€¦ · weyls second fundamental theorem r := c[zij j1 i p;1 j q] c-alg hom : r !c[(v∗)p vq]gl(v...

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Explicit Pieri Inclusions and Minimal Graded Free Resolutions of Modules of Covariants of Several Vectors and Covectors for a General Linear Group John A. Miller Baylor University Algebra Seminar Texas Tech University February 19, 2020 e-mail: [email protected] url: www.johnamiller.xyz John A. Miller (Baylor University) Pieri Inclusions and Syzygies 1

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Page 1: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Explicit Pieri Inclusions and Minimal Graded FreeResolutions of Modules of Covariants of SeveralVectors and Covectors for a General Linear Group

John A. Miller

Baylor University

Algebra SeminarTexas Tech UniversityFebruary 19, 2020

e-mail: [email protected]: www.johnamiller.xyz

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 1

Page 2: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

The following is joint work with Markus Hunziker and Mark Sepanskiof Baylor University.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 2

Page 3: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Syzygies

James Sylvester

syz·y·gy /’sizije/

from the Greek συζυγoς [syzygos] meaning “yoked together”

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 3

Page 4: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Syzygies

R = C [z1, . . . , zn] polynomial ring

M =⟨g1, . . . , gb0

⟩fin. gen. R-module

Rb0 free R-module with basis {ei : 1 ≤ i ≤ b0}

Consider the R-module map ε : Rb0 →M given by ei 7→ gi. Then

Rb0ϵ−→M −→ 0 is exact.

An element of ker ε is called a relation or first syzygy of M .

ker ε is called the module of first syzygies.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 4

Page 5: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Syzygies

R = C [z1, . . . , zn] polynomial ring

M =⟨g1, . . . , gb0

⟩fin. gen. R-module

Rb0 free R-module with basis {ei : 1 ≤ i ≤ b0}

Consider the R-module map ε : Rb0 →M given by ei 7→ gi. Then

Rb0ϵ−→M −→ 0 is exact.

An element of ker ε is called a relation or first syzygy of M .

ker ε is called the module of first syzygies.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 4

Page 6: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Syzygies

R = C [z1, . . . , zn] polynomial ring

M =⟨g1, . . . , gb0

⟩fin. gen. R-module

Rb0 free R-module with basis {ei : 1 ≤ i ≤ b0}

Consider the R-module map ε : Rb0 →M given by ei 7→ gi. Then

Rb0ϵ−→M −→ 0 is exact.

An element of ker ε is called a relation or first syzygy of M .

ker ε is called the module of first syzygies.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 4

Page 7: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Syzygies

The module of first syzygies, ker ε, is also a finitely generatedR-module.

Therefore, ∃ R-module map δ1 : Rb1 −→ Rb0 such that

Rb1δ1−→ Rb0

ϵ−→M −→ 0 is exact.

The module of second syzygies, ker δ1, is again finitely generated.

Therefore, ∃ R-module map δ2 : Rb2 −→ Rb1 such that

Rb2δ2−→ Rb1

δ1−→ Rb0ϵ−→M −→ 0 is exact.

etc.John A. Miller (Baylor University) Pieri Inclusions and Syzygies 5

Page 8: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Syzygies

Will this go on forever?

Syzygy Theorem, Hilbert 1890

For R = C [z1, . . . , zn] and M as above, ∃ exact sequence of freemodules (free resolution)

0 −→ Rbm δm−→ · · ·Rb1δ1−→ Rb0

ϵ−→M −→ 0,

where m ≤ n.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 6

Page 9: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Classical Invariant Theory

H a group, W finite dimensional H-module

{x1, . . . , xn} coordinate functions of W

Definition

The coordinate ring C [W ] = C [x1, . . . , xn] is the (graded) C-algebra of polynomials from W to C.

The grading on C [W ] is given by

C [W ] =⊕d∈Z≥0

C [W ]d

where C [W ]dare the homog. poly. of degree d.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 7

Page 10: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Classical Invariant Theory

H a group, W finite dimensional H-module

{x1, . . . , xn} coordinate functions of W

Definition

The coordinate ring C [W ] = C [x1, . . . , xn] is the (graded) C-algebra of polynomials from W to C.

The grading on C [W ] is given by

C [W ] =⊕d∈Z≥0

C [W ]d

where C [W ]dare the homog. poly. of degree d.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 7

Page 11: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Invariants

Definition

f ∈ C [W ] is called invariant (or H-invariant) if f (h · w) = f (w)for all h ∈ H,w ∈ W , i.e. f is constant on orbits.

Definition

The subalgebra

C [W ]H:= {f ∈ C[W ] | f is H-invariant}

of C [W ] is called the ring of invariants.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 8

Page 12: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Covariants

Definition

If U is another finite-dimensional H-representation, the moduleof covariants of W of type U is defined as the space

CovH(W,U) := {φ : Wpoly−→ U | φ (h · w) = h · φ (w)

∀ h ∈ H,w ∈ W} .

Note that if U = C is the trivial representation,

CovH(W,U) = C [W ]H.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 9

Page 13: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Fundamental Problem of CIT

Arthur Cayley David Hilbert Hermann Weyl

Fundamental Problem of CIT

Find generators and syzygies (relations) for the ring of invariantsC [W ]

Hand, more generally, for modules of covariants.

A partial solution to this problem for C [W ]Hwas given by Weyl in

1939.John A. Miller (Baylor University) Pieri Inclusions and Syzygies 10

Page 14: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Weyls First Fundamental Theorem

H = GL (V ) , W = V ∗ ⊕ · · · ⊕ V ∗︸ ︷︷ ︸p copies

⊕V ⊕ · · · ⊕ V︸ ︷︷ ︸q copies

= (V ∗)p ⊕ V q

For 1 ≤ i ≤ p and 1 ≤ j ≤ q, define

fij : (V∗)

p ⊕ V q → C

byfij (λ1, . . . , λp, v1, . . . , vq) = λi (vj) .

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 11

Page 15: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Weyls First Fundamental Theorem

H = GL (V ) , W = V ∗ ⊕ · · · ⊕ V ∗︸ ︷︷ ︸p copies

⊕V ⊕ · · · ⊕ V︸ ︷︷ ︸q copies

= (V ∗)p ⊕ V q

For 1 ≤ i ≤ p and 1 ≤ j ≤ q, define

fij : (V∗)

p ⊕ V q → C

byfij (λ1, . . . , λp, v1, . . . , vq) = λi (vj) .

Then, by construction, fij ∈ C [(V ∗)p ⊕ V q].

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 11

Page 16: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Weyls First Fundamental Theorem

H = GL (V ) , W = V ∗ ⊕ · · · ⊕ V ∗︸ ︷︷ ︸p copies

⊕V ⊕ · · · ⊕ V︸ ︷︷ ︸q copies

= (V ∗)p ⊕ V q

For 1 ≤ i ≤ p and 1 ≤ j ≤ q, define

fij : (V∗)

p ⊕ V q → Cby

fij (λ1, . . . , λp, v1, . . . , vq) = λi (vj) .

In fact, fij ∈ C[(V ∗)p ⊕ V q]GL(V ):

fij (g· (λ1, . . . , λp, v1, . . . , vq))

= fij (g · λ1, . . . , g · λp, g · v1, . . . , g · vq)= fij (λ1g

−1, . . . , λpg−1, g · v1, . . . , g · vq)

= λi (g−1g) · vj

= λivj = fij (λ1, . . . , λp, v1, . . . , vq)

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 11

Page 17: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Weyls First Fundamental Theorem

H = GL (V ) , W = V ∗ ⊕ · · · ⊕ V ∗︸ ︷︷ ︸p copies

⊕V ⊕ · · · ⊕ V︸ ︷︷ ︸q copies

= (V ∗)p ⊕ V q

For 1 ≤ i ≤ p and 1 ≤ j ≤ q, define

fij : (V∗)

p ⊕ V q → C

byfij (λ1, . . . , λp, v1, . . . , vq) = λi (vj) .

Theorem (FFT for GL (V ), Weyl 1939)

The basic invariants fij generate the invariant ring

C [(V ∗)p ⊕ V q]

GL(V )as a C-algebra.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 11

Page 18: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Weyls Second Fundamental Theorem

R := C [zij | 1 ≤ i ≤ p, 1 ≤ j ≤ q]

C-alg hom ε : R→ C [(V ∗)p ⊕ V q]

GL(V )given by

zij 7→ fij.

Then, by the FFT,

Rϵ−→ C [(V ∗)

p ⊕ V q]GL(V ) −→ 0

is exact.

Theorem (SFT for GL (V ), Weyl 1939)

The first syzygies of the invariant ring C[(V ∗)p ⊕ V q]GL(V ) are

generated by the (k+1)× (k+1) minors of the matrix z = (zij),where k = dimV .

Remark: The minimal graded free resolution of C [(V ∗)p ⊕ V q]

GL(V )

as an R-module was first computed by Lascoux in 1978.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 12

Page 19: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Weyls Second Fundamental Theorem

R := C [zij | 1 ≤ i ≤ p, 1 ≤ j ≤ q]

C-alg hom ε : R→ C [(V ∗)p ⊕ V q]

GL(V )given by

zij 7→ fij.

Then, by the FFT,

Rϵ−→ C [(V ∗)

p ⊕ V q]GL(V ) −→ 0

is exact.

Theorem (SFT for GL (V ), Weyl 1939)

The first syzygies of the invariant ring C[(V ∗)p ⊕ V q]GL(V ) are

generated by the (k+1)× (k+1) minors of the matrix z = (zij),where k = dimV .

Remark: The minimal graded free resolution of C [(V ∗)p ⊕ V q]

GL(V )

as an R-module was first computed by Lascoux in 1978.John A. Miller (Baylor University) Pieri Inclusions and Syzygies 12

Page 20: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Main Results

Hunziker–M–Sepanski

In the context of Hermann Weyl’s FT of invariant theory. for theclassical groups GL (V ), O (V ), and Sp (V ), we are able to:

Compute the syzygies of all modules of covariantsuniformly

Describe the differentials explicitly

Idea:

Via Howe duality: modules of covariants ←→ unitarizablehighest weight modules

These then have BGG resolutions in some parabolic category OHighest weight modules σ ⇝ “compressed” λ

Parametrize the corresponding minimal free resolution via λ

Parameters “unzip” to give Verma modules

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 13

Page 21: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Main Results

Hunziker–M–Sepanski

In the context of Hermann Weyl’s FT of invariant theory. for theclassical groups GL (V ), O (V ), and Sp (V ), we are able to:

Compute the syzygies of all modules of covariantsuniformly

Describe the differentials explicitly

Idea:

Via Howe duality: modules of covariants ←→ unitarizablehighest weight modules

These then have BGG resolutions in some parabolic category OHighest weight modules σ ⇝ “compressed” λ

Parametrize the corresponding minimal free resolution via λ

Parameters “unzip” to give Verma modules

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 13

Page 22: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Main Results

Hunziker–M–Sepanski

In the context of Hermann Weyl’s FT of invariant theory. for theclassical groups GL (V ), O (V ), and Sp (V ), we are able to:

Compute the syzygies of all modules of covariantsuniformly

Describe the differentials explicitly

Visualization:

0 −→ −→ ↗↘⊕ ↘↗ −→ · −→M −→ 0,

Today: Explain the set-up in the case H = GL (V ) and give anexample.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 13

Page 23: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Notation for GL (V ) Reps

H = GL (V ) where dim(V ) = k

{polynomial irreducible reps of H}xy bij

{λ = (λ1, . . . , λk) | λi ∈ Z, λ1 ≥ · · · ≥ λk ≥ 0}

λ = (λ1, . . . , λk) ⇝ Young diagram with λ1 boxes in the first row,

λ2 boxes in the second row, etc.

We will just write the Young diagram λ for the correspondingSchur-Weyl module Sλ (V ).

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 14

Page 24: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Notation for GL (V ) Reps

H = GL (V ) where dim(V ) = k

{polynomial irreducible reps of H}xy bij

{λ = (λ1, . . . , λk) | λi ∈ Z, λ1 ≥ · · · ≥ λk ≥ 0}

λ = (λ1, . . . , λk) ⇝ Young diagram with λ1 boxes in the first row,

λ2 boxes in the second row, etc.

We will just write the Young diagram λ for the correspondingSchur-Weyl module Sλ (V ).

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 14

Page 25: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Notation for GL (V ) Reps

H = GL (V ) where dim(V ) = k

{polynomial irreducible reps of H}xy bij

{λ = (λ1, . . . , λk) | λi ∈ Z, λ1 ≥ · · · ≥ λk ≥ 0}

λ = (λ1, . . . , λk) ⇝ Young diagram with λ1 boxes in the first row,

λ2 boxes in the second row, etc.

We will just write the Young diagram λ for the correspondingSchur-Weyl module Sλ (V ).

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 14

Page 26: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Notation for GL (V ) Reps

E.g.

λ = (4, 3, 1, 1) ⇝ Sλ (V )←→ λ =

A basis for Sλ (V ) is the set of semi-standard tableaux of shape λ.

That is, all fillings of λ with the alphabet [k] = {1, . . . , k} where thefilling is weakly increasing across rows and strictly increasing downcolumns.

E.g.1 1 1 12 2 234

1 2 3 32 3 434

Note that the tableau with all ones in the first row, all twos in thesecond row, etc. is the highest weight vector for Sλ (V ).

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 15

Page 27: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Notation for GL (V ) Reps

E.g.

λ = (4, 3, 1, 1) ⇝ Sλ (V )←→ λ =

A basis for Sλ (V ) is the set of semi-standard tableaux of shape λ.

That is, all fillings of λ with the alphabet [k] = {1, . . . , k} where thefilling is weakly increasing across rows and strictly increasing downcolumns.

E.g.1 1 1 12 2 234

1 2 3 32 3 434

Note that the tableau with all ones in the first row, all twos in thesecond row, etc. is the highest weight vector for Sλ (V ).

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 15

Page 28: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Notation for GL (V ) Reps

E.g.

λ = (4, 3, 1, 1) ⇝ Sλ (V )←→ λ =

A basis for Sλ (V ) is the set of semi-standard tableaux of shape λ.

That is, all fillings of λ with the alphabet [k] = {1, . . . , k} where thefilling is weakly increasing across rows and strictly increasing downcolumns.

E.g.1 1 1 12 2 234

1 2 3 32 3 434

Note that the tableau with all ones in the first row, all twos in thesecond row, etc. is the highest weight vector for Sλ (V ).

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 15

Page 29: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Notation for GL (V ) Reps

E.g.

λ = (4, 3, 1, 1) ⇝ Sλ (V )←→ λ =

A basis for Sλ (V ) is the set of semi-standard tableaux of shape λ.

That is, all fillings of λ with the alphabet [k] = {1, . . . , k} where thefilling is weakly increasing across rows and strictly increasing downcolumns.

E.g.1 1 1 12 2 234

1 2 3 32 3 434

Note that the tableau with all ones in the first row, all twos in thesecond row, etc. is the highest weight vector for Sλ (V ).

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 15

Page 30: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Notation for GL (V ) Reps

For general irreps of H = GL (V ) we have

H = {irreps of H}xy bij

{σ = (σ1, . . . , σk) | σi ∈ Z, σ1 ≥ · · · ≥ σk}

So, a general GL (V ) irrep is F (k)σ = Fσ for some

σ = (n1, . . . , ni, 0, . . . , 0,−mj, . . . ,−m1)︸ ︷︷ ︸k

E.g. σ = (3, 2, 2, 0, 0,−1,−4). We can associate to σ two Youngdiagrams:

σ+ = (3, 2, 2) = and σ− = (4, 1) =

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 16

Page 31: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Notation for GL (V ) Reps

For general irreps of H = GL (V ) we have

H = {irreps of H}xy bij

{σ = (σ1, . . . , σk) | σi ∈ Z, σ1 ≥ · · · ≥ σk}

So, a general GL (V ) irrep is F (k)σ = Fσ for some

σ = (n1, . . . , ni, 0, . . . , 0,−mj, . . . ,−m1)︸ ︷︷ ︸k

E.g. σ = (3, 2, 2, 0, 0,−1,−4). We can associate to σ two Youngdiagrams:

σ+ = (3, 2, 2) = and σ− = (4, 1) =

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 16

Page 32: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Notation for GL (V ) Reps

For general irreps of H = GL (V ) we have

H = {irreps of H}xy bij

{σ = (σ1, . . . , σk) | σi ∈ Z, σ1 ≥ · · · ≥ σk}

So, a general GL (V ) irrep is F (k)σ = Fσ for some

σ = (n1, . . . , ni, 0, . . . , 0,−mj, . . . ,−m1)︸ ︷︷ ︸k

E.g. σ = (3, 2, 2, 0, 0,−1,−4). We can associate to σ two Youngdiagrams:

σ+ = (3, 2, 2) = and σ− = (4, 1) =

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 16

Page 33: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to Covariants

H = GL (V ) , dim (V ) = k, W = (V ∗)p ⊕ V q

Let Σ ={σ ∈ H | CovGL(V ) ((V

∗)p ⊕ V q, Fσ) 6= 0

}.

Theorem (Kashiwara-Verge 1978)

σ ∈ Σ ⇐⇒ σ = (n1, . . . , ni, 0, . . . , 0,−mj, . . . ,−m1)

with 0 ≤ i ≤ q, 0 ≤ j ≤ p

Furthermore, for each σ ∈ Σ we can associate a partition λ (σ)to CovGL(V ) ((V

∗)p ⊕ V q, Fσ), namely

λ (σ) = (−k, . . . ,−k,−k −mj, . . . ,−k −m1︸ ︷︷ ︸p

;n1, . . . , ni, 0, . . . , 0︸ ︷︷ ︸q

).

Furthermore, the map σ 7→ λ(σ) is injective.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 17

Page 34: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to Covariants

H = GL (V ) , dim (V ) = k, W = (V ∗)p ⊕ V q

Let Σ ={σ ∈ H | CovGL(V ) ((V

∗)p ⊕ V q, Fσ) 6= 0

}.

Theorem (Kashiwara-Verge 1978)

σ ∈ Σ ⇐⇒ σ = (n1, . . . , ni, 0, . . . , 0,−mj, . . . ,−m1)

with 0 ≤ i ≤ q, 0 ≤ j ≤ p

Furthermore, for each σ ∈ Σ we can associate a partition λ (σ)to CovGL(V ) ((V

∗)p ⊕ V q, Fσ), namely

λ (σ) = (−k, . . . ,−k,−k −mj, . . . ,−k −m1︸ ︷︷ ︸p

;n1, . . . , ni, 0, . . . , 0︸ ︷︷ ︸q

).

Furthermore, the map σ 7→ λ(σ) is injective.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 17

Page 35: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Reduction of σ

σ = (n1, . . . , ni, 0, . . . , 0,−mj, . . . ,−m1)↓

λ (σ) = (−k, . . . ,−k,−k −mj, . . . ,−k −m1︸ ︷︷ ︸p

;n1, . . . , ni, 0, . . . , 0︸ ︷︷ ︸q

)

↓(λ+ ρ)

′= ( , . . . ,︸ ︷︷ ︸

p′

; , . . . ,︸ ︷︷ ︸q′

)

The terms in the resolution for M = CovGL(V ) ((V∗)

p ⊕ V q, Fσ) willbe parametrized by all Young diagrams contained in a p′ × q′

rectangle.

0 −→ −→ ↗↘⊕ ↘↗ −→ · −→M −→ 0

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 18

Page 36: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

dimV = k = 4, H = GL (V ), W = (V ∗)5 ⊕ V 5

M = CovGL(V )

((V ∗)

5 ⊕ V 5, F (4)σ

), p = q = 5.

Let σ = (3, 0,−1,−4).

σ = (3, 0− 1,−4)↓

λ = λ (σ) = (−k,−k,−k,−k −m2,−k −m1︸ ︷︷ ︸5

;n1, 0, 0, 0, 0︸ ︷︷ ︸5

)

↓λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

Where ρ = (p+ q − 1, . . . , 0) = (9, 8, 7, 6, 5, 4, 3, 2, 1, 0).

How to get from λ+ ρ to (λ+ ρ)′?

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 19

Page 37: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

dimV = k = 4, H = GL (V ), W = (V ∗)5 ⊕ V 5

M = CovGL(V )

((V ∗)

5 ⊕ V 5, F (4)σ

), p = q = 5.

Let σ = (3, 0,−1,−4).

σ = (3, 0− 1,−4)↓

λ = λ (σ) = (−k,−k,−k,−k −m2,−k −m1︸ ︷︷ ︸5

;n1, 0, 0, 0, 0︸ ︷︷ ︸5

)

↓λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

Where ρ = (p+ q − 1, . . . , 0) = (9, 8, 7, 6, 5, 4, 3, 2, 1, 0).

How to get from λ+ ρ to (λ+ ρ)′?

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 19

Page 38: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

dimV = k = 4, H = GL (V ), W = (V ∗)5 ⊕ V 5

M = CovGL(V )

((V ∗)

5 ⊕ V 5, F (4)σ

), p = q = 5.

Let σ = (3, 0,−1,−4).

σ = (3, 0− 1,−4)↓

λ = λ (σ) = (−4,−4,−4,−5,−8; 3, 0, 0, 0, 0)

↓λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

Where ρ = (p+ q − 1, . . . , 0) = (9, 8, 7, 6, 5, 4, 3, 2, 1, 0).

How to get from λ+ ρ to (λ+ ρ)′?

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 19

Page 39: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

dimV = k = 4, H = GL (V ), W = (V ∗)5 ⊕ V 5

M = CovGL(V )

((V ∗)

5 ⊕ V 5, F (4)σ

), p = q = 5.

Let σ = (3, 0,−1,−4).

σ = (3, 0− 1,−4)↓

λ = λ (σ) = (−4,−4,−4,−5,−8; 3, 0, 0, 0, 0)↓

λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

Where ρ = (p+ q − 1, . . . , 0) = (9, 8, 7, 6, 5, 4, 3, 2, 1, 0).

How to get from λ+ ρ to (λ+ ρ)′?

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 19

Page 40: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

dimV = k = 4, H = GL (V ), W = (V ∗)5 ⊕ V 5

M = CovGL(V )

((V ∗)

5 ⊕ V 5, F (4)σ

), p = q = 5.

Let σ = (3, 0,−1,−4).

σ = (3, 0− 1,−4)↓

λ = λ (σ) = (−4,−4,−4,−5,−8; 3, 0, 0, 0, 0)↓

λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

Where ρ = (p+ q − 1, . . . , 0) = (9, 8, 7, 6, 5, 4, 3, 2, 1, 0).

How to get from λ+ ρ to (λ+ ρ)′?

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 19

Page 41: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

To get (λ+ ρ)′:

1 Delete repeats on left & right(5, 4, 3, 1,−3; 7, 3, 2, 1, 0

)= (5, 4,−3; 7, 2, 0)

2 Delete any numbers on left < all numbers on right(5, 4, −3; 7, 2, 0

)= (5, 4; 7, 2, 0)

3 Delete any numbers on right > all numbers on left(5, 4; 7, 2, 0

)= (5, 4; 2, 0)

So (λ+ ρ)′= (5, 4; 2, 0), and p′ = q′ = 2.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 20

Page 42: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

To get (λ+ ρ)′:

1 Delete repeats on left & right(5, 4, 3, 1,−3; 7, 3, 2, 1, 0

)= (5, 4,−3; 7, 2, 0)

2 Delete any numbers on left < all numbers on right(5, 4, −3; 7, 2, 0

)= (5, 4; 7, 2, 0)

3 Delete any numbers on right > all numbers on left(5, 4; 7, 2, 0

)= (5, 4; 2, 0)

So (λ+ ρ)′= (5, 4; 2, 0), and p′ = q′ = 2.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 20

Page 43: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

To get (λ+ ρ)′:

1 Delete repeats on left & right(5, 4, 3, 1,−3; 7, 3, 2, 1, 0

)= (5, 4,−3; 7, 2, 0)

2 Delete any numbers on left < all numbers on right(5, 4, −3; 7, 2, 0

)= (5, 4; 7, 2, 0)

3 Delete any numbers on right > all numbers on left(5, 4; 7, 2, 0

)= (5, 4; 2, 0)

So (λ+ ρ)′= (5, 4; 2, 0), and p′ = q′ = 2.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 20

Page 44: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

To get (λ+ ρ)′:

1 Delete repeats on left & right(5, 4, 3, 1,−3; 7, 3, 2, 1, 0

)= (5, 4,−3; 7, 2, 0)

2 Delete any numbers on left < all numbers on right(5, 4, −3; 7, 2, 0

)= (5, 4; 7, 2, 0)

3 Delete any numbers on right > all numbers on left(5, 4; 7, 2, 0

)= (5, 4; 2, 0)

So (λ+ ρ)′= (5, 4; 2, 0), and p′ = q′ = 2.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 20

Page 45: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

λ+ ρ = (5, 4, 3, 1,−3; 7, 3, 2, 1, 0)

To get (λ+ ρ)′:

1 Delete repeats on left & right(5, 4, 3, 1,−3; 7, 3, 2, 1, 0

)= (5, 4,−3; 7, 2, 0)

2 Delete any numbers on left < all numbers on right(5, 4, −3; 7, 2, 0

)= (5, 4; 7, 2, 0)

3 Delete any numbers on right > all numbers on left(5, 4; 7, 2, 0

)= (5, 4; 2, 0)

So (λ+ ρ)′= (5, 4; 2, 0), and p′ = q′ = 2.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 20

Page 46: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

H = GL (V ) , k = 4, W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

⇝ (λ+ ρ)′= (5, 4; 2, 0), p′ = q′ = 2

0 −→ −→ ↗↘

⊕ ↘↗ −→ · −→ CovH(W,Fσ) −→ 0

Starting with the box on the bottom left, label diagonally viasuccessive differences in (λ+ ρ)

′= (5, 4; 2, 0):

5− 4 = 1 ,

4− 2 = 2 ,

2− 0 = 2

This “compressed” version needs to be unzipped.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 21

Page 47: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

H = GL (V ) , k = 4, W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

⇝ (λ+ ρ)′= (5, 4; 2, 0), p′ = q′ = 2

0 −→ −→ ↗↘

⊕ ↘↗ −→ · −→ CovH(W,Fσ) −→ 0

Starting with the box on the bottom left, label diagonally viasuccessive differences in (λ+ ρ)

′= (5, 4; 2, 0):

5− 4 = 1 ,

4− 2 = 2 ,

2− 0 = 2

This “compressed” version needs to be unzipped.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 21

Page 48: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

H = GL (V ) , k = 4, W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

⇝ (λ+ ρ)′= (5, 4; 2, 0), p′ = q′ = 2

0 −→ −→ ↗↘

⊕ ↘↗ −→ · −→ CovH(W,Fσ) −→ 0

Starting with the box on the bottom left, label diagonally viasuccessive differences in (λ+ ρ)

′= (5, 4; 2, 0):

5− 4 = 1 ,

4− 2 = 2 ,

2− 0 = 2

This “compressed” version needs to be unzipped.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 21

Page 49: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

H = GL (V ) , k = 4, W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

⇝ (λ+ ρ)′= (5, 4; 2, 0), p′ = q′ = 2

0 −→ 1 −→ 1

↗↘

⊕1

↘↗ −→ · −→ CovH(W,Fσ) −→ 0

Starting with the box on the bottom left, label diagonally viasuccessive differences in (λ+ ρ)

′= (5, 4; 2, 0):

5− 4 = 1 ,

4− 2 = 2 ,

2− 0 = 2

This “compressed” version needs to be unzipped.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 21

Page 50: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

H = GL (V ) , k = 4, W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

⇝ (λ+ ρ)′= (5, 4; 2, 0), p′ = q′ = 2

0 −→ 21 2 −→

21

↗↘

2

⊕21

↘↗ 2 −→ · −→ CovH(W,Fσ) −→ 0

Starting with the box on the bottom left, label diagonally viasuccessive differences in (λ+ ρ)

′= (5, 4; 2, 0):

5− 4 = 1 , 4− 2 = 2 ,

2− 0 = 2

This “compressed” version needs to be unzipped.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 21

Page 51: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

H = GL (V ) , k = 4, W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

⇝ (λ+ ρ)′= (5, 4; 2, 0), p′ = q′ = 2

0 −→ 2 21 2 −→

2 21

↗↘

2 2

⊕21

↘↗ 2 −→ · −→ CovH(W,Fσ) −→ 0

Starting with the box on the bottom left, label diagonally viasuccessive differences in (λ+ ρ)

′= (5, 4; 2, 0):

5− 4 = 1 , 4− 2 = 2 , 2− 0 = 2

This “compressed” version needs to be unzipped.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 21

Page 52: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

H = GL (V ) , k = 4, W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

⇝ (λ+ ρ)′= (5, 4; 2, 0), p′ = q′ = 2

0 −→ 2 21 2 −→

2 21

↗↘

2 2

⊕21

↘↗ 2 −→ · −→ CovH(W,Fσ) −→ 0

Starting with the box on the bottom left, label diagonally viasuccessive differences in (λ+ ρ)

′= (5, 4; 2, 0):

5− 4 = 1 , 4− 2 = 2 , 2− 0 = 2

This “compressed” version needs to be unzipped.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 21

Page 53: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

Unzip 2 21 :

σ = (3, 0,−1,−4)⇝ σ+ = (3) = σ− = (4, 1) =

Add boxes to the columns of σ− and σ+ as prescribed by thecolored diagram and its transpose, respectively.

2 21 = C [Mp×q]⊗

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 22

Page 54: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

Unzip 2 21 :

σ = (3, 0,−1,−4)⇝ σ+ = (3) = σ− = (4, 1) =

Add boxes to the columns of σ− and σ+ as prescribed by thecolored diagram and its transpose, respectively.

2 21 = C [Mp×q]⊗

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 22

Page 55: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

Unzip 2 21 :

σ = (3, 0,−1,−4)⇝ σ+ = (3) = σ− = (4, 1) =

Add boxes to the columns of σ− and σ+ as prescribed by thecolored diagram and its transpose, respectively.

2 21 = C [Mp×q]⊗

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 22

Page 56: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

Unzip 2 21 :

σ = (3, 0,−1,−4)⇝ σ+ = (3) = σ− = (4, 1) =

Add boxes to the columns of σ− and σ+ as prescribed by thecolored diagram and its transpose, respectively.

2 21 = C [M5×5]⊗

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 22

Page 57: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

Then the maps in the resolution look like

2 21 → 2

1 ,

which after unzipping looks like

C [M5×5]⊗

→ C [M5×5]⊗

Here, each tableau on the left is losing 2 boxes.

As a C [M5×5]-module map, it is enough to consider ⊗

→ C [M5×5]2 ⊗

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 23

Page 58: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

Then the maps in the resolution look like

2 21 → 2

1 ,

which after unzipping looks like

C [M5×5]⊗

→ C [M5×5]2 ⊗

Here, each tableau on the left is losing 2 boxes.

As a C [M5×5]-module map, it is enough to consider ⊗

→ C [M5×5]2 ⊗

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 23

Page 59: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

Then the maps in the resolution look like

2 21 → 2

1 ,

which after unzipping looks like

C [M5×5]⊗

→ C [M5×5]2 ⊗

Here, each tableau on the left is losing 2 boxes.

As a C [M5×5]-module map, it is enough to consider ⊗

→ C [M5×5]2 ⊗

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 23

Page 60: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

→ C [M5×5]2 ⊗

As a GL(5;C)×GL(5;C) rep, C [M5×5]2 decomposes into

C [M5×5]2 =(

⊗)⊕(⊗

)In general, C [M5×5]d decomposes as above with all shapes thathave d boxes and ≤ 5 rows.

Then, we end up with...

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 24

Page 61: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

→ C [M5×5]2 ⊗

As a GL(5;C)×GL(5;C) rep, C [M5×5]2 decomposes into

C [M5×5]2 =(

⊗)⊕(⊗

)In general, C [M5×5]d decomposes as above with all shapes thathave d boxes and ≤ 5 rows.

Then, we end up with...

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 24

Page 62: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

→ (⊗

)⊗

As a GL(5;C)×GL(5;C) rep, C [M5×5]2 decomposes into

C [M5×5]2 =(

⊗)⊕(⊗

)In general, C [M5×5]d decomposes as above with all shapes thathave d boxes and ≤ 5 rows.

Then, we end up with...

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 24

Page 63: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

→ (⊗

)⊗

Computing the syzygies for CovGL(V )

((V ∗)

5 ⊕ V 5, F (4)σ

)then

comes down to finding C[M5×5]-module and GL(V )-equivariantPieri inclusions, such as

→ ⊗ and → ⊗

For the rest of this talk: updated description of Pieri inclusions.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 24

Page 64: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Example

→ (⊗

)⊗

Computing the syzygies for CovGL(V )

((V ∗)

5 ⊕ V 5, F (4)σ

)then

comes down to finding C[M5×5]-module and GL(V )-equivariantPieri inclusions, such as

→ ⊗ and → ⊗

For the rest of this talk: updated description of Pieri inclusions.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 24

Page 65: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

The Pieri Rule

Theorem (Pieri Rule)

Let µ be a partition corresponding to a Schur-Weyl module Sµ (V )and ν = (1, . . . , 1) be a partition of m. Then we have an isomor-phism of GL (V )-modules

Sν (V )⊗ Sµ (V ) ∼=⊕λ

Sλ (V )

where the sum is over all λ ⊃ µ obtained by adding m boxes toµ with no two boxes in the same row. Similarly,

S(m) (V )⊗ Sµ (V ) ∼=⊕λ

Sλ (V )

where the sum is over all λ ⊃ µ obtained by adding m boxes toµ with no two boxes in the same column.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 25

Page 66: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

The Pieri Rule - One Box

⊗ ∼= ⊕ ⊕

⊗ ∼= ⊕ ⊕ ⊕

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 26

Page 67: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

The Pieri Rule - Two Boxes

⊗ ∼= ⊕ ⊕ ⊕ ⊕

⊗ ∼= ⊕ ⊕ ⊕

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 27

Page 68: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

The Pieri Rule - Three Boxes

⊗ ∼= ⊕ ⊕ ⊕ ⊕ ⊕

⊗ ∼= ⊕ ⊕ ⊕

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 28

Page 69: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Pieri Inclusions

From the Pieri rule we get maps

Sλ (V )→ Sν (V )⊗ Sµ (V ) ,

called Pieri inclusions, unique up to non-zero scalar multiple.

E.g.

−→ ⊗ −→ ⊗

These Pieri inclusions were first given by Olver in his thesis (1982)via iterating one box removal, and made more explicit by Sam(2009) and Sam and Weyman (2012).

We give a general closed form description of Pieri inclusionsremoving m boxes and show this rule is more efficient.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 29

Page 70: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Pieri Inclusions

From the Pieri rule we get maps

Sλ (V )→ Sν (V )⊗ Sµ (V ) ,

called Pieri inclusions, unique up to non-zero scalar multiple.

E.g.

−→ ⊗ −→ ⊗

These Pieri inclusions were first given by Olver in his thesis (1982)via iterating one box removal, and made more explicit by Sam(2009) and Sam and Weyman (2012).

We give a general closed form description of Pieri inclusionsremoving m boxes and show this rule is more efficient.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 29

Page 71: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Olver’s description of Pieri inclusions

ΦO =∑J

(−1)|J|JcJ

The J are the ways to remove the indicated box up and out of thediagram, |J | is the number of rows used, and the cJ depend on therows used.

E.g.,ΦO−→ ⊗ ,

ΦO

(123

)=?

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ −1

21 ⊗ 2

3 ,3

2

1 ⇝ 1

21 ⊗ 3

2

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30

Page 72: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Olver’s description of Pieri inclusions

ΦO =∑J

(−1)|J|JcJ

The J are the ways to remove the indicated box up and out of thediagram, |J | is the number of rows used, and the cJ depend on therows used.

E.g.,ΦO−→ ⊗ , ΦO

(123

)=?

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ −1

21 ⊗ 2

3 ,3

2

1 ⇝ 1

21 ⊗ 3

2

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30

Page 73: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Olver’s description of Pieri inclusions

ΦO =∑J

(−1)|J|JcJ

The J are the ways to remove the indicated box up and out of thediagram, |J | is the number of rows used, and the cJ depend on therows used.

E.g.,ΦO−→ ⊗ , ΦO

(123

)=?

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ −1

21 ⊗ 2

3 ,3

2

1 ⇝ 1

21 ⊗ 3

2

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30

Page 74: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Olver’s description of Pieri inclusions

ΦO =∑J

(−1)|J|JcJ

The J are the ways to remove the indicated box up and out of thediagram, |J | is the number of rows used, and the cJ depend on therows used.

E.g.,ΦO−→ ⊗ , ΦO

(123

)=?

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ −1

21 ⊗ 2

3 ,3

2

1 ⇝ 1

21 ⊗ 3

2

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30

Page 75: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Olver’s description of Pieri inclusions

ΦO =∑J

(−1)|J|JcJ

The J are the ways to remove the indicated box up and out of thediagram, |J | is the number of rows used, and the cJ depend on therows used.

E.g.,ΦO−→ ⊗ , ΦO

(123

)=?

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ −1

21 ⊗ 2

3 ,

3

2

1 ⇝ 1

21 ⊗ 3

2

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30

Page 76: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Olver’s description of Pieri inclusions

ΦO =∑J

(−1)|J|JcJ

The J are the ways to remove the indicated box up and out of thediagram, |J | is the number of rows used, and the cJ depend on therows used.

E.g.,ΦO−→ ⊗ , ΦO

(123

)=?

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ −1

21 ⊗ 2

3 ,3

2

1 ⇝ 1

21 ⊗ 3

2

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30

Page 77: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Olver’s description of Pieri inclusions

ΦO =∑J

(−1)|J|JcJ

The J are the ways to remove the indicated box up and out of thediagram, |J | is the number of rows used, and the cJ depend on therows used.

E.g.,ΦO−→ ⊗ , ΦO

(123

)=?

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ −1

21 ⊗ 2

3 ,3

2

1 ⇝ 1

21 ⊗ 3

2

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30

Page 78: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Olver’s description of Pieri inclusions

ΦO =∑J

(−1)|J|JcJ

The J are the ways to remove the indicated box up and out of thediagram, |J | is the number of rows used, and the cJ depend on therows used.

E.g.,ΦO−→ ⊗ , ΦO

(123

)=?

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ −1

21 ⊗ 2

3 ,3

2

1 ⇝ −1

21 ⊗ 2

3

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30

Page 79: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Olver’s description of Pieri inclusions

ΦO =∑J

(−1)|J|JcJ

The J are the ways to remove the indicated box up and out of thediagram, |J | is the number of rows used, and the cJ depend on therows used.

E.g.,ΦO−→ ⊗ , ΦO

(123

)= − 3 ⊗ 1

2 + 2 ⊗ 13 − 1 ⊗ 2

3

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ −1

21 ⊗ 2

3 ,3

2

1 ⇝ −1

21 ⊗ 2

3

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30

Page 80: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Olver’s description of Pieri inclusions

ΦO =∑J

(−1)|J|JcJ

The J are the ways to remove the indicated box up and out of thediagram, |J | is the number of rows used, and the cJ depend on therows used.

E.g.,ΦO−→ ⊗ , ΦO

(123

)= − 3 ⊗ 1

2 + 2 ⊗ 13 − 1 ⊗ 2

3

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ −1

21 ⊗ 2

3 ,3

2

1 ⇝ −1

21 ⊗ 2

3

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 30

Page 81: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Blocks of a diagram

Block notation: λ = (wh11 , w

h22 . . . , w

hN−1N−1 , w

hNN ) where wi < wi+1

and each wi appears as a part of λ exactly hi times.

...

(1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31

Page 82: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Blocks of a diagram

Block notation: λ = (wh11 , w

h22 . . . , w

hN−1N−1 , w

hNN ) where wi < wi+1

and each wi appears as a part of λ exactly hi times.

...

(1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31

Page 83: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Blocks of a diagram

Block notation: λ = (wh11 , w

h22 . . . , w

hN−1N−1 , w

hNN ) where wi < wi+1

and each wi appears as a part of λ exactly hi times.

...

(1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31

Page 84: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Blocks of a diagram

Block notation: λ = (wh11 , w

h22 . . . , w

hN−1N−1 , w

hNN ) where wi < wi+1

and each wi appears as a part of λ exactly hi times.

...

(1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31

Page 85: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Blocks of a diagram

Block notation: λ = (wh11 , w

h22 . . . , w

hN−1N−1 , w

hNN ) where wi < wi+1

and each wi appears as a part of λ exactly hi times.

w1h1

...

(1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31

Page 86: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Blocks of a diagram

Block notation: λ = (wh11 , w

h22 . . . , w

hN−1N−1 , w

hNN ) where wi < wi+1

and each wi appears as a part of λ exactly hi times.

w2h2

...

(1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31

Page 87: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Blocks of a diagram

Block notation: λ = (wh11 , w

h22 . . . , w

hN−1N−1 , w

hNN ) where wi < wi+1

and each wi appears as a part of λ exactly hi times.

...

wN−1

hN−1

(1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31

Page 88: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Blocks of a diagram

Block notation: λ = (wh11 , w

h22 . . . , w

hN−1N−1 , w

hNN ) where wi < wi+1

and each wi appears as a part of λ exactly hi times.

...

wN

hN (1, 1) = (12):

(6, 6, 6, 6, 3, 3, 3, 1) = (11, 33, 64):

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 31

Page 89: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

New description of Pieri inclusions

Φ =∑P

(−1)|P |P

H(P )

The P are the ways to remove the indicated box up and out of thediagram with a row skipping restriction. The coefficients H(P )are similar to the cJ , but depend only on the blocks used.

E.g.,Φ−→ ⊗ ,

Φ

(123

)=?

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ − 1 ⊗ 23

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 32

Page 90: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

New description of Pieri inclusions

Φ =∑P

(−1)|P |P

H(P )

The P are the ways to remove the indicated box up and out of thediagram with a row skipping restriction. The coefficients H(P )are similar to the cJ , but depend only on the blocks used.

E.g.,Φ−→ ⊗ , Φ

(123

)=?

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ − 1 ⊗ 23

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 32

Page 91: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

New description of Pieri inclusions

Φ =∑P

(−1)|P |P

H(P )

The P are the ways to remove the indicated box up and out of thediagram with a row skipping restriction. The coefficients H(P )are similar to the cJ , but depend only on the blocks used.

E.g.,Φ−→ ⊗ , Φ

(123

)=?

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ − 1 ⊗ 23

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 32

Page 92: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

New description of Pieri inclusions

Φ =∑P

(−1)|P |P

H(P )

The P are the ways to remove the indicated box up and out of thediagram with a row skipping restriction. The coefficients H(P )are similar to the cJ , but depend only on the blocks used.

E.g.,Φ−→ ⊗ , Φ

(123

)=?

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ − 1 ⊗ 23

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 32

Page 93: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

New description of Pieri inclusions

Φ =∑P

(−1)|P |P

H(P )

The P are the ways to remove the indicated box up and out of thediagram with a row skipping restriction. The coefficients H(P )are similar to the cJ , but depend only on the blocks used.

E.g.,Φ−→ ⊗ , Φ

(123

)=?

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ − 1 ⊗ 23

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 32

Page 94: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

New description of Pieri inclusions

Φ =∑P

(−1)|P |P

H(P )

The P are the ways to remove the indicated box up and out of thediagram with a row skipping restriction. The coefficients H(P )are similar to the cJ , but depend only on the blocks used.

E.g.,Φ−→ ⊗ , Φ

(123

)= − 3 ⊗ 1

2 + 2 ⊗ 13 − 1 ⊗ 2

3

3

2

1 ⇝ − 3 ⊗ 12 ,

3

2

1 ⇝ 2 ⊗ 13

3

2

1 ⇝ − 1 ⊗ 23

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 32

Page 95: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Pieri inclusions removing two boxes

↪→ ⊗ : 7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

( )

65432 41 2

⇝ 56 ⊗

1 22 434

,

65432 41 2

⇝ 34 ⊗

1 22 456

,

65432 41 2

⇝ −1

434 ⊗

1 22 456

,

65432 41 2

⇝ −1

431 ⊗

4 12 645

,

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33

Page 96: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Pieri inclusions removing two boxes

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

( )

65432 41 2

⇝ 56 ⊗

1 22 434

,

65432 41 2

⇝ 34 ⊗

1 22 456

,

65432 41 2

⇝ −1

434 ⊗

1 22 456

,

65432 41 2

⇝ −1

431 ⊗

4 12 645

,

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33

Page 97: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Pieri inclusions removing two boxes

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

( )

65432 41 2

⇝ 56 ⊗

1 22 434

,

65432 41 2

⇝ 34 ⊗

1 22 456

,

65432 41 2

⇝ −1

434 ⊗

1 22 456

,

65432 41 2

⇝ −1

431 ⊗

4 12 645

,

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33

Page 98: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Pieri inclusions removing two boxes

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

( )

65432 41 2

⇝ 56 ⊗

1 22 434

,

65432 41 2

⇝ 34 ⊗

1 22 456

,

65432 41 2

⇝ −1

434 ⊗

1 22 456

,

65432 41 2

⇝ −1

431 ⊗

4 12 645

,

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33

Page 99: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Pieri inclusions removing two boxes

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

( )

65432 41 2

⇝ 56 ⊗

1 22 434

,

65432 41 2

⇝ 34 ⊗

1 22 456

,

65432 41 2

⇝ −1

434 ⊗

1 22 456

,

65432 41 2

⇝ −1

431 ⊗

4 12 645

,

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33

Page 100: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Pieri inclusions removing two boxes

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

( )

65432 41 2

⇝ 56 ⊗

1 22 434

,

65432 41 2

⇝ 34 ⊗

1 22 456

,

65432 41 2

⇝ −1

434 ⊗

1 22 456

,

65432 41 2

⇝ −1

431 ⊗

4 12 645

,

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33

Page 101: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Pieri inclusions removing two boxes

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

( )

65432 41 2

⇝ 56 ⊗

1 22 434

,

65432 41 2

⇝ 34 ⊗

1 22 456

,

65432 41 2

⇝ −1

434 ⊗

1 22 456

,

65432 41 2

⇝ 1

413 ⊗

1 42 645

,

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33

Page 102: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Pieri inclusions removing two boxes

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

( )

65432 41 2

⇝ 56 ⊗

1 22 434

,

65432 41 2

⇝ 34 ⊗

1 22 456

,

65432 41 2

⇝ −1

434 ⊗

1 22 456

,

65432 41 2

⇝ 1

413 ⊗

1 42 645

,

65432 41 2

⇝ − 1

5 · 423 ⊗

1 54 436

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33

Page 103: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Pieri inclusions removing two boxes

↪→ ⊗ :

1 22 43456

7→∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗P

( )

65432 41 2

⇝ 56 ⊗

1 22 434

,

65432 41 2

⇝ 34 ⊗

1 22 456

,

65432 41 2

⇝ −1

434 ⊗

1 22 456

,

65432 41 2

⇝ 1

413 ⊗

1 42 645

,

65432 41 2

⇝ 1

2023 ⊗

1 34 456

− 1

1023 ⊗

1 43 546

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 33

Page 104: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Complexity of the descriptions

In both the old and new descriptions, the number of terms in thePieri inclusion acting on λ = (w

h11 , . . . , w

hNN ) depends on the

number of paths acting on the diagram.

Old description: paths given by the number of choices of rows inthe diagram, where you must choose the first (bottom most) row.

2h1−1 ·N∏i=2

2hi ,

New description: paths given by the number of choices of rows inthe diagram, where you must choose the first row and cannot skiprows within blocks.

h1 ·N∏i=2

(hi + 1) .

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 34

Page 105: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Complexity of the descriptions

In both the old and new descriptions, the number of terms in thePieri inclusion acting on λ = (w

h11 , . . . , w

hNN ) depends on the

number of paths acting on the diagram.

Old description: paths given by the number of choices of rows inthe diagram, where you must choose the first (bottom most) row.

2h1−1 ·N∏i=2

2hi ,

New description: paths given by the number of choices of rows inthe diagram, where you must choose the first row and cannot skiprows within blocks.

h1 ·N∏i=2

(hi + 1) .

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 34

Page 106: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Complexity of the descriptions

In both the old and new descriptions, the number of terms in thePieri inclusion acting on λ = (w

h11 , . . . , w

hNN ) depends on the

number of paths acting on the diagram.

Old description: paths given by the number of choices of rows inthe diagram, where you must choose the first (bottom most) row.

2h1−1 ·N∏i=2

2hi ,

New description: paths given by the number of choices of rows inthe diagram, where you must choose the first row and cannot skiprows within blocks.

h1 ·N∏i=2

(hi + 1) .

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 34

Page 107: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Computation Time in Macaulay2

Old description:

New description:

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 35

Page 108: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Computation Time in Macaulay2

Old description:

New description:

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 35

Page 109: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Computation Time in Macaulay2

Old description:

New description:

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 35

Page 110: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to the syzygies example

Recall:

W = (V ∗)5 ⊕ V 5, σ = (3, 0,−1,−4)

0 −→ 2 21 2 −→

2 21

↗↘

2 2

⊕21

↘↗ 2 −→ · −→ CovH(W,Fσ) −→ 0

And after unzipping, the maps look like ⊗

→ (⊗

)⊗

and are induced by Pieri inclusions.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 36

Page 111: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to the syzygies example

Consider the image of the highest weight vector:

5 5 5 54 43 321

⊗1 1 12 2345

→ ?

We will apply the Pieri inclusion map removing two boxes to eachterm separately, then put them together.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 37

Page 112: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to the syzygies example

Φ

1 1 12 2345

=∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗ P

( )

Some terms in the sum are (pre-straightening):

54321

21 1

⇝ 45 ⊗

1 1 12 23

,

54321

21 1

⇝ 1

5

(13 ⊗

1 4 12 25

)

54321

21 1

⇝ − 1

15

(31 ⊗

2 1 12 54

)

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 38

Page 113: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to the syzygies example

Φ

1 1 12 2345

=∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗ P

( )

Some terms in the sum are (pre-straightening):

54321

21 1

⇝ 45 ⊗

1 1 12 23

,

54321

21 1

⇝ 1

5

(13 ⊗

1 4 12 25

)

54321

21 1

⇝ − 1

15

(31 ⊗

2 1 12 54

)

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 38

Page 114: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to the syzygies example

Φ

1 1 12 2345

= 45 ⊗

1 1 12 23

− 35 ⊗

1 1 12 24

− 2

3

(25 ⊗

1 1 12 43

)2

3

(25 ⊗

1 1 12 34

)− 3

5

(15 ⊗

1 1 42 23

)− 2

5

(15 ⊗

1 1 22 34

)+

2

5

(15 ⊗

1 1 22 43

)+

3

5

(15 ⊗

1 1 32 24

)+ 3

4 ⊗1 1 12 25

+ · · ·

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 39

Page 115: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to the syzygies example

Φ

5 5 5 54 43 321

=∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗ P

Some terms in the sum are (pre-straightening):

12345

345 5 5

⇝ 43 ⊗

5 5 5 54321

,

12345

345 5 5

⇝ −1

3

53 ⊗

4 5 5 54321

12345

345 5 5

⇝ −1

3

45 ⊗

5 5 3 54321

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 40

Page 116: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to the syzygies example

Φ

5 5 5 54 43 321

=∑

2-paths P

(−1)|P |

H (P )Pout

( )⊗ P

Some terms in the sum are (pre-straightening):

12345

345 5 5

⇝ 43 ⊗

5 5 5 54321

,

12345

345 5 5

⇝ −1

3

53 ⊗

4 5 5 54321

12345

345 5 5

⇝ −1

3

45 ⊗

5 5 3 54321

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 40

Page 117: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to the syzygies example

Φ

5 5 5 54 43 321

= 4

43 ⊗

5 5 5 54321

− 16

3

53 ⊗

5 5 5 44321

+

16

3

54 ⊗

5 5 5 34321

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 41

Page 118: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to the syzygies example

5 5 5 54 43 321

⊗1 1 12 2345y

∑2-paths P,2-paths Q

(−1)|P |+|Q|

H(P)H (Q)

(⊗

)⊗

P

⊗Q

E.g. one term in the image is

8

5

(43 ⊗

15

)⊗

5 5 5 54321

⊗1 1 22 34

∈ C[M5×5]2 ⊗

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 42

Page 119: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to the syzygies example

Still need 43 ⊗

15 ∈ C[M5×5]2, given by a minor:

z11 z12 z13 z14 z15z21 z22 z23 z24 z25z31 z32 z33 z34 z35z41 z42 z43 z44 z45z51 z52 z53 z54 z55

43 ⊗

15 = z31z45 − z35z41.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 43

Page 120: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to the syzygies example

Still need 43 ⊗

15 ∈ C[M5×5]2, given by a minor:

z11 z12 z13 z14 z15z21 z22 z23 z24 z25

3 z31 z32 z33 z34 z354 z41 z42 z43 z44 z45

z51 z52 z53 z54 z55

43 ⊗

15 = z31z45 − z35z41.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 43

Page 121: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to the syzygies example

Still need 43 ⊗

15 ∈ C[M5×5]2, given by a minor:

1 5

z11 z12 z13 z14 z15z21 z22 z23 z24 z25

3 z31 z32 z33 z34 z354 z41 z42 z43 z44 z45

z51 z52 z53 z54 z55

43 ⊗

15 = z31z45 − z35z41.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 43

Page 122: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to the syzygies example

Still need 43 ⊗

15 ∈ C[M5×5]2, given by a minor:

1 5

z11 z12 z13 z14 z15z21 z22 z23 z24 z25

3 z31 z32 z33 z34 z354 z41 z42 z43 z44 z45

z51 z52 z53 z54 z55

43 ⊗

15 = z31z45 − z35z41.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 43

Page 123: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

Back to the syzygies example

Still need 43 ⊗

15 ∈ C[M5×5]2, given by a minor:

1 5

z11 z12 z13 z14 z15z21 z22 z23 z24 z25

3 z31 z32 z33 z34 z354 z41 z42 z43 z44 z45

z51 z52 z53 z54 z55

43 ⊗

15 = z31z45 − z35z41.

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 43

Page 124: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

“Unzip” each term.

Compute the differentials similarly.

Generally, for any W = (V ∗)p ⊕ V q and any σ.

0 −→ 2 21 2 −→

2 21

↗↘

2 2

⊕21

↘↗ 2 −→ · −→ CovH(W,Fσ) −→ 0

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 44

Page 125: €¦ · Weyls Second Fundamental Theorem R := C[zij j1 i p;1 j q] C-alg hom : R !C[(V∗)p Vq]GL(V ) given by zij 7!fij: Then, by the FFT, R !ϵ C[(V∗)p Vq]GL(V ) ! 0 is exact

References

Peter Olver, Differential Hyperforms I, University of MinnesotaMathematics Report, available www.math.umn.edu/~olver/Steven Sam, Computing inclusions of Schur modules, Journal ofSoftware for Algebra and Geometry, arXiv:0810.4666

Steven Sam, Jerzy Weyman, Pieri resolutions for classical groups,Journal of Algebra, arXiv:0907.4505

Markus Hunziker, John Miller, Mark Sepanski, Explicit PieriInclusions, arXiv:1911.11045

Markus Hunziker, John Miller, Mark Sepanski, Minimal GradedFree Resolutions of Modules of Covariants For Classical Groups, comingsoon

John A. Miller (Baylor University) Pieri Inclusions and Syzygies 45