C > 0 such that for all non-negative

k ∈ 2Z, ∣∣∣∣℘g(k)− (k/2)N−2

(N − 2)! e1!e2! · · · el!

∣∣∣∣ ≤ C (1 + k)N−3 .

2. Fix ξ ∈ Z. There exists a constant Cξ > 0 such that for all k ∈ Z satisfying

k − ξ ∈ 2Z,

∣∣∣∣℘g(k + ξ)− (k/2)N−2

(N − 2)! e1!e2! · · · el!

∣∣∣∣ ≤ Cξ (1 + k)N−3 .

Proof. |µ | = (∑

ki)− 1 = |Φ+ | − 1 = N − 1. Use Lemma 2.12 and Proposition 2.9

(1) with M = N − 1.

For (2), note that µ1 = k1 − 1 = k2 = l − 1 ≥ 0 by Theorem 1.12 (3) and the

assumption on the rank of g. Thus, `(µ) ≥ 2 and we can apply Proposition 2.9

(2).

Let

Pg(k) =

(k/2)N−2

(N−2)! e1!e2!···el!, if k ≥ 0;

0, otherwise.

Notice that Pg(nk) = nN−2 Pg(k) for all n > 0. Heckman defines the asymptotic

branching function by replacing ℘g with its asymptotic counterpart Pg. Define bg :

34

R× E → R by

bg(k, λ) =∑

w∈Wg

sgn(w) Pg

(wλ− k

). (2.5)

Then

bg(nk, nλ) = nN−2bg(k, λ).

Recall that the principal branching multiplicity is

Bg(k, λ) =∑

w∈Wg

sgn(w) ℘g (Lw (k, λ)) (2.6)

with Lw(k, λ) defined in (1.13). The next theorem specializes [Hec82, Lemma 3.6] to

the current situation.

Theorem 2.14. Let g be a simple complex Lie algebra of rank l ≥ 2. Let N be the

number of positive roots of g and let e1, . . . , el be the exponents of g. There exists a

constant C > 0 such that for all λ ∈ P+(g) and k ≥ 0 with λ− k ∈ 2Z we have

|Bg(k, λ)− bg(k, λ) | ≤ C(1 + λ

)N−3.

Proof. By (2.6) and (2.5),

|Bg(k, λ)− bg(k, λ) | ≤∑w∈W

∣∣∣℘g(w(λ + ρ)− ρ− k)− Pg(wλ− k)∣∣∣

=∑w∈W

∣∣∣℘g(wλ− k + w(ρ)− ρ)− Pg(wλ− k)∣∣∣ .

For each w ∈ W , apply Lemma 2.13 with ξ = w(ρ)− ρ to find constants Cw > 0 such

35

that

|Bg(k, λ)− bg(k, λ) | ≤∑w∈W

Cw(1 + |wλ− k|)N−3

≤ C0(1 + λ + k)N−3.

where C0 > 0 is constant.

If λ− k ≥ 0, then

|Bg(k, λ)− bg(k, λ) | ≤ C0(1 + 2λ)N−3 ≤ C(1 + λ)N−3,

for some constant C > 0.

If λ− k < 0, Bg(k, λ) = 0 so that for n ≥ 0,

| bg(k, λ) | = 1

nN−2| bg(nk, nλ) | ≤ C ′

nN−2(1 + nλ + nk)N−3.

As n →∞, | bg(k, λ) | → 0. Thus, bg(k, λ) = 0 when λ− k < 0.

2.3 Principal branching for sp(2, C)

In this section we describe in detail the principal branching rule in the rank two

symplectic case. In particular, we obtain a quadratic estimate of the branching mul-

tiplicities which will be useful in Section 5.2.

Let K = SL(2, C) and let V = F 3 be the four-dimensional irreducible representa-

tion of K with ordered basis x30, 3 x2

0x1, 3 x0x21, x3

1 . By 1.2, V has a skew-symmetric

non-degenerate K-invariant form Γ, which in terms of the given basis may be taken

36

to be

J =

0 0 0 1

0 0 −3 0

0 3 0 0

−1 0 0 0

(See [GW98, 5.1.22]). Then GΓ = Sp(F 3, Γ) is the rank two symplectic group and

the embedding ρ3 : K → GΓ is explicitly given by

a b

c d

7→

a3 3 a2c 3 ac2 c3

a2b a + 3 abc 3 bc2 + 2 c c2d

ab2 3 b2c + 2 b 3 bcd + d cd2

b3 3 b2d 3 bd2 d3

,

where ad− bc = 1. By definition,

GΓ =

g ∈ GL(F 3) | ρ3(g)t J ρ3(g) = J

.

Let g = Lie(GΓ) = sp(2, C) and k = Lie(K). By Corollary 1.11 the image of the

differential dρ3 : k → g is a principal TDS of g.

The exponents of g are λ := (3 ≥ 1) and the conjugate partition is then λ] =

(2, 1, 1). Indeed, g has 2 simple roots of height 1; 1 positive root with height 2; and

the unique highest root has height 3. By (1.11), the partition function ℘g for the

principal branching multiplicities is determined by the generating function

∑k≥0

℘g(k) qk =1− q2

(1− q2)2(1− q4)(1− q6)=

1

(1− q2)(1− q4)(1− q6).

While we could apply the asymptotic methods of this chapter, the rank is sufficiently

37

small to find an exact formula for ℘g(k). In fact,

℘g(n) =

[

(n+6)2

48

], if n ≥ 0 is even;

0, otherwise

(2.7)

where [n] denotes the nearest integer to n [Slo].

Let h be a Cartan subalgebra for g, Φ := Φ(h), and let E be the real span of Φ.

There exists a basis ε1, ε2 for E such that

∆ := α1 := ε1 − ε2, α2 := 2 ε2 (2.8)

is a base of simple roots for Φ and the vectors

Φ+ := α1, α2, α3 := α1 + α2, α4 := 2 α1 + α2

form a set of positive roots for Φ. The Killing form restricted to E is then the usual

inner product ( , ) relative to the basis ε1, ε2 . Let ρ := 12

∑α∈Φ+ α = 2 ε1 +ε2. The

dominant weight lattice is given by

P+(g) = Z≥0 $1 ⊕ Z≥0 $2,

where

$1 = ε1 and $2 = ε1 + ε2 (2.9)

are the fundamental weights. The finite-dimensional irreducible representations V λ2 =

V λ of g are parametrized by λ = l $1 + m $2, l,m ∈ Z≥0.

The Weyl group W := Wg has order 8 and is generated by the simple reflections

38

si ∈ GL(E) corresponding to the simple roots αi, i = 1, 2. Precisely,

s1(a ε1 + b ε2) = b ε1 + a ε2

s2(a ε1 + b ε2) = a ε1 − b ε2

Index the elements of W by the symbols wi, i = 1, . . . , 8 as follows

w1 = 1, w2 = s1, w3 = s2,

w4 = s1s2, w5 = s2s1, w6 = s1s2s1,

w7 = s2s1s2, w8 = s1s2s1s2 = s2s1s2s1.

The Bruhat order < on W determined by the simple reflections s1 and s2 is depicted

in Figure 2.2 by giving the Hasse diagram of the poset (W, <).

Let δ ∈ E such that (δ, α1) = 2 = (δ, α2). By (2.8)

δ = 3ε1 + ε2. (2.10)

Recall that for w ∈ W , Lw : R× E → R is defined by

Lw(k, λ) = w(λ + ρ)− ρ− k,

where k ∈ R, λ ∈ E, and λ = (δ, λ). We prefer to work in the coordinates of the

fundamental weights $1 and $2. By (2.9) and (2.10),

(δ,$1) = (δ, ε1) = (3ε1 + ε2, ε1) = 3

39

and

(δ,$2) = (δ, ε2) = (3ε1 + ε2, ε1 + ε2) = 4.

Thus,

l $1 + m $2 = 3 l + 4 m. (2.11)

To simplify notation, write

Lw(k, l, m) := Lw(k, λ), where λ = l $1 + m $2.

For i = 1, . . . , 8 we abuse notation and write Li for Lwiwhen the context requires.

For each w ∈ W , the value of Lw(k, l, m) appears next to the corresponding node for

w in Figure 2.2.

Definition 2.15. For k, l, m ≥ 0 define

A(k, l, m) = w ∈ W | Lw(k, l, m) ≥ 0 .

We set A(k, l, m) = ∅ when any of k, l, m are negative. A subset S ⊂ W is called

admissible if S = A(k, l, m) for some k, l, m ≥ 0.

Lemma 2.16. The non-empty admissible subsets of W are in the list

1 , 1, 2 , 1, 3 , 1, 2, 3 , 1, 2, 3, 4 , 1, 2, 3, 5 , (2.12)

where we have identified wi ∈ W with its index i.

Proof. A(k, l, m) = T ∗δ (k, λ) where λ = l $1 + m $2. Thus an admissible sub-

set is an order ideal of W . By Figure 2.2, we observe that L8(k, l, m), L7(k, l, m),

and L6(k, l, m) are strictly negative when k, l, m ≥ 0 from which we conclude that

40

L2 : l + 4m− 2− k

L4 : l − 2m− 8− k

L6 : −3l − 2m− 12− k

L1 : 3l + 4m − k

L8 : −3l − 4m − 14 − k

L3 : 3l + 2m− 2− k

L5 : −l + 2m− 6− k

L7 : −l − 4m− 12− k

Figure 2.2: Hasse diagram for the Weyl group of sp(2, C) with nodes labeled byLw(k, l, m)

w8, w7, w6 never appear in an admissible subset.

It follows by direct calculation that

Lw(k, l, m) + Lw′(k, l, m) = −2k − 14 < 0

whenever w′ = w8w. In particular, w4 and w5 never appear in an admissible subset

together as w4 = w8w5. Therefore, any admissible subset must be contained in either

M1 := 1, 2, 3, 4 or M2 := 1, 2, 3, 5 . It is possible to demonstrate k, l, m ≥ 0 with

A(k, l, m) appearing in (2.12) and these are exactly the order ideals contained in M1

or M2.

Denote the principal branching multiplicity of Proposition 1.13 by

B(k, l, m) := Bg(k, (l + m, m)). (2.13)

41

Proposition 2.17. For integers k, l, m ≥ 0, B(k, l, m) = 0 whenever k − l /∈ 2Z;

Proof. Let λ = l$1 + m$2. By 1.13, Bg(k, λ) = 0 when λ − k is odd. By (2.11),

λ− k = 3l + 4m− k which is even if and only if l and k have matching parity.

To obtain a strong estimate of the branching multiplicities, we replace ℘g(n) with

the quadratic (n + 6)2/48. For this to be effective, particular attention must be paid

to which elements of the Weyl group are relevant in the computation.

Definition 2.18. For S ⊂ W , let

QS(k, l, m) =∑w∈S

sgn(w)1

48(Lw(k, l, m) + 6)2 . (2.14)

Lemma 2.19. There exists a constant C > 0 such that for all integers k, l, m ≥ 0,

k − l ∈ 2Z,

|B(k, l, m)−QS(k, l, m) | ≤ C, (2.15)

where S = A(k, l, m).

Proof. Recall that [ n ] denotes the nearest integer to n. Then

|B(k, l, m)−QS(k, l, m) | ≤∑w∈S

∣∣∣∣ (Lw(k, l, m) + 6)2

48−[(Lw(k, l, m) + 6)2

48

] ∣∣∣∣≤∑w∈S

1 = |S | ≤ |W | = 8.

Remark 2.20. Empirically, we have observed that the error in (2.15) has the following

tight bound

|B(k, l, m)−QS(k, l, m) | ≤ 11/12,

for k, l, m ≥ 0, k − l ∈ 2Z.

42

Remark 2.21. Let U ⊂ GΓ and N ⊂ K be the naturally chosen maximal unipotent

subgroups. The algebra

A := C [ N\GΓ/U ] := f ∈ C [ GΓ ] | f(ngu) = f(g) for all n ∈ N, g ∈ GΓ, u ∈ U

is the appropriate model for understanding the branching rule and was studied ex-

tensively by Papageorgiou in [Pap98]. In particular, a minimal system of generators

is determined for A is determined. Moreover, the author gives a formula for the

branching rule albeit a recursive one.

43

Chapter 3

Graded multiplicities in Type G

3.1 The symmetric pair (G2, so4)

In this section we review the construction of the Cartan involution which gives rise

to the algebraic symmetric pair (G2, so4). The primary reference is [Kna02, Chapter

VI]. Let g denote the complex simple Lie algebra of type G2 and let h be a Cartan

subalgebra of g. Set Φ := Φ (h). Then the real span of Φ is a 2-dimensional Euclidean

space E ⊂ h∗. Consider R3 with standard basis ε1, ε2, ε3 . Then E may be identified

with the plane

(x1, x2, x3) ∈ R3 | x1ε1 + x2ε2 + x3ε3 = 0 .

Define the following vectors in E:

α1 =ε1 − ε2, α2 =− 2ε1 + ε2 + ε3,

α3 =α1 + α2, α4 =2α1 + α2, (3.1)

α5 =3α1 + α2, α6 =3α1 + 2α2.

Then Φ may be taken as Φ+ ∪ (−Φ+) where Φ+ = αi | 1 ≤ i ≤ 6 [Kna02, C.2].

The set ∆ = α1, α2 is then a base of simple roots for Φ.

44

The root space decomposition for g is

g = h⊕∑α∈Φ

gα

where

gα = X ∈ g | [ H, X ] = α(H)X for all H ∈ h .

Let κ : g × g → C denote the Killing form of g. Since κ is non-degenerate on h we

can identify h with h∗. In particular, for each root α ∈ h∗ let Hα ∈ h satisfying

κ(H, Hα) = α(H).

The Killing form determines a non-degenerate bilinear form on h∗ according to

( α, β ) = κ(Hα, Hβ), for α, β ∈ h∗.

The realization (3.1) of the root system is constructed so that ( , ) agrees with the

usual inner product on E ⊂ R3 relative to the basis ε1, ε2, ε3 .

For each root α ∈ Φ, let

hα =2Hα

( α, α ).

To simplify notation write α−i for the negative root −αi, where 1 ≤ i ≤ 6. Similarly,

set hi = hαiand h−i = h−αi

= −hi.

For i = 1, . . . , 6 there exist non-zero root vectors Xi ∈ gαiand X−i ∈ g−αi

satisfying

(I) [ hi, hj ] = 0, for 1 ≤ i, j ≤ 6;

(II) [ hi, Xj ] = αj(hi) Xj, for 1 ≤ i ≤ 6, 1 ≤ | j | ≤ 6;

(III) [ Xi, X−i ] = hi, for 1 ≤ i ≤ 6;

45

(IV) [ Xi, Xj ] = 0, when αi + αj /∈ Φ, j 6= −i;

(V) [ Xi, Xj ] = N(i, j) Xk, when αi + αj = αk ∈ Φ.

The structure constants N(i, j) satisfy

(i) N(i, j) = −N(−i,−j);

(ii) N(i, j) = ε(i, j)(1 + p) where ε(i, j) = ±1 and p ≥ 0 is the maximal integer

satisfying αj − pαi ∈ Φ, i 6= j.

The choice of signs ε(i, j) is not uniquely determined. By [Sam90, Section 2.8, Prop.

A],

(a) ε(i, j) = −ε(j, i), 1 ≤ | i | , | j | ≤ 6;

(b) ε(i, j) = ε(j, k) = ε(k, i), for pairwise independent roots αi, αj, αk ∈ Φ satisfying

αi + αj + αk = 0

N(i, j) 1 2 3 4 5 6

1 1 2 3

2 −1 1

3 −2 −3

4 −3 3

5 −1

6

N(i,−j) 1 2 3 4 5 6

1 3 2 1

2 −1 1

3 −3 1 −2 −1

4 −2 2 −1 1

5 −1 1 −1

6 −1 1 −1 1

Table 3.1: Structure constants N(i, j) for G2.

In Table 3.1 the constants N(i, j), 1 ≤ i, j ≤ 6 are provided with the same choice

of signs as [Dok98]. This determines N(−i,−j), 1 ≤ i, j ≤ 6 by (i). To determine

the mixed case N(i,−j), 1 ≤ i, j ≤ 6 one uses (b). The remaining structure of g is

encoded by the constants αi(hj) which appear in Table 3.2.

The Cartan involutions of g are classified by its Vogan diagram. For Type G the

diagram appears in Figure 3.1 [Kna02, VI.10] and it determines a unique involution

46

αj(hi) α1 α2 α3 α4 α5 α6

h1 = HL 2 −3 −1 1 3 0

h2 −1 2 1 0 −1 1

h3 −1 3 2 1 0 3

h4 1 0 1 2 3 3

h5 1 −1 0 1 2 1

h6 = HR 0 1 1 1 1 2

Table 3.2: Structure constants αj(hi) for G2.

θ : g → g. The “unpainted” node of Figure 3.1 corresponds to the short simple root

α1 while the “painted” node corresponds to the long simple root α2.

By the proof of [Kna02, Theorem 6.88], θ |h is the identity; θ(X1) = X1; and

θ(X2) = −X2. For αi ∈ Φ, θ(Xi) = aiXi, for some constants ai = ±1. The ai satisfy

ai+j = aiaj whenever αi, αj and αi + αj are roots. By (3.1),

θ(X1) = X1, θ(X2) = −X2,

θ(X3) = −X3, θ(X4) = −X4, (3.2)

θ(X5) = −X5, θ(X6) = X6,

and θ(X−i) is similarly determined.

Let k (resp. p) denote the +1-eigenspace (resp. −1-eigenspace) of θ.

k = X ∈ g | θ(X) = X ,

p = X ∈ g | θ(X) = −X . (3.3)

Then g = k⊕ p, k is a Lie subalgebra of g, and p is a representation of k.

Figure 3.1: Vogan diagram for G2.

47

We have

k = gα1 ⊕ g−α1 ⊕ h⊕ gα6 ⊕ g−α6 = kL ⊕ kR

where

kL := gα1 ⊕ Ch1 ⊕ g−α1∼= sl(2, C),

kR := gα6 ⊕ Ch6 ⊕ g−α6∼= sl(2, C).

Indeed, the roots α1 and α6 are orthogonal so that the algebras kL and kR commute

and k ∼= kL ⊕ kR∼= so(4, C) as Lie algebras. Set

XL = X1, XR = X6,

HL = h1, HR = h6, (3.4)

YL = X−1, YR = X−6.

Then (HL, XL, YL) and (HR, XR, YR) are standard triples in g spanning kL and kR,

respectively.

By (3.2)

p =5∑

i=2

gαi⊕ g−αi

. (3.5)

The action of k on p can be organized as a 4 × 2 matrix with the arrows showing

which root space the corresponding operator moves a root vector.

0 ⇐ XR 0

gα5 g−α2

⇑ gα4 g−α3 YL

XL gα3 g−α4 ⇓

gα2 g−α5

0 YR ⇒ 0

The k-representation p can be identified with the representation dρ3,1 : k → GL(F 3,1).

48

The map φ : p → F 3,1 sending

X5 7→ x30y0, X−2 7→ x3

0y1,

X4 7→ 3 x20x1y0, X−3 7→ 3 x2

0x1y1,

X3 7→ 3 x0x21y0, X−4 7→ 3 x0x

21y1, (3.6)

X2 7→ x31y0 X−5 7→ x3

1y1

provides the isomorphism of k-representations. To see this, one compares the action

of the operators dρ3,1(k) with the structure constants of g in Tables 3.1 and 3.2. To

illustrate, compare the action of XL on the root vectors Xi ∈ p with the action of

dρ3,1(XL) on φ(Xi).

[ XL, X5 ] = 0, dρ3,1(XL)(x30y0) = 0,

[ XL, X4 ] = 3 X3, dρ3,1(XL)(3 x20x1y0) = 3 x3

0y0 = 3 φ(X3),

[ XL, X3 ] = 2 X4, dρ3,1(XL)(3 x0x21y0) = 6 x2

0x1y0 = 2 φ(X4),

[ XL, X2 ] = X3, dρ3,1(XL)(x31y0) = 3 x0x

21y0 = φ(X3).

Let G be the adjoint group of g and let Kθ be the subgroup of G whose elements

commute with θ

Kθ = g ∈ G | Ad(g).θ(X) = θ(Ad(g).X), for all X ∈ g .

Then Kθ is the connected Lie group SO(4, C) [GW98, Section 12.4.3]. Both k and

p are Kθ-invariant. It will be convenient to work with the simply connected cover

K := SL(2, C)× SL(2, C) ∼= Spin(4, C) of Kθ. We have

0 −→ Z2 −→ Kπ−→ Kθ −→ 0,

49

where Z2∼= 〈 (±I,±I) 〉.

The next task is to determine a maximal toral subalgebra of p. By 1.16, these arise

as the centralizers of regular semisimple elements of p. It is known that the symmetric

pair (g, k) has real rank equal to two and as such any maximal toral subalgebra is

two-dimensional [GW98, 12.4.3].

Lemma 3.1. Let Z = X5 +X−5 +√−1 (X3 + X−3). Then Z is a regular semisimple

element of p and the centralizer pZ is the maximal toral subalgebra of p spanned by

X3 + X−3 and X5 + X−5.

Proof. Both Z5 := X5 + X−5 and Z3 := X3 + X−3 are semisimple by 1.6. Since the

roots α5 and α3 are orthogonal [ Z5, Z3 ] = 0. In particular, ad(Z3) and ad(Z5) are

simultaneously diagonalizable. Hence, Z = Z5 +√−1 Z3 is semisimple in g. Direct

computation in G2 shows pZ is two-dimensional and spanned by Z5, Z3 . By Lemma

1.15, Z is regular in p.

Let a = pZ be the maximal toral subalgebra of p found in Lemma 3.1. Let Mθ be

the centralizer of a in Kθ

Mθ = g ∈ Kθ | Ad(g)(X) = X, for all X ∈ a .

The next theorem describes the lift of Mθ to K = SL(2, C)× SL(2, C).

Theorem 3.2. Let M = π−1(Mθ). Then M is isomorphic to the eight element non-

abelian group of basic unit quaternions

H := 〈±1,±i,±j,±k 〉/〈 i2 = j2 = k2 = ijk = −1 〉.

50

Explicitly, the embedding H → SL(2, C)× SL(2, C) is determined by

±1 7→ ±

1 0

0 1

,

1 0

0 1

±i 7→ ±

i 0

0 −i

,

i 0

0 −i

±j 7→ ±

0 1

−1 0

,

0 1

−1 0

±k 7→ ±

0 i

i 0

,

0 i

i 0

Proof. By Lemma 1.16, Mθ is the stabilizer in Kθ of Z. Hence, M is the stabilizer of

Z in K or equivalently the stabilizer of φ(Z) ∈ F 3,1. By (3.6),

f := φ(Z) = x30y0 + x3

1y1 + 3√−1(x0x

21y0 + x2

0x1y1

).

Let (g, h) ∈ K, det g = det h = 1. The equation ρ3,1((g, h))(f) − f = 0 is solved for

the entries in the matrices g and h resulting in the given embedding.

The algebra C [ p ]K of K-invariant polynomial functions on p is well-known to

be a polynomial algebra generated by two invariants u2 and u6 of degree 2 and 6,

respectively [GW98, Section 12.4.3]. This fact, however, will be observed in the

sequel. By Theorem 1.20 we have

C [ p ] = H⊗ C [ u2, u6 ] , (3.7)

51

where H is the space of harmonic functions

H = f ∈ C [ p ] | ∂(u2)f = 0 = ∂(u6)f . (3.8)

Let Hd = H ∩ C [ p ]d denote the harmonic functions of degree d. We view Hd as

a representation of K and denote the graded multiplicities defined in (1.16) by

fd(k, l) := dimC HomK

(F k,l,Hd

)= dimC HomKθ

(F k,l,Hd

), (3.9)

for integers d, k, l ≥ 0. In Section 3.2 we give a formula for the graded multiplicities

in terms of the branching rule studied in Section 2.3.

The q-multiplicities defined in (1.17) are denoted

pk,l(q) :=∑d≥0

fd(k, l) qd , (3.10)

where q is an indeterminate. In Section 3.3, we give explicit formulae for the q-

multiplicities as rational expressions in q.

Since Hd is first and foremost a representation of Kθ∼= SO(4, C) we observe the

following fact.

Lemma 3.3. If k − l /∈ 2Z then fd(k, l) = 0 for all d ≥ 0. In particular, pk,l(q) = 0

when the integers k and l have different parity.

Proof. The representations of SO(4, C) which lift to Spin(4, C) are precisely the F k,l

with k − l ∈ 2Z.

Let T denote the maximal torus of K consisting of the diagonal matrices

T =

s 0

0 s−1

,

t 0

0 t−1

| for s, t ∈ C×

.

52

For a representation V of T , the character of V is the rational function ch(V ) :

T → C×

ch(V )(s, t) =∑k,l∈Z

[dim V (k, l)] sktl,

where V (k, l) is the weight space

V (k, l) =

v ∈ V |(diag(s, s−1), diag(t, t−1)

).v = sktlv

.

The character of an irreducible K-representation F k,l is denoted χk,l = ch(F k,l) and

is given by

χk,l = χk,l(s, t) =(sk+1 − s−k−1)(tl+1 − t−l−1)

(s− s−1)(t− t−1). (3.11)

For example, the character of p = F 3,1 is

χ3,1 = s3t +s3

t+ st +

s

t+

t

s+

1

st+

t

s3+

1

s3t. (3.12)

By (3.12), the non-zero weight spaces of p are all one dimensional with characters

sktl, where k ∈ −3,−1, 1, 3 and l ∈ −1, 1 . The following is Lemma 1.21 for the

pair (G2, so(4, C)).

Lemma 3.4. ∑k,l≥0

pk,l(q) χk,l =(1− q2)(1− q6)∏

i∈−3,−1,1,3 j∈−1,1

(1− qsitj). (3.13)

3.2 An alternating formula for graded multiplici-

ties

In this section we use an application of Howe duality to derive an alternating for-

mula for the graded multiplicities in Type G. For convenience write K = KL ×KR

where KL and KR are both isomorphic to SL(2, C). Fix n > 0 and let V be the

53

irreducible representation F 2n−1⊗F 1 of K. By 1.2, F 2n−1 possesses a non-degenerate

KL-invariant skew-symmetric form Γ := Γ2n−1 which determines an embedding of KL

into a symplectic group. We have

K = KL ×KR → Sp2n×GL2, (3.14)

where Sp2n := Sp(F 2n−1, Γ) is the rank n symplectic group preserving Γ and GL2 =

GL(2, C). Viewed as a representation of Sp2n×GL2 we have V ∼= C2n⊕C2n ∼= M2n×2

with the groups acting via the matrix multiplication:

(g, h).X = gXh for all g ∈ Sp2n, h ∈ GL2 and X ∈ M2n×2.

Consider the algebra C [ V ] of polynomial functions on V . By [How95, Theorem

3.8.5.3],

C [ V ] ∼= H ⊗ C [ V ]Sp2n

where H is the space of symplectic harmonic functions, i.e. the polynomials annihi-

lated by constant coefficient differential operators which commute with the action of

Sp2n

H :=

f ∈ C [ V ] | ∂(θ)f = 0 for all θ ∈ C [ V ]Sp2n

. (3.15)

The First Fundamental Theorem of Invariant Theory for Sp2n describes the invariants.

By [GW98, Theorem 4.2.2],

C [ V ]Sp2n = C[C2n ⊕ C2n

]Sp2n = C [ ξ2 ]

where ξ2 is (up to scalar) the quadratic invariant determined by the form Γ. Specif-

ically, if X ∈ M2n×2, then ξ2(X) is given by contracting the columns of X via the

symplectic form Γ. Under the action of GL2, ξ2 transforms according to the determi-

54

nant. Thus,

H = f ∈ C [ V ] | ∂(ξ2)f = 0 .

The space H is graded by degree

H =⊕d≥0

Hd,

where Hd := H ∩ C [ V ]d. By [How95, 3.8.5.3], each graded component of H is

multiplicity-free under the action of Sp2n×GL2:

Hd =⊕

λ=(λ1≥λ2≥0)λ1+λ2=d

V λn ⊗F λ

2 . (3.16)

Notice, that only the representations λ with `(λ) ≤ 2 appear in (3.16).

Let g = sp(n, C). By restricting to K = KL × KR ⊂ Sp2n×GL2 we obtain the

following lemma.

Lemma 3.5. The K-representation F k,l occurs in Hd with multiplicity Bg (k, (l + m, m))

where d = l + 2m

Hd =⊕

k,l,m≥0d=l+2m

Bg (k, (l + m, m)) F k,l.

Proof. Write λ = (λ1 ≥ λ2) = (l + m, m) with l,m ≥ 0. If the weight λ occurs in

(3.16) then d = λ1 + λ2 = l + 2m. Restricting to KL ×GL2 we have

V λn ⊗F l

2 =⊕

k,l,m≥0

Bg (k, (l + m, m)) F k⊗F λ2 . (3.17)

The GL2-representation F λ2 restricts to the SL2-representation with highest weight

λ1 − λ2 = l. Hence, (3.17) becomes

V λn ⊗F l

2 =⊕

k,l,m≥0

Bg (k, (l + m, m)) F k⊗F l. (3.18)

55

To obtain a formula for the graded multiplicities (3.9) we specialize Lemma 3.5

to the case n = 2. Then g = sp(2, C) and V = M4×2 is in fact the K-representation

p ∼= F 3,1 in the G2 Cartan decomposition (3.3). Moreover, the Sp4 × GL2-invariant

ξ2 is the K-invariant u2. By direct computation, Bg(0, (3, 3)) = 1. Thus, when l = 0

and m = 3 there is a trivial KL-representation in degree d = 2l + m = 6. This proves

the existence of a K-invariant function u6 ∈ H6.

Theorem 3.6. Let B(k, l, m) := Bsp(2,C) (k, (l + m, m)). For integers k, l, m ≥ 0

fd(k, l) = B(k, l, m)−B(k, l, m− 3), (3.19)

where d = l + 2m.

Proof. The harmonic polynomials (3.8) are contained in the symplectic harmonics

(3.15)

H = f ∈ C [ p ] | ∂(u6)f = 0 = ∂(u2)f

= f ∈ H | ∂(u6)f = 0 . (3.20)

View u6 : Hd−6 → Hd and ∂(u6) : Hd → Hd−6 as linear operators. We obtain an

orthogonal decomposition1

Hd = ker ∂(u6)⊕ img u6.

By (3.20), ker ∂(u6) = Hd. Therefore,

Hd = Hd ⊕(u6 · Hd−6

). (3.21)

1〈f, g〉 := ∂(g)f is a Hermitian inner product on each Hd; the bar denotes complex conjugation.The operators u6 and ∂(u6) are adjoint with respect to this form.

56

By Lemma 3.5, the multiplicity of F k,l in H is B(k, l, m) where d = l + 2m. Since u6

is K-invariant, the multiplicity of F k,l in u6 · Hd−6 is B(k, l, m′) where d−6 = l+2m′.

Thus, m′ = m− 3. By (3.21),

fd(k, l) = B(k, l, m)−B(k, l, m′) = B(k, l, m)−B(k, l, m− 3).

yielding (3.19).

3.3 a priori formulae for the q-multiplicities

In this section we give explicit formulae for the q-multiplicities in Type G. Recall

that these are the polynomials (3.10)

pk,l(q) =∑d≥0

fd(k, l) qd, k, l ≥ 0,

where

fd(k, l) = dimC HomK

(F k,l,Hd

).

Surprisingly, we do not make use of Theorem 3.6 to achieve this result. In fact, we

will do not make any attempt at deriving these formulae, but rather give the answer a

priori and verify it is correct by comparing the appropriate generating functions. Each

q-multiplicity is presented as a rational expression; the general form is summarized

by the following theorem.

Theorem 3.7. For k, l ≥ 0,

pk,l(q) =pk,l(q)

(1− q2)(1− q4)(3.22)

where pk,l(q) is a polynomial divisible by (1−q2)(1−q4). If pk,l(q) 6= 0 then k−l ∈ 2Z.

57

Note the parity condition has already been observed. Before describing the nu-

merators pk,l(q) in (3.22), we need to define some auxiliary expressions.

Definition 3.8. Let bnc denote the floor of n. For k, l ∈ Z≥0 define

Ak(q) =

1 + q2 + q4 + q6 if k is even

q + 2q3 + q5 if k is odd

(3.23)

Bk,l(q) =

0 if k is odd

1− q2 if k and l are both even

q6(1− q2) if k is even and l is odd

(3.24)

L0(l) =

⌊l

6

⌋+

0 if l = 1 mod 6

1 otherwise

(3.25)

L2(l) =

⌊l

6

⌋+

0 if l = 0 mod 6

1 if l = 2, 3 mod 6

2 otherwise

(3.26)

L6(l) =

⌊l

6

⌋+

−1 if l = 0, 3 mod 6

1 if l = 4 mod 6

0 otherwise

(3.27)

L8(l) =

⌊l

6

⌋+

1 if l = 3, 5 mod 6

0 otherwise

(3.28)

58

K1(k) = L8(k) (3.29)

K2(k) =

⌊k

6

⌋+

0 if k = 0, 3 mod 6

1 if k = 1, 2, 5 mod 6

2 if k = 4 mod 6

(3.30)

Proposition 3.9. In the notation of Theorem 3.7,

pk,k+2l(q) = qk+2l[K1 (1− q8) + K2 q2(1− q4) + Bk,l(q)− q2l+4(1− q2k+2)

](3.31)

p3k+l,k+l(q) = qk+l[L0 + L2 q2 + q4 − L6 q6 − L8 q8 + q4k+2l+6 − qk+1Ak+1(q)

](3.32)

p2k+3l,l(q) = qk+l[Ak(q)− ql+1Ak+l+1(q)− qk+2(1− q4l+4)

](3.33)

where Ki := Ki(k) and Li := Li(l).

The proof of Proposition 3.9 and hence Theorem 3.7 is straightforward, but re-

quires a considerable amount of computation which is carried out in the following

two sections. The use of a computer algebra system to verify the correctness of the

calculation was indispensable. Appendix A contains Maple code which parallels the

following calculations.

3.3.1 Step 1. The generating function

The purpose of this section is to compute a closed form for the expression

W(X, Y ) :=∑k,l≥0

pk,l(q)

(1− q2)(1− q4)XkY l. (3.34)

First, the function n 7→⌊

n6

⌋is determined by the generating function

F(Z) :=∑n≥0

⌊n

6

⌋Zn =

Z6

(1− Z)(1− Z6). (3.35)

59

It is now straightforward to determine generating functions for the auxiliary expres-

sions (3.25)-(3.30).

L0(Y ) :=∑l≥0

L0(l) Y l

= F(Y ) +1 + Y 2 + Y 3 + Y 4 + Y 5

1− Y 6(3.36)

L2(Y ) :=∑l≥0

L2(l) Y l

= F(Y ) +2Y + Y 2 + Y 3 + 2Y 4 + 2Y 5

1− Y 6(3.37)

L6(Y ) :=∑l≥0

L6(l) Y l

= F(Y ) +−1− Y 3 + Y 4

1− Y 6(3.38)

L8(Y ) :=∑l≥0

L8(l) Y l

= F(Y ) +Y 3 + Y 5

1− Y 6(3.39)

K1(X) :=∑k≥0

K1(k) Xk

= L8(X) (3.40)

K2(X) :=∑k≥0

K2(k) Xk

= F(X) +X + X2 + 2X4 + X5

1−X6(3.41)

Next we determine generating functions for the various forms of Ak(q) and Bk,l(q)

that appear in Proposition 3.9. Since they all heavily depend on various parity condi-

60

tions it is useful to define the following intermediate expression to aid in programming

D(a, b, c, d ; X, Y ) :=a + bY + cX + dXY

(1−X2)(1− Y 2). (3.42)

The coefficient of XkY l in D(a, b, c, d; X, Y ) can be viewed as the (k, l)-entry in the

table below where the uppermost left corner is the (0, 0)-entry.

a c a c a

b d b d b

a c a c a

b d b d b

In particular,

D(a, b, 0, 0; X, 0) =∑k≥0

k even

a Xk +∑k≥0

k odd

b Xk.

In terms of D, we have

A1(X) :=∑k≥0

Ak+1(q) Xk

= D(q + 2q3 + q5, 1 + q2 + q4 + q6, 0, 0; X, 0) (3.43)

A2(X) :=∑k≥0

Ak(q) Xk

= D(1 + q2 + q4 + q6, q + 2q3 + q5, 0, 0; X, 0) (3.44)

A3(X) :=∑k,l≥0

Ak+l+1(q) XkY l

= D(q + 2q3 + q5, 1 + q2 + q4 + q6, 1 + q2 + q4 + q6, q + 2q3 + q5; X, Y )

(3.45)

B(X, Y ) :=∑k,l≥0

Bk,l(q) XkY l

= D(1− q2, 0, q6(1− q2), 0; X, Y ). (3.46)

61

We can now find generating functions for the three cases defined in Proposition

3.9. Define

P1(X, Y ) :=∑k,l≥0

pk,k+2l(q)XkY l

P2(X, Y ) :=∑k,l≥0

p3k+l,k+l(q)XkY l

P3(X, Y ) :=∑k,l≥0

p2k+3l,l(q)XkY l.

To compute P1 we first expand (3.31) and isolate any factors that are dependent

on k and l:

pk,k+2l(q) = (1− q8)qk+2lK1(k)

+ q2(1− q4)qk+2lK2(k)

+ qk+2lBk,l(q)

− q4qk+4l

+ q6q3k+4l.

Next, sum these coefficients for XkY l, k, l ≥ 0.

P1(X, Y ) = (1− q8)K1(qX)

1− q2Y

+ q2(1− q4)K2(qX)

1− q2Y

+ B(qX, q2Y )

− q4 1

(1− qX)(1− q4Y )

+ q6 1

(1− q3X)(1− q4Y ).

(3.47)

62

Similarly, from (3.32) and (3.33)

P2(X, Y ) =L0(qY )

1− qX+ q2L2(qY )

1− qX+

q4

(1− qY )(1− qX)− q6L6(qY )

1− qX

− q8L8(qY )

1− qX+

q6

(1− q5X)(1− q3Y )− qA1(q

2X)

1− qY,

(3.48)

and

P3(X, Y ) =A2(qX)

1− qY− qA3(qX, q2Y )− q2

(1− q2X)(1− qY )+

q6

(1− q2X)(1− q5Y ).

(3.49)

The equations (3.31)-(3.33) defining the pk,l overlap on the two “diagonals” (k, k)

and (3k, k).

Lemma 3.10 (Diagonals). 1. Equation (3.31) with l = 0 agrees with (3.32) with

k = 0.

2. Equation (3.32) with l = 0 agrees with (3.33) with k = 0.

Proof. It suffices to check this at the level of generating functions, i.e. show that

P1(Z, 0) = P2(0, Z) and P2(Z, 0) = P3(0, Z). This is performed in Appendix A.

Define

∆1,1(Z) =∑k≥0

pk,k(q) Zk = P2(0, Z) = P1(Z, 0), (3.50)

and

∆3,1(Z) =∑k≥0

p3k,k(q) Zk = P3(0, Z) = P2(Z, 0). (3.51)

By Lemma 3.10, both expressions are well-defined. We can now close the expression

in (3.34) by making the appropriate substitutions into equations (3.47)-(3.51) and

taking into account the Inclusion-Exclusion Principle for the overlapping diagonals.

63

Lemma 3.11. The closed form for (3.34) is

(1− q2)(1− q4)W(X, Y ) = P1(XY, Y 2) + P2(X3Y,XY ) + P3(X

2, X3Y )

−∆1,1(XY )−∆3,1(X3Y ).

(3.52)

3.3.2 Step 2: Averaging over the Weyl group

To complete the proof of Proposition 3.9, we need to compare (3.52) and (3.13) after

the appropriate algebraic modifications. Recall that

∑k,l≥0

pk,l(q) χk,l =(1− q2)(1− q6)∏

i∈−3,−1,1,3 j∈−1,1

(1− qsitj).

By (3.11),

∑w1,w2∈±1

(−1)w1w2

∑k,l≥0

pk,l(q) sw1(k+1)tw2(l+1) =(s− s−1)(t− t−1) (1− q2)(1− q6)∏

i∈−3,−1,1,3 j∈−1,1

(1− qsitj).

We need to average W(X, Y ) over the Weyl group Z2 × Z2∼= (±1,±1) of K in a

similar fashion and make the needed substitutions. The following lemma concludes

the proof of Proposition 3.9 and Theorem 3.7.

Lemma 3.12.

∑w1,w2∈±1

(−s)w1(−t)w2 W(sw1 , tw2) =(s− s−1)(t− t−1) (1− q2)(1− q6)∏

i∈−3,−1,1,3 j∈−1,1

(1− qsitj). (3.53)

Proof. The left-hand side is computed using Lemma 3.11. This is shown to agree

with the right-hand side in Appendix A.

64

3.4 Hilbert series for C [ G2 ]SO(4,C)

We now have obtained a generating function for the q-multiplicities

W(X, Y ) =∑k,l≥0

pk,l(q)XkY l (3.54)

which can be closed to a rational expression. We derive some important combinatorial

data from (3.54).

The first observation will be useful in Chapter 4. Let U be the subgroup of upper

triangular unipotent matrices in K

U =

1 u1

0 1

,

1 u2

0 1

| u1, u2 ∈ C

. (3.55)

Proposition 3.13.

chq

(C [ p ]U

)(s, t) =

W(s, t)

(1− q2) (1− q6). (3.56)

Proof. By highest-weight theory(F k,l

)Uis one dimensional with character sktl. Com-

bine (3.54) with separation of variables (3.7).

To obtain the Hilbert series of C [ G2 ]SO(4,C) we first need to appeal to the classical

situation of spherical harmonics.

Theorem 3.14 (Classical spherical harmonics). Let K0 = SL(2, C) and let k0 =

Lie(K0). Then

C [ k0 ] = C [ k0 ]K0 ⊗H0,

where H0 = ⊕d≥0H0d is a graded K0-representation. Moreover, C [ k0 ]K0 is a polyno-

mial ring generated by a quadratic invariant and the q-character of H0 as a represen-

65

tation of K0 is given by

chq H0 =∑d≥0

qd χ2d.

Proof. Note that K0 is locally isomorphic to SO(3, C) and k0∼= so(3, C). The adjoint

representation of k0 is then the standard representation V = C3 of SO(3, C). Let

x1, x2, x3 be a basis for V ∗. Then r2 := x21 +x2

2 +x23 ∈ C [ V ] is SO(3, C)-invariant.

The space of spherical harmonic functions is defined as

H0 = f ∈ C [ V ] | ∆(f) = 0 ,

where ∆ = ∂2

∂x21

+ ∂2

∂x22

+ ∂2

∂x23

is the usual Laplacian. By [Vin89, Section 9.4],

C [ V ] ∼= C[r2]⊗H0,

where H0d := H0 ∩ C [ V ]d is the irreducible SO(3, C)-representation of dimension

2d + 1. Therefore, in terms of K0-representations, chH0d = χ2d.

Corollary 3.15.

C [ k ] ∼= C [ k ]K ⊗H(k)

where H(k) is a graded K-representation. Moreover, C [ k ]K is a polynomial ring

minimally generated by two quadratic invariants and

chq H(k) =∑k,l≥0

qk+l χ2k,2l. (3.57)

Proof. We have k = kL⊕kR and apply 3.14 to each factor of sl(2, C). The correspond-

ing q-characters can then be multiplied since χk,0 · χ0,l = χk,l, for all k, l ≥ 0.

66

Lemma 3.16.

∑k,l≥0

qk+l p2k,2l(q) =W(√

q,√

q)

+W(−√q,

√q)

2(3.58)

Proof. By 3.7, p2k,2l+1(q) = 0 for all k, l ≥ 0. Hence,

W (√

q,√

q) +W (−√q,√

q) = 2∑k,l≥0

p2k,l(q) (√

q)2k(√

q)l

= 2∑k,l≥0

p2k,2l(q) (√

q)2k+2l

= 2∑k,l≥0

p2k,2l(q) qk+l.

Theorem 3.17. Let g be the complex Lie algebra G2. Then

Hilb(C [ g ]SO(4,C)

)=

h(q)

(1− q2)2 (1− q4)2 (1− q6)2 (1− q9) (1− q10), (3.59)

where h(q) is the palindromic polynomial

h(q) = q29 + q27 + q25 + 4q23 + q22 + 5q21 + 5q20 + 5q19 + 7q18 + 9q17 + 8q16

+10q15 + 10q14 + 8q13 + 9q12 + 7q11 + 5q10 + 5q9 + 5q8 + q7 + 4q6 + q4 + q2 + 1.

In particular, C [ g ]SO(4,C) is Gorenstein.

Proof. We decompose under K. By Theorem 1.20 and Corollary 3.15,

C [ g ]K = C [ k⊕ p ]K = C [ k ]K ⊗ C [ p ]K ⊗ (H (k)⊗H (p))K . (3.60)

67

By 3.10 and 3.57,

chq H (k)⊗H (p) = chq H (k) · chq H (p)

=∑

m,p≥0

qm+p χ2m,2p ·∑k,l≥0

pk,l(q) χk,l

=∑

m,p,k,l≥0

qm+ppk,l(q)∑

0≤s≤min2m,k0≤t≤min2p,l

χ2m+k−2s,2p+l−2t

where the last equality is the Clebsch-Gordan rule for tensor products of K-representations.

In particular, the character χ0,0 appears if and only if min2m, k = (2m + k)/2 and

min2p, l = (2p + l/2); if and only if k = 2m and l = 2p. Thus,

Hilb((H (k)⊗H (p))K

)=∑k,l≥0

qk+l p2k,2l(q).

Combining Lemma 3.16 with the q-characters for the remaining terms in (3.60) we

obtain

Hilb(C [ g ]K

)=

1

(1− q2)2· 1

(1− q2)(1− q6)·W(√

q,√

q)

+W(−√q,

√q)

2.

After some minor algebraic manipulation the resulting expression is equal to (3.59).

By a theorem of Hochster and Roberts, C [ g ]SO(4,C) is Cohen-Macaulay [Eis95,

p.467]. Since h(q) is palindromic, C [ g ]SO(4,C) is Gorenstein by a result of Stanley

[Eis95, Exercise 21.19].

68

Chapter 4

Invariant theory

4.1 Covariants of double binary forms

In this section we review the theory of covariants for double binary forms. See [Olv99,

Ch. 10] for a classical treatment although our treatment is inspired by [Dol03, Ch.

5]. For a vector space V , let P(V ) be the space of polynomial functions on V . Then

P(V ) = ⊕d≥0Pd(V ) is graded by degree.

Let K = SL(2, C)×SL(2, C). The irreducible K-representation Fm,p = SmC2⊗SpC2

is viewed as the space of binary (m, p)-forms of degree m in the variables x0, x1 and

degree p in the variables y0, and y1. A typical f ∈ Fm,p is of the form

f = f(x0, x1, y0, y1) =m∑

i=0

p∑j=0

(m

i

)(p

j

)ai,j xm−i

0 xi1 yp−j

0 yj1 (4.1)

with ai,j ∈ C.

Definition 4.1. A covariant of Fm,p of degree d and order (k, l) is a K-equivariant

map J : Fm,p → F k,l given by homogeneous polynomial coordinates of degree d. In

terms of coordinates,

J(f) =k∑

s=0

l∑t=0

(k

s

)(l

t

)bs,t x

k−s0 xs

1 yl−t0 yt

1 (4.2)

69

where f is as in (4.1) and each bs,t is a homogeneous polynomial of degree d in the

coordinates ai,j of f .

Equivalently, a covariant J : Fm,p → F k,l of degree d and order (k, l) is an element

of the space

Cov(Fm,p)(d; k, l) :=[Pd (Fm,p)⊗ F k,l

]K(4.3)

where f ⊗ w ∈ Pd (Fm,p) ⊗ F k,l is identified with J : v 7→ f(v) · w, for v ∈ Fm,p. A

covariant of order (0, 0) is an invariant of Fm,p.

Let W = (F 1,0 ⊕ F 0,1)∗. The polynomial functions on W provides a model for all

finite-dimensional irreducible representations of K. For d ≥ 0,

Pd(W ) ∼= Sd(F 1,0 ⊕ F 0,1)

∼=⊕

k+l=d

Sk(C2) ⊗Sl(C2)

∼=⊕

k+l=d

F k,l.

Thus P(W ) ∼= ⊕k,l≥0Fk,l.

Definition 4.2. The algebra of covariants of Fm,p is the algebra

Cov(Fm,p) := P (Fm,p ×W )K ∼= [P (Fm,p)⊗ P (W )]K . (4.4)

Consider D ≥ 0,

PD(Fm,p ×W )K ∼=⊕

d+e=D

⊕e=k+l

[Pd(F

m,p)⊗ F k,l]K

∼=⊕

d+e=D

⊕e=k+l

Cov(Fm,p)(d; k, l),

so that Cov(Fm,p) is a triply-graded algebra whose homogeneous components are

Cov(F k,l)(d; k, l) for d, k, l ≥ 0.

70

Let U be the subgroup of K of upper triangular unipotent matrices (3.55).

Definition 4.3. The algebra P(Fm,p)U is called the algebra of semiinvariants of

Fm,p.

Lemma 4.4. There is an isomorphism of triply graded algebras

Cov(Fm,p) ∼= P(Fm,p)U . (4.5)

Proof. Fix a non-zero U -invariant vector w0 ∈ W . Consider the map

Ψ : P(Fm,p)⊗ P(W ) → P(Fm,p)

sending f⊗g 7→ g(w0)f , where f ∈ P(Fm,p) and g ∈ P(W ). Restriction to Cov(Fm,p)

gives the isomorphism.

We emphasize that a covariant is a certain kind of function between represen-

tations of K. As such, it should be evaluated at a binary form which we typically

denote by f as in (4.1). We make a slight abuse of notation in the case of a degree 1

covariant of Fm,p and order (m, p). Such a covariant is unique up to scalar being a

multiple of the identity Fm,p → Fm,p which we denote by the generic form f itself.

It is possible to combine covariants via transvection. Define the differential oper-

ators

ΩL =∂2

∂u1∂x0

− ∂2

∂x1∂u0

and

ΩR =∂2

∂v1∂y0

− ∂2

∂y1∂v0

.

Let J1 ∈ Cov(Fm,p)(d1; k1, l1) and J2 ∈ Cov(Fm,p)(d2; k2, l2). The (s, t)-transvectant

71

of J1 and J2 is the covariant (J1, J2)s,t given by

(J1, J2)s,t(f) = Ωs

L ΩtR [J1(f(x0, x1, y0, y1)) · J2(f(u0, u1, v0, v1))]

∣∣∣u0=x0,u1=x1v0=y0,v1=y1

Then (J1, J2)s,t is a covariant of degree d1 + d2 and order (k1 + k2 − 2s, l1 + l2 − 2t).

A famous result of Gordan says that Cov(Fm,p) is generated as an algebra by a

finite number of iterated transvectants of the base form f . In particular, Cov(Fm,p) is

finitely generated as an algebra by homogeneous elements. A generating set has been

determined in the cases Cov(F 1,1), Cov(F 2,1), Cov(F 2,2) by Peano [Pea82],[Tur22].

The cases Cov(F 3,1) and Cov(F 4,1) were treated by J.A. Todd [Tod46a],[Tod46b].

The generators for the (3, 1) case are given in Table 4.1. Each covariant is denoted

by the symbol Jd,k,l identifying it as a covariant of degree d and order (k, l). Luckily,

the order and degree of each generator is a unique triple (d, k, l) and no confusion can

arise. The base form is also denoted f := J1,3,1 : F 3,1 → F 3,1. In some cases we have

introduced a constant coefficient for computational purposes.

72

Deg

ree

1J

1,3

,1:=

f

2J

2,4

,0:=

1 18(f

,f)1

,1J

2,2

,2:=

1 72(f

,f)2

,0J

2,0

,0:=

1 72(f

,f)3

,1

3J

3,5

,1:=

1 6(f

,J2,4

,0)1

,0J

3,3

,3:=

1 3(f

,J2,2

,2)1

,0J

3,1

,1:=

1216(f

,J2,4

,0)3

,0

4J

4,4

,0:=

(f,J

3,1

,1)0

,1J

4,2

,2:=

(f,J

3,1

,1)1

,0J

4,0

,4:=

1 2(J

2,2

,2,J

2,2

,2)2

,0

5J

5,3

,1:=

(J2,2

,2,J

3,1

,1)0

,1J

5,1

,3:=

(J2,2

,2,J

3,1

,1)1

,0

6J

6,6

,0:=

(J3,5

,1,J

3,1

,1)0

,1J

6,4

,2:=

(J3,3

,3,J

3,1

,1)0

,1J

6,2

,4:=

(J3,3

,3,J

3,1

,1)1

,0J

6,0

,0:=

1 2(J

3,1

,1,J

3,1

,1)1

,1

7J

7,1

,3:=( f

,J3,1

,12) 2,0

8J

8,0

,4:=( J

2,2

,2,J

3,1

,12) 2,0

9J

9,1

,5:=( J

3,3

,3,J

3,1

,12) 2,0

12J

12,0

,6:=( J

3,3

,3,J

3,1

,13) 3,0

Tab

le4.

1:Todd’s

gener

ator

sfo

rC

ov(F

3,1).

73

4.2 Syzygies for the binary (3, 1)-forms

Fix m, p ≥ 0. Suppose we know a finite generating set

G = Ji ∈ Cov(Fm,p)(di; ki, li) | i = 1, . . . , r

of homogeneous covariants for Cov(Fm,p). Let

SG = C [ X1, . . . , Xr ]

be the triply-graded polynomial algebra with the degree1 of Xi equal to (di, ki, li),

i = 1, . . . , r. Write SG(d, k, l) for the homogeneous elements of SG of degree (d, k, l).

There exists a presentation for Cov(Fm,p)

0 → IG → SGπ−→ Cov(Fm,p) → 0, (4.6)

such that π(Xi) = Ji for i = 1, . . . , r and IG is a finitely generated homogeneous ideal

of SG. Write IG(d, k, l) = SG(d, k, l)∩ IG. We call an element of IG a syzygy2 of Fm,p.

For a triply graded vector space W = ⊕d,k,l≥0 Wd,k,l we denote the multivariate

Hilbert series of W by

H(W )(q, s, t) =∑

d,k,l≥0

[ dimC Wd,k,l ] qdsktl.

By (4.6),

H(Cov(Fm,p))(q, s, t) = H(SG)(q, s, t)−H(IG)(q, s, t).

Suppose F is a finite subset of triples from (Z≥0)3. For each (d, k, l) ∈ F it is a

simple linear algebra problem to compute a basis for the space IG(d, k, l) and hence,

1In this section the entire triple (d, k, l) is referred to simply as degree.2A syzygy is then a polynomial identity in the covariants. In homological terms, the ideal IG is

called the first syzygy.

74

determine a set of generators for the ideal

IF := 〈 IG(d, k, l) | (d, k, l) ∈ F 〉 ⊂ IG.

Unfortunately, it is not clear a priori when IF = IG. If, however, the Hilbert series

for Cov(Fm,p) is known in advance then a dimension count gives

IF = IG ⇐⇒ H(SG/IF) = H(SG/IG).

We now apply this to the case of the binary (3, 1)-forms. Recall from Chapter

3, that F 3,1 is the −1-eigenspace p in the Cartan decomposition g = k ⊕ p for the

symmetric algebraic pair (G2, so(4, C)). By Lemma 4.4, Cov(F 3,1) ∼= C [ p ]U and by

Proposition 3.13

H(Cov(F 3,1)) =W(s, t)

(1− q2)(1− q6). (4.7)

We proceed as follows:

Step 1. Let G be the twenty covariant generators Ji := Jdi,ki,li given in Table 4.1.

Then SG is a triply-graded polynomial algebra in the generators Xi := Xdi,ki,li .

Step 2. Let M(d, k, l) be the set of monomials of degree (d, k, l) in the Xi. Compu-

tationally, this may be obtained by finding the coefficient Cd,k,l of qdsktk in

the series

1∏20i=1(1− qdiskitliXi)

.

Then M(d, k, l) consists of the additive terms in Cd,k,l.

Step 3. Choose a finite subset F of (Z≥0)3. For each (d, k, l) ∈ F compute a basis

for IG(d, k, l) as follows:

75

(a) Form the linear combination

P (X1, . . . , X20) =∑

M∈M(d,k,l)

AM M

in unknowns AM . Then

π(P (X1, . . . , X20)) = P (J1, . . . , J20) : F 3,1 → F k,l (4.8)

is a covariant of F 3,1 of degree d and order (k, l).

(b) Evaluate (4.8) at a sufficiently generic form (to be explained momentar-

ily) f ∈ F 3,1 and set P (J1, . . . , J20)(f) = P (J1(f), . . . , J20(f)) = 0.

(c) Collect like terms in the x0, x1, y0, y1. The coefficients of each term is a

linear equation in the AM which we solve simultaneously to determine

a basis for the space IG(d, k, l).

When done we have a generating set for the ideal IF .

Step 4. Compute the Hilbert series for SG/IF and compare with (4.7). If they agree

then IF = IG and we have a complete set of syzygies for F 3,1.

The next lemma makes precise what constitutes a sufficiently generic form.

Lemma 4.5. A covariant J : F 3,1 → Fm,p is identically zero if and only if it van-

ishes on a maximal toral subalgebra a ⊂ p = F 3,1. In particular, J ∈ Cov(F 3,1) is

identically zero if and only if

J(a(x30 y0 + x3

1 y1) + b(x0 x21 y0 + x2

0 x1 y1)) = 0

for all a, b ∈ C.

Proof. Since a covariant is K-equivariant, it vanishes at a particular f ∈ F 3,1 if

and only if it vanishes on the orbit K · f . Every semisimple element of p = F 3,1

76

is conjugate to an element of any fixed maximal toral subalgebra a ⊂ p. Since the

semisimple elements are dense in p it is enough to check vanishing on a. Lemma 3.1

gives one such choice for a.

Theorem 4.6. Let

F = (5, 7, 3), (6, 6, 2), (6, 4, 4), (6, 6, 6), (6, 8, 4), (6, 10, 2), (7, 5, 5), (7, 7, 3),

(7, 5, 3), (7, 9, 1), (8, 6, 4), (8, 4, 4), (8, 4, 6), (8, 8, 2), (8, 6, 2), (9, 3, 5),

(9, 5, 3), (9, 7, 1), (9, 5, 7), (9, 7, 5), (9, 9, 3), (9, 11, 1), (10, 4, 6), (10, 6, 4),

(10, 8, 2), (10, 4, 4), (10, 2, 6), (10, 6, 2), (11, 3, 7), (11, 5, 5), (11, 3, 5), (11, 5, 3),

(11, 7, 3), (11, 9, 1), (12, 2, 6), (12, 4, 4), (12, 4, 8), (12, 6, 6), (12, 8, 4), (12, 10, 2),

(12, 12, 0), (13, 3, 7), (13, 5, 5), (13, 3, 5), (13, 7, 3), (14, 2, 8), (14, 2, 6), (14, 4, 6),

(14, 6, 4), (15, 1, 7), (15, 3, 9), (15, 5, 7), (15, 7, 5), (16, 2, 8), (16, 4, 6), (17, 1, 9),

(17, 3, 7), (18, 2, 10), (18, 4, 8), (18, 6, 6), (19, 1, 9), (21, 1, 11), (24, 0, 12) .

Then IF = IG and this ideal is minimally generated by the 104 syzygies that appear

in Appendix B.2.

The proof of Theorem 4.6 requires a significant amount of computer computation.

Section B.1 has source code for a C++ program called syzygies that implements

Step 3. It accepts as input (d, k, l) and returns a basis for IG(d, k, l). The commu-

tative algebra software Macaulay2 [Gra] calls this external program for each (d, k, l)

appearing in F . The results are combined to form the ideal IF and a minimal set

of generators for this ideal are computed. The 104 resulting generators appear in

Section B.2. Next, the Hilbert series of SG/IF is computed. This takes several hours

to complete on a modern computer platform. The result is compared to the known

Hilbert series (4.7) for Cov(F 3,1). By observing that they are equal, the theorem is

proved. Section B.2 demonstrates the necessary input to Macaulay2.

77

Chapter 5

Asymptotic methods for graded

multiplicities

5.1 The Brion polytope

Let K be a connected semisimple algebraic Lie group with maximal torus T . Let

k = Lie(K) and h = Lie(T ). Then h is a Cartan subalgebra of g. Let Φ := Φ(h)

and choose a set of positive roots Φ+ for Φ. Let E denote the real span of Φ. For

λ ∈ P+(k), let L(λ) denote the finite-dimensional irreducible representation of K with

highest weight λ. We write λ∗ ∈ P+(k) for the highest weight of the contragredient

representation L(λ)∗.

Fix a finite-dimensional irreducible representation V of K and suppose X ⊂ V

is a K-invariant affine cone-variety. Then K acts on the algebra C [ X ] of regular

functions on X and the space of functions C [ X ]d is K-stable. Let fd(λ) denote the

multiplicity of λ ∈ P+(k) in C [ X ]d

fd(λ) = dimC HomK (L(λ), C [ X ]d) .

Example 5.1. If (g, k) is an algebraic symmetric pair with Cartan decomposition

g = k⊕p then the nullconeN ⊂ p is an affine cone-variety. By Lemma 1.19, restriction

78

provides an isomorphism of C [N ] with the harmonic functions H ⊂ C [ p ]. Thus,

fd(λ) is precisely the graded multiplicity (1.16).

Definition 5.2. The Brion polytope of X is

Bri(X) =

λ

d∈ P+(k)⊗Z Q | ∃ λ ∈ P+(k) and d > 0 such that fd(λ

∗) 6= 0

.

Naming Bri(X) after M. Brion is adopted from [Smi04]. Brion proved that the

closure of Bri(X) in E is a convex polytope whenever X is irreducible [Bri87]. We

provide our own proof of this fact in the following context. For each m > 0, let

Gm = (Z≥0)m viewed as an additive semigroup

Theorem 5.3. Suppose

A =⊕

(d,λ)∈Z×Gm

A(d, λ)

is a Z×Gm-graded integral domain over C which is finitely generated by homogeneous

elements Xi ∈ A(di, λi) where di > 0, λi ∈ Gm, for i = 1, . . . , r. Let

C =

λ

d∈ Qm | ∃ λ ∈ G and d > 0 such that A(d, λ) 6= 0

Then the closure of C in Rm is the convex hull of the points ξi := λi/di ∈ Qm,

i = 1, . . . , r.

Proof. Let S = Hull(ξ1, . . . , ξr)∩Qm. Suppose 0 6= A(d, λ) for some d > 0. Choose a

monomial Xn11 · · ·Xnr

r ∈ A such that d =∑r

i=1 nidi and λ =∑r

i=1 niλi. Let cj =njdj

d,

for j = 1, . . . , r. Then∑r

j=1 cj = 1 and

λ

d=

r∑i=1

ni

dλi =

r∑j=1

cj ξj ∈ S.

This proves C ⊂ S.

79

Conversely, suppose ξ =∑r

i=1 ci ξi ∈ S where∑r

i=1 ci = 1 and each ci = mi/li is

rational with mi ∈ Z≥0 and li ∈ N. Let d = lcm(l1d1, . . . , lrdr) and let

ni =cid

di

=mid

lidi

∈ Z≥0, for i = 1, . . . , r. (5.1)

Since ci = nidi/d,

d = dr∑

i=1

ci =r∑

i=1

ni di.

Since A is a domain, 0 6= Xn11 · · ·Xnr

r ∈ A(d, λ) where λ =∑r

i=1 ni λi. By (5.1),

λ

d=

r∑i=1

ni λi

d=

r∑i=1

ci λi

di

=r∑

i=1

ci ξi = ξ

which proves S ⊂ C.

5.1.1 The Brion polytope for Type G

In this section we return to the symmetric pair (g, k) := (G2, so4). In the notation

of the previous section, we take K = SL(2, C) × SL(2, C); V = p where g = k ⊕ p is

the Cartan decomposition; and X = N is the nullcone of p. The Brion polytope can

then be viewed as the set

Bri(N ) = (k/d, l/d) ∈ Q2 | ∃ k, l ≥ 0 and d > 0 with fd(k, l) 6= 0 ,

where fd(k, l) is the graded multiplicity (3.9).

Proposition 5.4. The nullcone

N = X ∈ p | u2(X) = 0 = u6(X)

for the pair (G2, so4) is irreducible. Moreover, the closure of Bri(N ) in R2 is a convex

polytope.

80

Proof. As Kθ = SO(4, C) is connected, irreducibility follows from Lemma 1.17. The

algebra of semiinvariants A = C [N ]U is then a triply graded integral domain. The

homogeneous component A(d, k, l) is non-zero if and only if fd(k, l) 6= 0, in which

case A(d, k, l) is one dimensional. Convexity follows from 5.3 applied to A.

The boundary of the Brion polytope can be completely determined from Todd’s

generators for the covariants of the binary (3, 1)-forms. This is because

C [N ]U ∼= (C [ p ] /〈u2, u6 〉)U ∼= C [ p ]U /〈u2, u6 〉 ∼= Cov(F 3,1

)/〈 J2,0,0, J6,0,0 〉.

Thus, for all of the non-invariant covariant generators Jd,k,l that appear in Table 4.1

we plot the point (x, y) = (k/d, l/d) and take the convex hull. Observe that distinct

covariant generators may give rise to the same point in Bri(N ). The resulting polytope

appears in Figure 5.1 and is bounded by the lines

x = 0, y = 0, y = 1, x + 2y = 1, x− y = 2. (5.2)

12

1

y=l/d

y=l/d

1 2 3

x = k/dx = k/d

Figure 5.1: The Brion polytope for the nullcone of (G2, so(4, C)).

81

5.2 Density of the graded multiplicities in Type G

Let Ω denote the interior of the closure of Bri(N ) in R2. The goal of this section is

to construct a piecewise-linear function g : Ω → R which describes the asymptotic

density of the graded multiplicities fd(k, l) for the pair (G2, so(4, C)). In order to

achieve this we take advantage of two earlier results:

1. the quadratic estimate (2.14) for the principal branching multiplicities B(k, l, m)

for sp(2, C);

2. the formula (3.19) for the graded multiplicities

fd(k, l) = B(k, l, m)−B(k, l, m− 3), where d = l + 2m.

We briefly recall the relevant notation from Section 2.3. The dominant integral

weights for sp(2, C) are of the form λ = l $1 + m $2, l,m ≥ 0 where $1 and $2 are

the fundamental weights. The Weyl group of sp(2, C) is denoted by W . For w ∈ W ,

k, l, m ∈ R

Lw(k, l, m) = w(λ + ρ)− ρ− k, where λ = l $1 + m $2 (5.3)

and explicit formulae for each Lw are provided in Figure 2.2. For real numbers

k, l, m ≥ 0,

A(k, l, m) = w ∈ W | Lw(k, l, m) ≥ 0 .

If any of k, l, m are negative then we set A(k, l, m) = ∅. For S ⊂ W and k, l, m ∈ R,

QS(k, l, m) =∑w∈S

sgn(w)1

48(Lw(k, l, m) + 6)2 .

82

The following convention will be used for the remainder of this section.

Convention 1. Whenever the symbols d, l, m appear together it should be

understood that

d = l + 2m. (5.4)

With this in mind, define

gd(k, l) = QS(k, l, m)−QS(k, l, m− 3),

where S = A(k, l, m). Thus,

gd(k, l) =∑

w∈A(k,l,m)

1

48sgn(w)

((Lw (k, l, m) + 6)2 − (Lw (k, l, m− 3) + 6)2 ) . (5.5)

Set S = A(k, l, m) and S ′ = A(k, l, m− 3). When S = S ′, Lemma 2.19 immediately

yields

| fd(k, l)− gd(k, l) | ≤ C

for some constant C. Things are more delicate when S 6= S ′ as we are unable to

appeal to 2.19 directly. Fortunately, the terms in (5.5) that may or may not appear

due to differences between S and S ′ contribute little to the overall calculation.

Lemma 5.5. For each w ∈ W there exists a constant Cw > 0 such that for all k, l, m,

S = A(k, l, m), S ′ = A(k, l, m− 3) we have

1. If w ∈ S\S ′ then

−Cw ≤ Lw(k, l, m− 3) < 0.

2. If w′ ∈ S ′\S then

0 ≤ Lw′(k, l, m− 3) < Cw′ .

83

Proof. By (5.3),

Lw(k, l, m− 3) = Lw(k, l, m)− 3w ($2) = Lw(k, l, m)− εwCw,

where Cw := 3∣∣∣w ($2)

∣∣∣ and εw = ±1 is chosen accordingly. If w ∈ S\S ′, then

Lw(k, l, m) ≥ 0, but Lw(k, l, m− 3) < 0. Thus,

0 > Lw(k, l, m− 3) = Lw(k, l, m)− εwCw ≥ 0− εwCw ≥ −εwCw.

The second case is similar.

Proposition 5.6. There exists a constant C > 0, such that for all integers k, l, d ≥ 0,

k − l ∈ 2Z,

| fd(k, l)− gd(k, l) | ≤ C.

Proof. Let S = A(k, l, m) and S ′ = A(k, l, m − 3). Using 2.19, there is a constant

C ′ > 0 such that

| fd(k, l)− gd(k, l) | ≤ |B(k, l, m)−QS(k, l, m) |+|B(k, l, m− 3)−QS(k, l, m− 3) |

≤ C ′ + |B(k, l, m− 3)−QS(k, l, m− 3) |

= C ′ + |B(k, l, m− 3)−QS′(k, l, m− 3) + QS′(k, l, m− 3)−QS(k, l, m− 3) |

≤ C ′ + |B(k, l, m− 3)−QS′(k, l, m− 3) |+ |QS(k, l, m− 3)−QS′(k, l, m− 3) |

≤ 2C ′ + |QS(k, l, m− 3)−QS′(k, l, m− 3) | .

Estimate the second term by applying the Inclusion-Exclusion Principle:

S ∪ S ′ = (S ∩ S ′) ∪ (S\S ′) ∪ (S ′\S).

84

Since terms from the intersection cancel,

|QS(k, l, m− 3)−QS′(k, l, m− 3) | =∣∣QS\S′(k, l, m− 3)−QS′\S(k, l, m− 3)

∣∣≤

∑w∈S\S′

(Cw + 6)2

48+

∑w′∈S\S′

(Cw′ + 6)2

48

≤ 2∑w∈W

(Cw + 6)2

48.

It is straightforward to compute gd(k, l) for each of the possible A(k, l, m) ⊂ W

that are admissible. For example, if k, l, m ≥ 0 such that S := A(k, l, m) = 1 then

using Figure 2.2 and (5.5),

gd(k, l) = QS(k, l, m)−QS(k, l, m− 3)

=1

48

((L1(k, l, m) + 6)2 − (L1(k, l, m− 3) + 6)2

)=

1

48

((3l + 4m− k + 6)2 − (3l + 4(m− 3)− k + 6)2

)=

1

2(3l + 4m− k) .

Substituting m = (d− l)/2, gd(k, l) = 12(l + 2d− k). The remaining cases appear in

Table 5.1. Observe that the resulting expression is linear in d, k, and l despite the

quadratic estimate of the branching rule.

Admissible subset Estimate of graded multiplicityS = A(k, l, m) gd(k, l)

1 12(l + 2d− k)

1, 2 1 + l

1, 3 14(3d− k − 1)

1, 2, 3 14(2l − d + k + 3)

1, 2, 3, 4 12(1 + k)

1, 2, 3, 5 0

Table 5.1: Piecewise linear approximation for the graded multiplicities.

85

We extend the earlier convention as follows:

Convention 2. Whenever the symbols x, y, d and k, l, m appear together it

should be understood that k, l, m are dependent on x, y, d via the equations

x = k/d, y = l/d, d = l + 2m. (5.6)

Equivalently,

k = dx, l = dy, m = (d− dy)/2. (5.7)

Define ` : R2 → R by

`w(x, y) = limd→∞

Lw(k, l, m)

d,

for (x, y) ∈ R2. Each `w(x, y) can be computed from the formula for Lw(k, l, m) in

Figure 2.2. For example, if w = w2 ∈ W

Lw(k, l, m) = l + 4m− 2− k

= dy + 2d− 2dy − 2− dx

= 2d− dx− dy − 2.

Thus,

`w(x, y) = limd→∞

(2− x− y − 2/d) = 2− x− y.

The remaining `w are given in Table 5.2.

Next, we define the asymptotic analogue of the sets A(k, l, m). For (x, y) ∈ R2,

let

A(x, y) = w ∈ W | `w(x, y) ≥ 0 . (5.8)

86

w ∈ W `w(x, y)

w1 y − x + 2

w2 2− x− y

w3 2y − x + 1

w4 2y − x− 1

w5 −x− 2y + 1

w6 −x− 2y − 1

w7 −x− 2

w8 −x− y − 2

Table 5.2: `w(x, y) for w ∈ W .

The classes A(x, y) partition Ω. For S ⊂ W , define

RS = (x, y) ∈ Ω | A(x, y) = S

Each non-empty RS is a subset of Ω whose closure in Ω is a convex polytope bounded

by lines `w(x, y) = 0 and the boundary of the Brion polytope. If the line `w(x, y) = 0

is in the closure of RS then it is contained in RS if and only if w /∈ S. Moreover,

Ω =⊎RS, (5.9)

where the union is over the admissible subsets S in the list 1 , 1, 2 , 1, 3 ,

1, 2, 3 , 1, 2, 3, 4 . The union is clearly disjoint by (5.8). Figure 5.2 shows the

partitioning (5.9).

Define g : Ω → R

g(x, y) = limd→∞

gd(k, l)

d,

for all (x, y) ∈ Ω. Then

g(x, y) = limd→∞

1

d( QS(k, l, m)−QS(k, l, m− 3) ) (5.10)

87

R1

R1,2

R1,3

R1,2,3

R1,2,3,4

12

1

yy

1 2 3xx

Figure 5.2: Partitioning of the Brion polytope.

where unfortunately S = A(k, l, m) varies with d. The next lemma overcomes this

issue.

Lemma 5.7. For (x, y) ∈ R2,

g(x, y) = limd→∞

1

d( QS(k, l, m)−QS(k, l, m− 3) ) ,

where S = A(x, y) can be taken independent of d in the limit.

Proof. Fix (x, y) ∈ Ω. For each w ∈ W , Lw(k, l, m) = Lw(dx, dy, (d − dy)/2) is a

linear polynomial in d. There are then three possibilities for w:

1. Lw(k, l, m) is constant for all d. In this case, the term corresponding to w

in (5.10) contributes nothing in the limit and may therefore be included or

excluded as required.

2. Lw(k, l, m) →∞ as d →∞. Then w ∈ A(k, l, m) for all d sufficiently large and

`w(x, y) > 0. Hence, w ∈ A(x, y).

3. Lw(k, l, m) → −∞ as d → ∞. Then w /∈ A(k, l, m) for all d sufficiently large

and `w(x, y) < 0. Hence, w /∈ A(x, y).

88

Lemma 5.7 allows us to use Table 5.1 to compute g(x, y) on eachRS. For example,

suppose A(x, y) = 1, 3 . Then

g(x, y) = limd→∞

gd(k, l)

d

= limd→∞

1

4

3d− k − 1

d

= limd→∞

1

4(3− x− 1/d)

=1

4(3− x).

The remaining cases are exhibited in Table 5.3. Figure 5.3 shows the graph of the

equation z = g(x, y).

Admissible subset Density of graded multiplicitiesS ⊂ W g(x, y) for (x, y) ∈ RS

1 12(y − x + 2)

1, 2 y

1, 3 14(3− x)

1, 2, 3 14(2y − 1 + x)

1, 2, 3, 4 x2

Table 5.3: Density of the graded multiplicities.

Recall that for each d ≥ 0, fd(k, l) is defined only for integers k, l ≥ 0. We set

fd(k, l) = 0, whenever k or l is not an integer. Define f : Ω → R by

f(x, y) = lim supd→∞

fd(k, l)

d,

for all (x, y) ∈ Ω. Define

Ωsupp =

(k

d,l

d

)∈ Ω | k, l, d ∈ Z≥0, d > 0, k − l ∈ 2Z

. (5.11)

89

Then Ωsupp is a dense subset of Ω. The piecewise-linear function g describes the

asymptotic density of the graded multiplicities in the following sense.

Theorem 5.8. For (x, y) ∈ Ωsupp, f(x, y) = g(x, y).

Proof. By Proposition 5.6,

gd(k, l)− C < fd(k, l) < gd(k, l) + C (5.12)

whenever k, l, d ∈ Z≥0 and k − l ∈ 2Z.

Fix (x, y) ∈ Ωsupp. Write x = k1/d1 and y = l1/d1, where k1, l1, d1 ∈ Z≥0, d1 > 0,

k1 − l1 ∈ 2Z, and d1 does not divide both k1 and l1.

f(x, y) = lim supd→∞

fd(k, l)

d

= lim supd→∞

fd

(dk1

d1, d l1

d1

)d

= lim supd→∞

fd(dk1, dl1)

d

by (5.12),

≤ lim supd→∞

gd(dk1, dl1)

d

= lim supd→∞

gd

(dd1

k1

d1, dd1

l1d1

)d

= lim supd→∞

gd (dd1x, dd1y)

d

= lim supd→∞

gd (dx, dy)

d

= g(x, y)

Therefore, f(x, y) ≤ g(x, y). The reverse inequality is similar.

90

1.0

0.75

0.5y

0.25

3.02.5

x

2.01.51.00.50.0

0.00.0

0.1

0.2

0.3

0.4

0.5

Figure 5.3: Graph of the density function z = g(x, y).

91

Appendix A

Computer proof of Lemmas 3.10, 3.12

In this section we implement the generating functions defined in 3.3.1 in Maple. Thisis then used to prove Lemma 3.10 and Lemma 3.12.Equations (3.35)-(3.41):

GFF := Z -> Z^6/(1-Z)/(1-Z^6):

GFL0 := Y -> GFF(Y) + (1+Y^2+Y^3+Y^4+Y^5)/(1-Y^6):

GFL2 := Y -> GFF(Y) + (2*Y+Y^2+Y^3+2*Y^4+2*Y^5)/(1-Y^6):

GFL6 := Y -> GFF(Y) + (-1-Y^3+Y^4)/(1-Y^6):

GFL8 := Y -> GFF(Y) + (Y^3+Y^5)/(1-Y^6):

GFK1 := X -> GFL8(X):

GFK2 := X -> GFF(X) + (X+X^2+2*X^4+X^5)/(1-X^6):

Equations (3.42)-(3.46):

GFD := (a,b,c,d,X,Y) -> (a+b*X+c*Y+d*X*Y)/(1-X^2)/(1-Y^2):

GFA1 := X -> GFD(q+2*q^3+q^5, 1+q^2+q^4+q^6, 0, 0, X, 0):

GFA2 := X -> GFD(1+q^2+q^4+q^6, q+2*q^3+q^5, 0, 0, X, 0):

GFA3 := (X,Y) -> GFD(q+2*q^3+q^5,

1+q^2+q^4+q^6,

1+q^2+q^4+q^6,

q+2*q^3+q^5, X, Y):

GFB := (X,Y) -> GFD(1-q^2, 0, q^6*(1-q^2), 0, X, Y):

Equations (3.47)-(3.49):

GFP1 := (X,Y) - (1-q^8)*GFK1(q*X)/(1-q^2*Y)

+ q^2*(1-q^4)*GFK2(q*X)/(1-q^2*Y)

+ GFB(q*X,q^2*Y)

- q^4/(1-q*X)/(1-q^4*Y)

+ q^6/(1-q^3*X)/(1-q^4*Y):

GFP2 := (X,Y) -> GFL0(q*Y)/(1-q*X)

+ q^2*GFL2(q*Y)/(1-q*X)

+ q^4/(1-q*X)/(1-q*Y)

- q^6*GFL6(q*Y)/(1-q*X)

92

- q^8*GFL8(q*Y)/(1-q*X)

+ q^6/(1-q^5*X)/(1-q^3*Y)

- q*GFA1(q^2*X)/(1-q*Y):

GFP3 := (X,Y) -> GFA2(q*X)/(1-q*Y)

- q*GFA3(q*X,q^2*Y)

- q^2/(1-q^2*X)/(1-q*Y)

+ q^6/(1-q^2*X)/(1-q^5*Y):

Equations (3.50) and (3.51):

GFD11 := Z -> GFP1(Z,0):

GFD31 := Z -> GFP2(Z,0):

To verify the Diagonal Lemma compute:

simplify(GFP2(0,Z) - GFP1(Z,0));

simplify(GFP3(0,Z) - GFP2(Z,0));

Each returns zero which proves Lemma 3.10.Equation (3.52):

GFW := (X,Y)->

(GFP1(X*Y,Y^2)

+ GFP2(X^3*Y,X*Y)

+ GFP3(X^2,X^3*Y)

- GFD11(X*Y)

- GFD31(X^3*Y))/(1-q^2)/(1-q^4):

Right hand side of (3.53):

Qchar := (s-1/s)*(t-1/t)*(1-q^2)*(1-q^6):

for i in -3,-1,1,3 do

for j in -1,1 do

Qchar := Qchar/(1-q*s^i*t^j):

od:

od:

Left hand side of (3.53):

Avg := 0:

for w1 in -1,1 do

for w2 in -1,1 do

Avg := Avg + w1 * w2 * s^w1 * t^w2 * GFW(s^w1,t^w2):

od:

od:

Now subtract

Difference := simplify(Avg - Qchar);

which returns zero proving Lemma 3.12.

93

Appendix B

Syzygies for (3, 1)-covariants

B.1 Program for computing syzygies

The following C++ source code computes a basis for the syzygies of the binary (3, 1)-forms in a fixed degree. Symbolic computation is performed using the C++ libraryGiNac [GiN] To illustrate the usage of the program, we compute the syzygies in degree(5, 7, 3).

$ ./syzygies 5 7 3

-X_(3,1,1)*X_(1,3,1)^2-X_(2,4,0)*X_(3,3,3)+X_(2,2,2)*X_(3,5,1),0

For programming simplicity the output is always terminated by 0. In this caseIG(5, 7, 3) is one-dimensional and the covariants satisfy the polynomial identity

−J3,1,1J21,3,1 − J2,4,0J3,3,3 + J2,2,2J3,5,1 = 0.

94

#include <iostream>#include <list>#include <fstream>#include <sstream>#include <string>#include <ginac/ginac.h>using namespace std;using namespace GiNaC;

symbol a("a"),b("b"),q("q"),s("s"),t("t"),X("X");symbol x0("x_0"),x1("x_1"),y0("y_0"),y1("y_1");symbol u0("u_0"),u1("u_1"),v0("v_0"),v1("v_1");symbol infty("infty");

// input: a form f in symbols xx0, xx1, yy0, yy1// output: apply the omega operator to fexomega(const ex &f,

const symbol xx0, const symbol xx1,const symbol yy0, const symbol yy1)

return f.diff(xx0).diff(yy1) - f.diff(yy0).diff(xx1);

// input: two forms f and g in x0,x1,y0,y1// output: the (m,n)-transvection of f and gextransvect(const ex &f, const ex &g,

const int m, const int n)ex ret = f * g.subs(lst(x0==u0,x1==u1,y0==v0,y1==v1));for (int i = 1; i <= n; i++)ret = omega(ret,y0,y1,v0,v1);

for (int i = 1; i <= m; i++)ret = omega(ret,x0,x1,u0,u1);

return ret.subs(lst(u0==x0,u1==x1,v0==y0,v1==y1));

// input: a symbol X and indexes d,k,l// output: the indexed symbol X_d,k,lexinline sym_gen(const symbol X,

const int d,const int k,const int l)

return indexed(X,idx(d,infty),idx(k,infty),idx(l,infty));

95

// input: a monomial term in variables x0,x1,y0,y1,a,b// output: the coefficient of the termexinline mycoeff(const ex &term)return term.subs(lst(x0==1,x1==1,y0==1,y1==1,a==1,b==1));

intmain(int argc, char* argv[])// Create a list of tuples (d,m,p)// for each generatorlist<lst> gens;lst g131(1,3,1);lst g240(2,4,0),g222(2,2,2),g200(2,0,0);lst g351(3,5,1),g333(3,3,3),g311(3,1,1);lst g440(4,4,0),g422(4,2,2),g404(4,0,4);lst g531(5,3,1),g513(5,1,3);lst g660(6,6,0),g642(6,4,2),g624(6,2,4),g600(6,0,0);lst g713(7,1,3);lst g804(8,0,4);lst g915(9,1,5);lst g1206(12,0,6);gens.push_back(g131);gens.push_back(g240);gens.push_back(g222);gens.push_back(g200);gens.push_back(g351);gens.push_back(g333);gens.push_back(g311);gens.push_back(g440);gens.push_back(g422);gens.push_back(g404);gens.push_back(g531);gens.push_back(g513);gens.push_back(g660);gens.push_back(g642);gens.push_back(g624);gens.push_back(g600);gens.push_back(g713);gens.push_back(g804);gens.push_back(g915);gens.push_back(g1206);

// Compute each covariant generator as// an expression in a,b,x0,x1,y0,y1ex ff = a*pow(x0,3)*y0 + b*pow(x0,2)*x1*y1

+ b*x0*pow(x1,2)*y0 + a*pow(x1,3)*y1;ex FF = transvect(ff,ff,1,1)/18;ex hh = transvect(ff,ff,2,0)/72;ex I2 = transvect(ff,ff,3,1)/72;ex jj = transvect(ff,FF,1,0)/6;ex tt = transvect(ff,hh,1,0)/3;

96

ex pp = transvect(ff,FF,3,0)/216;ex DD = transvect(hh,hh,2,0)/2;ex I6 = transvect(pp,pp,1,1)/2;

// Associate to each indexed symbol X_d,k,l// an expression for the corresponding covariantexmap cov_map;cov_map[sym_gen(X,1,3,1)] = ff;cov_map[sym_gen(X,2,4,0)] = FF;cov_map[sym_gen(X,2,2,2)] = hh;cov_map[sym_gen(X,2,0,0)] = I2;cov_map[sym_gen(X,3,5,1)] = jj;cov_map[sym_gen(X,3,3,3)] = tt;cov_map[sym_gen(X,3,1,1)] = pp;cov_map[sym_gen(X,4,4,0)] = transvect(ff,pp,0,1);cov_map[sym_gen(X,4,2,2)] = transvect(ff,pp,1,0);cov_map[sym_gen(X,4,0,4)] = DD;cov_map[sym_gen(X,5,3,1)] = transvect(hh,pp,0,1);cov_map[sym_gen(X,5,1,3)] = transvect(hh,pp,1,0);cov_map[sym_gen(X,6,6,0)] = transvect(jj,pp,0,1);cov_map[sym_gen(X,6,4,2)] = transvect(tt,pp,0,1);cov_map[sym_gen(X,6,2,4)] = transvect(tt,pp,1,0);cov_map[sym_gen(X,6,0,0)] = I6;cov_map[sym_gen(X,7,1,3)] = transvect(ff,pow(pp,2),2,0);cov_map[sym_gen(X,8,0,4)] = transvect(hh,pow(pp,2),2,0);cov_map[sym_gen(X,9,1,5)] = transvect(tt,pow(pp,2),2,0);cov_map[sym_gen(X,12,0,6)] = transvect(tt,pow(pp,3),3,0);

// Associate to an indexed symbol X_d,k,l// an indexed variable in M2 notationexmap str_map;str_map[sym_gen(X,1,3,1)] = symbol("X_(1,3,1)");str_map[sym_gen(X,2,4,0)] = symbol("X_(2,4,0)");str_map[sym_gen(X,2,2,2)] = symbol("X_(2,2,2)");str_map[sym_gen(X,2,0,0)] = symbol("X_(2,0,0)");str_map[sym_gen(X,3,5,1)] = symbol("X_(3,5,1)");str_map[sym_gen(X,3,3,3)] = symbol("X_(3,3,3)");str_map[sym_gen(X,3,1,1)] = symbol("X_(3,1,1)");str_map[sym_gen(X,4,4,0)] = symbol("X_(4,4,0)");str_map[sym_gen(X,4,2,2)] = symbol("X_(4,2,2)");str_map[sym_gen(X,4,0,4)] = symbol("X_(4,0,4)");str_map[sym_gen(X,5,3,1)] = symbol("X_(5,3,1)");str_map[sym_gen(X,5,1,3)] = symbol("X_(5,1,3)");str_map[sym_gen(X,6,6,0)] = symbol("X_(6,6,0)");str_map[sym_gen(X,6,4,2)] = symbol("X_(6,4,2)");str_map[sym_gen(X,6,2,4)] = symbol("X_(6,2,4)");str_map[sym_gen(X,6,0,0)] = symbol("X_(6,0,0)");

97

str_map[sym_gen(X,7,1,3)] = symbol("X_(7,1,3)");str_map[sym_gen(X,8,0,4)] = symbol("X_(8,0,4)");str_map[sym_gen(X,9,1,5)] = symbol("X_(9,1,5)");str_map[sym_gen(X,12,0,6)] = symbol("X_(12,0,6)");

// Read command line for (d,k,l)size_t d = atoi(argv[1]);size_t k = atoi(argv[2]);size_t l = atoi(argv[3]);

// The list of all monomials of degree (d,k,l)list<ex> *mons = new(list<ex>);

// Hilbert series of the triply-graded polynomial algebraex PHS = 1;for (list<lst>::const_iterator i = gens.begin();

i != gens.end();i++)

PHS = PHS/(1

-pow(q,i->op(0))*pow(s,i->op(1))*pow(t,i->op(2))*indexed(X,

idx(i->op(0),infty),idx(i->op(1),infty),idx(i->op(2),infty)));

// get the coeff of q^d s^k t^l in PHSex se;se = PHS.series(q==0,d+1).coeff(q,d);se = se.series(s==0,k+1).coeff(s,k);se = se.series(t==0,l+1).coeff(t,l);se = se.expand();

// separate into terms and push into monsif (is_a<add>(se))for (const_iterator i = se.begin(); i != se.end(); i++)mons->push_back(*i);

elsemons->push_back(se);

// give each monomial a coeffcient A[index] and sum into mono_eqn// keep track of the coeffcients in vars;// number of monomials in num_monssize_t index = 0;

98

size_t num_mons = 0;ex mono_eqn=0;lst vars;for (list<ex>::const_iterator i = mons->begin();

i != mons->end();i++)

stringstream ss;ss << "A" << index;symbol newA(ss.str());ss.str("");mono_eqn += newA*(*i);vars.append(newA);index++;num_mons++;

// substitute each covariant J[d,k,l] into X[d,k,l]// expand the result and collect coefficientsex cov_eqn = mono_eqn.subs(cov_map,subs_options::no_pattern);cov_eqn = cov_eqn.expand();cov_eqn = cov_eqn.collect(lst(a,b,x0,x1,y0,y1),true);

// remove coefficients, each linear in the A[index]// push into lin_eqns list and solvelst lin_eqns;if (cov_eqn != 0)if (is_a<add>(cov_eqn))for (const_iterator i = cov_eqn.begin(); i != cov_eqn.end(); i++)lin_eqns.append(mycoeff(*i)==0);

elselin_eqns.append(mycoeff(cov_eqn)==0);

ex lin_solns = lsolve(lin_eqns,vars);

// determine a basis for the soln spacelist<lst> basis;for (const_iterator i = lin_solns.begin();

i != lin_solns.end();i++)

if (i->lhs() == i->rhs()) lst v;v.append(i->lhs() == 1);for (lst::const_iterator j = vars.begin(); j != vars.end(); j++)if (*j != i->rhs())v.append(ex(*j==0));

basis.push_back(v);

99

// specialise the rhs of lin_solns for each basis element// and then sub this into mono_eqn to recover a syzygyfor (list<lst>::const_iterator i = basis.begin();

i != basis.end();i++)

ex syzygy = mono_eqn.subs(lin_solns).subs(*i);cout << syzygy.subs(str_map) << ",";

// A zero to terminate the output.cout << "0" << endl;return 0;

100

B.2 Computing the Hilbert series in Macaulay2

The following transcript demonstrates how to use the syzygy program of the previoussection to compute the Hilbert series of IF using Macaulay2.

# The triply-graded polynomial algebra with generators X_(d,k,l)

R = QQ[

X_(1,3,1),X_(2,4,0),X_(2,2,2),X_(3,5,1),X_(3,3,3),X_(3,1,1),

X_(4,4,0),X_(4,2,2),X_(4,0,4),X_(5,3,1),X_(5,1,3),X_(6,6,0),

X_(6,4,2),X_(6,2,4),X_(7,1,3),X_(8,0,4),X_(9,1,5),X_(12,0,6),

X_(2,0,0),X_(6,0,0),

Degrees=>

1,3,1,2,4,0,2,2,2,3,5,1,3,3,3,3,1,1,4,4,0,

4,2,2,4,0,4,5,3,1,5,1,3,6,6,0,6,4,2,6,2,4,

7,1,3,8,0,4,9,1,5,12,0,6,2,0,0,6,0,0

];

# Candidate degrees for a minimal generating set of syzygies

F = 5,7,3,6,6,2,6,4,4,6,6,6,6,8,4,6,10,2,7,5,5,

7,7,3,7,5,3,7,9,1,8,6,4,8,4,4,8,4,6,8,8,2,

8,6,2,9,3,5,9,5,3,9,7,1,9,5,7,9,7,5,9,9,3,

9,11,1,10,4,6,10,6,4,10,8,2,10,4,4,10,2,6,10,6,2,

11,3,7,11,5,5,11,3,5,11,5,3,11,7,3,11,9,1,12,2,6,

12,4,4,12,4,8,12,6,6,12,8,4,12,10,2,12,12,0,13,3,7,

13,5,5,13,3,5,13,7,3,14,2,8,14,2,6,14,4,6,14,6,4,

15,1,7,15,3,9,15,5,7,15,7,5,16,2,8,16,4,6,17,1,9,

17,3,7,18,2,10,18,4,8,18,6,6,19,1,9,21,1,11,

24,0,12;

# Call the external syzygies program to process degree (d,k,l)

syzygies = (X) -> (

command := concatenate("!./syzygies ",toString(X#0)," ",

toString(X#1)," ",

toString(X#2));

value concatenate("ideal(", get command,")"));

# compute syzygies for every degree in F

li = apply(F,syzygies);

I_F = ideal(R);

for i in li do

if i != 0 then

I_F = I_F + i

I_F = trim I_F;

hs = hilbertSeries I_F;

101

The resulting Hilbert series agrees with (4.7). It takes serveral hours for Macaulay2

to compute the Hilbert series.The remainder of this section lists the resulting 104 syzygies of the binary (3, 1)-

forms which minimally generate IG. For typographical reasons, we write Xdkl when-ever d, k, l are single digits.

102

X2 131X

311−

X222X

351+

X240X

333

3X222X

440−

X240X

422−

3X131X

531

3X333X

311−

2X222X

422+

3X131X

513

3X351X

311−

2X240X

422−

3X131X

531

X2 222X

200−

X222X

422+

X240X

404+

X131X

513

4X3 222+

X2 131X

404+

X2 333

3X2 131X

222X

200−

12X

240X

2 222−

X2 131X

422−

3X351X

333

X2 131X

240X

200−

4X2 240X

222+

X2 131X

440−

X2 351

3X222X

333X

200−

3X333X

422+

3X351X

404+

2X131X

624

X222X

351X

200−

X240X

333X

200+

3X333X

440−

X351X

422−

2X131X

642

X131X

2 311−

X222X

531−

X240X

513

6X2 222X

311+

X333X

422−

X131X

624

6X240X

222X

311+

6X333X

440−

X351X

422−

3X131X

642

2X2 240X

311+

X351X

440−

X131X

660

9X333X

531−

3X351X

513−

6X222X

642+

2X240X

624

X222X

422X

200−

3X131X

513X

200−

X2 422+

3X440X

404+

X131X

713

3X131X

311X

404−

3X333X

513+

2X222X

624

3X131X

311X

422−

6X351X

513−

3X222X

642+

X240X

624

2X131X

311X

440−

X351X

531−

X222X

660+

X240X

642

36X

222X

2 311+

4X2 422−

3X131X

713

3X240X

2 311+

3X2 131X

600+

X440X

422

3X131X

222X

311X

200−

3X351X

513−

3X222X

642−

X240X

624

2X131X

240X

311X

200+

3X351X

531−

3X222X

660−

X240X

642

4X311X

624−

X222X

713+

3X131X

804

4X422X

513−

X222X

713−

3X131X

804

6X422X

531−

12X

440X

513−

6X311X

642+

X240X

71336

X131X

222X

600+

24X

440X

513+

12X

311X

642+

X240X

713

2X131X

240X

600−

X440X

531+

X311X

660

4X222X

513X

200−

4X404X

531−

X222X

713+

X131X

804

6X222X

531X

200+

6X240X

513X

200−

6X440X

513−

6X311X

642−

X240X

713

X131X

422X

404+

6X2 222X

513+

X333X

624

6X131X

440X

404+

18X

2 222X

531+

6X240X

222X

513+

3X333X

642+

X351X

624

6X2 131X

513X

200+

18X

2 222X

531−

6X240X

222X

513−

X2 131X

713+

3X333X

642−

X351X

624

103

3X2 131X

531X

200−

2X131X

440X

422−

6X240X

222X

531+

6X2 240X

513+

3X333X

660−

3X351X

642

X131X

240X

422X

200+

X131X

440X

422−

6X240X

222X

531−

12X

2 240X

513−

6X333X

660+

3X351X

642

X131X

240X

440X

200+

X131X

2 440−

2X2 240X

531−

X351X

660

8X422X

624−

3X333X

713−

3X131X

915

18X

222X

642X

200+

6X240X

624X

200+

72X

131X

333X

600+

36X

404X

660−

2X422X

642+

22X

440X

624+

X351X

713

48X

333X

513X

200−

8X222X

624X

200−

24X

404X

642−

9X333X

713+

3X131X

915

18X

351X

513X

200−

6X240X

624X

200−

36X

131X

333X

600−

18X

404X

660−

8X422X

642−

2X440X

624−

5X351X

713

X351X

531X

200+

X222X

660X

200−

X240X

642X

200+

6X131X

351X

600+

X422X

660−

X440X

642

4X2 422X

200−

3X131X

713X

200−

108X

2 222X

600−

36X

531X

513−

9X240X

804

24X

222X

311X

513+

X333X

713−

X131X

915

18X

240X

311X

513−

18X

131X

333X

600−

4X422X

642+

2X440X

624−

X351X

713

36X

222X

311X

531+

4X422X

642−

8X440X

624+

X351X

713

3X240X

311X

531+

3X131X

351X

600+

X422X

660

X2 311X

404+

X2 513−

X222X

804

4X2 311X

422+

12X

2 222X

600−

3X240X

804

3X2 131X

200X

600−

3X2 311X

440+

X440X

422X

200−

12X

240X

222X

600+

3X2 531

4X513X

624−

3X333X

804−

X222X

915

X131X

311X

713−

4X531X

624−

3X351X

804−

X240X

915

36X

2 311X

513−

9X131X

804X

200−

36X

131X

404X

600+

X422X

713

12X

2 311X

531−

4X131X

422X

600−

X440X

713

4X311X

422X

404−

3X333X

804+

X222X

915

12X

311X

440X

404−

6X513X

642+

6X531X

624−

3X351X

804+

X240X

915

8X311X

2 422+

36X

222X

333X

600−

12X

531X

624−

18X

351X

804−

3X240X

915

4X131X

311X

513X

200−

2X513X

642−

2X531X

624−

3X351X

804−

X240X

915

X131X

311X

531X

200+

6X222X

351X

600−

6X240X

333X

600+

3X513X

660+

X531X

642

X240X

311X

422X

200+

X311X

440X

422−

6X222X

351X

600+

12X

240X

333X

600−

6X513X

660−

3X531X

642

X240X

311X

440X

200+

X311X

2 440+

2X240X

351X

600−

X531X

660

X513X

713−

2X422X

804−

X311X

915

3X222X

804X

200+

12X

222X

404X

600−

X422X

804−

X311X

915

3X240X

804X

200+

12X

240X

404X

600+

12X

131X

513X

600+

X531X

713+

3X440X

804

3X131X

404X

713+

36X

2 222X

804+

8X2 624−

3X333X

915

36X

222X

2 513−

36X

2 222X

804−

4X2 624+

3X333X

915

104

8X440X

422X

404+

36X

222X

531X

513+

12X

240X

222X

804+

4X642X

624+

X351X

915

4X440X

2 422+

18X

222X

2 531+

27X

240X

531X

513−

3X131X

440X

713+

9X2 240X

804+

9X351X

333X

600+

3X660X

624

3X131X

222X

713X

200+

72X

222X

531X

513−

X131X

422X

713+

36X

240X

222X

804−

3X351X

915

X131X

240X

713X

200−

48X

2 440X

404−

108X

222X

2 531−

96X

240X

531X

513+

X131X

440X

713−

48X

2 240X

804−

12X

2 642−

16X

660X

624

2X131X

311X

660X

200+

12X

2 240X

222X

600−

3X240X

2 531+

3X2 351X

600−

X660X

642

X240X

2 440X

200−

4X3 240X

600+

X3 440−

X2 660

12X

4 311−

4X222X

422X

600+

12X

131X

513X

600+

3X440X

804

9X333X

804X

200+

36X

333X

404X

600−

X624X

713−

X422X

915+

X131X

12,0

,6

8X531X

624X

200−

3X351X

804X

200+

2X240X

915X

200−

36X

351X

404X

600+

12X

131X

624X

600−

X642X

713+

3X440X

915

18X

222X

311X

804+

X624X

713−

X422X

915

12X

240X

311X

804−

3X351X

804X

200−

12X

351X

404X

600−

12X

131X

624X

600−

X642X

713−

X440X

915

X2 311X

713+

12X

222X

513X

600+

3X531X

804

36X

311X

2 513+

X624X

713−

X131X

12,0

,6

12X

311X

2 531+

24X

333X

440X

600−

8X351X

422X

600−

12X

131X

642X

600−

X660X

713

3X624X

804−

3X513X

915+

X222X

12,0

,6

36X

2 311X

804−

12X

422X

804X

200−

48X

422X

404X

600+

X2 713

3X311X

404X

713−

3X513X

915+

2X222X

12,0

,6

X311X

422X

713+

72X

333X

513X

600−

36X

222X

624X

600+

9X642X

804−

6X531X

915−

2X240X

12,0

,6

X311X

440X

713+

12X

351X

513X

600−

4X240X

624X

600+

3X660X

804

3X131X

311X

804X

200+

36X

333X

513X

600−

24X

222X

624X

600+

3X642X

804−

3X531X

915−

X240X

12,0

,6

3X222X

311X

713X

200+

72X

333X

513X

600−

12X

222X

624X

600+

9X642X

804−

3X531X

915−

2X240X

12,0

,6

18X

513X

804X

200+

72X

404X

513X

600−

3X713X

804−

2X311X

12,0

,6

X422X

404X

713+

18X

222X

513X

804+

X624X

915

4X2 422X

713−

6X440X

404X

713−

3X131X

2 713+

36X

240X

513X

804+

72X

333X

624X

600+

6X642X

915+

2X351X

12,0

,6

36X

3 513−

36X

222X

513X

804−

X624X

915+

X333X

12,0

,6

6X2 131X

713X

600−

36X

2 531X

513+

2X440X

422X

713−

36X

240X

531X

804+

12X

351X

624X

600+

3X660X

915

3X131X

311X

915X

200−

3X440X

404X

713+

18X

240X

513X

804−

3X642X

915−

2X351X

12,0

,6

18X

513X

915X

200−

6X222X

12,0

,6X

200+

72X

404X

624X

600−

3X713X

915+

2X422X

12,0

,6

108X

311X

513X

804+

3X713X

915−

4X422X

12,0

,6

24X

222X

624X

200X

600+

36X

311X

531X

804+

6X531X

915X

200+

2X240X

12,0

,6X

200−

9X333X

713X

600+

15X

131X

915X

600+

2X440X

12,0

,6

18X

311X

404X

804X

200+

72X

311X

2 404X

600−

3X804X

915+

2X513X

12,0

,6

12X

311X

422X

804X

200+

6X2 311X

915X

200−

X311X

2 713+

18X

333X

804X

600+

6X222X

915X

600+

2X531X

12,0

,6

105

108X

2 513X

804−

108X

222X

2 804−

3X2 915+

4X624X

12,0

,6

X404X

2 713+

36X

222X

2 804+

X2 915

6X311X

422X

915X

200−

4X131X

311X

12,0

,6X

200−

108X

2 222X

804X

600+

27X

240X

2 804−

24X

2 624X

600+

36X

333X

915X

600+

2X642X

12,0

,6

54X

311X

2 804+

9X804X

915X

200+

36X

404X

915X

600−

X713X

12,0

,6

9X404X

713X

804X

200+

36X

2 404X

713X

600+

54X

513X

2 804+

X915X

12,0

,6

81X

404X

2 804X

2 200+

648X

2 404X

804X

200X

600+

1296

X3 404X

2 600+

81X

3 804+

X2 12,0

,6

48X

440X

422X

804X

200+

24X

311X

440X

915X

200+

576X

240X

2 513X

600+

12X

131X

422X

713X

600

−43

2X240X

222X

804X

600−

X440X

2 713+

108X

2 531X

804−

48X

642X

624X

600+

36X

351X

915X

600+

8X660X

12,0

,6

106

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108

Curriculum Vitae

Anthony Paul van GroningenPlace of Birth: Kew, Victoria, AustraliaEducation

• Doctor of Philosophy in Mathematics, May 2007University of Wisconsin—MilwaukeeDissertation Title: Graded multiplicities of the nullconefor the algebraic symmetric pair of type G

• Master of Science in Mathematics, May 2000University of Wisconsin—Milwaukee

• Bachelor of Science in Applied Science, May 1998Major: Computer Science; Minor: MathematicsUniversity of Wisconsin—Milwaukee

Major Professor Date

Major Professor Date