REPRESENTATION THEORY AND SYMPLECTIC QUOTIENT

SINGULARITIES

GWYN BELLAMY

Abstract. The first part of this article is an informal introduction to the representation

theory of the symmetric group - it is intended for the working mathematician who knows no

representation theory. In the second part we explain, more generally, how representation theory

can be used to study symplectic quotient singularities. Namely, one can use representation

theory to decide when these singular spaces admit crepant resolutions.

Introduction

Representation theory has a long rich history going back to the pioneering work of Frobenius

at the end of the 19th century on the representation theory of finite groups. Since then it has

grown into a broad field covering the representation theory of finite dimensional algebras, rep-

resentation theory of infinite dimensional algebras, Lie theory, geometric representation theory,

and, of course, the representation theory of group algebras. Despite its long history, the field is

more active today than ever. This is partly due to its applicability to other areas of mathemat-

ics such as algebraic geometry, knot theory, integrable systems, and combinatorics, as well as

to theoretical physics, computer science and chemistry. In this article we focus on one of these

applications, namely to symplectic algebraic geometry.

This article has two distinct parts. The first is an informal introduction to the representation

theory of the symmetric group. Not only do we introduce the reader to the basic ideas of field,

but we also try to give them an idea of the current state of affairs and the important open

problems in the area. This part should be accessible to anyone who has taken the first two

years of an undergraduate mathematics degree. In the second part, we assume a bit more of

the reader; namely an understanding of the basic facts in algebraic geometry. We describe in

this section the motivation behind, and progress made, in the classification of finite symplectic

groups whose corresponding quotient singularity admits crepant resolutions. This programme

is now almost complete, and we summarize the current state of affairs.

Acknowledgements. I would like to thank Travis Schedler for all our (very!) fruitful conver-

sations about symplectic singularities over the years. I also thank David Evans for suggesting

I write this article in the first place.

1. Part I: The symmetric group

Representation theory was first developed in the context of group representations. This was

systematically developed in the work of Frobenius during the latter half of the 19th century.

Since then it has developed into an important part of pure mathematics, with applications

to many other fields of science (for instance in chemistry). To motivate the representation

theory of finite groups, we consider here the key example of the symmetric group. Recall that

the symmetric group Sn is the group of all permutations of a set of n objects. e.g. in cycle

1

notation,

S3 = {1, (12), (23), (13), (123), (132)}. (1.0.1)

We are interested in representations of the group Sn. That is, we want to consider all the

possible ways to represent Sn as a matrix group. For example, the group above has a natural

2-dimensional representation:{(1 0

0 1

),

(−1 1

0 1

),

(1 0

1 −1

),

(0 −1

−1 0

),

(0 −1

1 −1

),

(−1 1

−1 0

)}, (1.0.2)

where each of the permutations in (1.0.1) is sent to the corresponding matrix in (1.0.2). To give

the formal definition of a representation, we fix a field K. We assume throughout that K = Kis algebraically closed.

Definition 1.1. A representation of Sn is a finite dimensional K-vector space V with an action

Sn × V → V, (σ, v) 7→ σ · v

such that

(a) The map σ · − : V → V is linear for each σ ∈ Sn.

(b) 1 · v = v, for all v ∈ V .

(c) σ1 · (σ2 · v) = (σ1σ2) · v, for all σ1, σ2 ∈ Sn and v ∈ V .

For example, take V to be a 2-dimensional vector space over K, with basis {v1, v2}, and define

(12) · v1 = −v1, (12) · v2 = v1 + v2,

(23) · v1 = v1 + v2, (23) · v2 = −v2.

Since every element of S3 is a product of some number of the transpositions (1, 2) and (2, 3),

the above formulas, together with property (c) of the definition of representation, define the

action of every element of S3 on V . In this way, each σ ∈ S3 can be expressed as an explicit

2× 2 matrix. We recover the matrix representation of (1.0.2). We want to be able to describe

“all” possible representations of the group Sn. Obviously, this is too vague as stated. But one

key fact that I hope to convey is that the answer depends heavily on the characteristic char(K)

of the field K.

1.1. Irreducible representations. To make the problem more precise, we first need a way to

break up representations into their simplest constituents. The correct notion here is that of an

irreducible representation.

Definition 1.2. A subrepresentation W of V is a subspace W ⊆ V such that σ ·w ∈W for all

σ ∈ Sn and w ∈W . The representation V is said to be irreducible if the only subrepresentations

of V are {0} or V .

One should think of representations as molecules - they are built up in some complicated way

from atoms (the irreducible representations). Thus, we are naturally lead to ask: What are the

atoms? How can we glue the atoms to make molecules? First we describe some basic examples.

Example 1.3 (The trivial representation). We take triv = K{v0}, the one-dimensional vector

space with basis v0, and

σ · v0 = v0, ∀ σ ∈ Sn.

The trivial representation is the “simplest” representation.

2

Example 1.4 (The sign representation). We take sgn = K{w0} with

σ · w0 = (−1)`(σ)w0, ∀ σ ∈ Sn.

where `(σ) is the length of the permutation σ.

Notice that triv = sgn if and only if char(K) = 2. Since these representations are one-

dimensional, they are necessarily irreducible. In general, an irreducible representation need not

be one-dimensional though.

1.2. The reflection representation. Let V be a vector space with basis {x1, . . . , xn}, and

inner product defined by 〈xi, xj〉 = δi,j . We make V into a representation of Sn by the action

σ · xi = xσ(i), ∀ σ ∈ Sn.

Then V (the permutation representation) is not irreducible. But the subspace

h =

{a1x1 + · · ·+ anxn ∈ V

∣∣ ∑i

ai = 0

}is an irreducible subrepresentation, called the reflection representation. The transpositions

(i, j) ∈ Sn act by

(ij) · v = v − 〈v, xi − xj〉(xi − xj).

Notice that this is precisely the formula for an orthogonal reflection. This realizes Sn as

the symmetries of the n-simplex e.g. S3 is the symmetry group of the triangle and S4 the

symmetries of the tetrahedron.

Returning to general representations, we are interested in two questions:

(A) What are the irreducible representations of the symmetric group Sn?

(B) How can one decompose a general representation into irreducible representations?

Before we can even begin to answer these questions, we need to decide when two represen-

tations are the same. We will say that the representations V and W of Sn are isomorphic if

there exists a vector space isomorphism φ : V →W such that

φ(σ · v) = σ · φ(v)

for all v ∈ V and σ ∈ Sn. In representation theory, we are interested in classifying representa-

tions up to isomorphism.

If the characteristic of the field K is zero, or greater than n, then we have a complete answer to

(A) and (B). The answer for (B) is Maschke’s Theorem [16, §8] - every molecule is an atom! This

is a general result in the theory of representations of finite groups over a field of characteristic

zero, which says that every representation is the direct sum of its irreducible subrepresentations.

Moreover, it is easily seen that each group has, up to isomorphism, only finitely many irreducible

representations.

If the characteristic of the field K is between 2 and n, then

(A) There exists a partial answer (as we’ll see).

(B) Is a hopeless situation!

Though we can, in principal, classify the atoms that appear in (A), it seems to be a hopeless

task to describe precisely how the atoms glue (B). In a precise mathematical sense, this is a

“wild” problem.

3

1.3. Partitions. The classification of irreducible representations for the symmetric group in

characteristic zero (or characteristic greater than n), is given in terms of wonderful combinatorial

objects called partitions. A partition of n is a non-increasing sequence of positive integers

λ = (λ1 ≥ λ2 ≥ · · · ≥ 0) such that λ1 + λ2 + · · · = n; in this case we write λ ` n. For instance,

the partitions of 5 are:

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

Theorem 1.5 (I. Schur). If char(K) = 0, then the irreducible representations of Sn are

parametrized by partitions of n,

λ ` n ↔ Vλ.

By “parametrize”, we mean that:

• Each irreducible V is isomorphic to Vλ for some λ ` n.

• If Vλ is isomorphic to Vµ then λ = µ.

Remark 1.6. Representation theory in general, and the representation theory of the symmetric

group in particular, was first systematically developed by Fredinand Frobenius and his PhD

student Issai Schur (with many important contributions by Richard Brower) in the period

1880-1905. See [8] for a very comprehensive guide to the historical development of the field.

1.4. Young diagrams. To help us better visualize partitions, and explore their deeper combi-

natorics, we represent each partition as a Young diagram:

µ = (4, 2, 2) = or λ = (3, 2, 1) = .

For example, we have seen:

triv = Vλ, where λ = = (n),

h = Vµ, where µ = = (n− 1, 1),

sgn = Vρ, where ρ = = (1, . . . , 1).

Implicitly, we take n = 5 throughout.

1.5. A basis. Having given an abstract parametrization of the irreducible representations of

the symmetric group, it raises the obvious question: What do these representations actually

look like? The most complete answer to such a question would be to give an explicit basis of

each representation and describe how each element of the group acts on each basis element.

In order to do this, we must introduce a bit more combinatorics. In particular, we need the

notion of standard tableaux. A standard tableau is a filling of λ by {1, . . . , n} such that numbers

increase along rows and columns. For instance,

1 3 5 62 3

∈ Std(4, 2), but 4 3 2 61 5

/∈ Std(4, 2).

Theorem 1.7 (A. Young). The representation Vλ has a basis {vT | T ∈ Std(λ)}.4

Using Young’s Theorem, together with some clever combinatorics, it is also possible to derive

a formula for the dimension of the representation. First, we need one more combinatorial

definition. Choose a box � in a Young diagram. Including the box itself, count how many

boxes there are directly below and directly to the right of that box. This is called the hook

length of �, denoted h(�). For instance, in the following Young diagram, the entry in each box

is the hook length of that box:

λ =

7 5 4 26 4 3 14 2 11

= (4, 4, 3, 1). (1.5.1)

Corollary 1.8. Let λ be a partition of n. Then,

dimVλ = n!∏�∈λ

1

h(�).

For instance, if λ is the partition (4, 4, 3, 1) of 12 as in (1.5.1) then the above formula tells us

that dimVλ = 5940.

Example 1.9. If we take λ = (4, 2) then the set Std(4, 2) equals:

1 2 3 45 6

1 2 3 54 6

1 2 4 53 6

1 3 4 52 6

1 2 3 64 5

1 2 4 63 5

1 3 4 62 5

1 2 5 63 4

1 3 5 62 4

Hence, dimVλ = 9. One can check that the hook length formula also gives 9 in this case.

In general, it is a difficult combinatorial problem to compute how a given permutation σ will

act on a basis element vT . However, in characteristic zero, one can use the powerful theory of

characters to compute effectively the representations of Sn. We will not explain this here, but

refer the interested reader to [16] or [23].

1.6. Positive characteristic. If char(K) = p ≤ n, then the representation theory of the sym-

metric group is much (much!) more complicated. As noted above in answer to question (B),

Maschke’s Theorem fails in this case and it is extremely difficult (and in general an open prob-

lem) to describe how an arbitrary representation is built up from the irreducible representations.

If we focus instead on the irreducible representations, then it is possible to parametrize them,

but very difficult to say much more. In this situation the irreducible representations are still

parametrized by the partitions of n, but not all partitions now occur (more precisely, several

partitions might label the same irreducible module, so we must find a way of choosing just one

of these partitions). Brauer gave a way of parametrizing the irreducible representations of Sn

in positive characteristic in his work on modular representation theory in the 1930’s; see [5]. A

more precise answer was given by Robinson [22].

Theorem 1.10 (R. Brauer, G. Robinson). The irreducible representations of Sn are parametrized

by p-regular partitions.

5

A partition is p-regular if at most p− 1 rows have the same length. For instance, if

λ =

then λ is 5-regular, but not 3-regular. Based on Theorem 1.7 and Corollary 1.8, it is natural to

ask:

• What is a basis of Vλ if λ is p-regular?

Or, easier,

• What is the dimension of Vλ if λ is p-regular?

In his 1990 paper [15] (about 60 years after the work of Brauer!) James gave a precise,

though technical, conjectural algorithm for computing the dimension of the irreducible modules

in positive characteristic. However, much more recently, remarkable work of Williamson [25]

has shown that James’ conjecture is (hopelessly) false. As of yet, there does not seem to be any

good guess as to what the correct answer should be.

We have given here a very superficial introduction to the subject of group representations.

For a more comprehensive introduction see [16] or [23]. In particular, we should mention that

these sources describe the character theory of finite groups (mentioned briefly above), which is

a fundamental concept when dealing with representations.

1.7. Contribution of Welsh mathematicians. Welsh mathematicians have played a surpris-

ingly important role in the development of the representation theory of the symmetric group.

The author is not qualified to give a detailed, or systematic, account of this, but would like to

mention in particular the work of D. E. Littlewood, A. Morris and A. Richardson.

Whilst at University College Swansea, Littlewood and Richardson began a fruitful collab-

oration to develop the connection between the representation theory of the symmetric group

and the theory of symmetric functions. It was at this time that they discovered their famous

combinatorial “Littlewood-Richardson rule” for computing the multiplicity of an irreducible rep-

resentation Vλ in an induced module IndSn+m

Sn×SmVµ⊗Vν . This multiplicity cλµ,ν is now called the

“Littlewood-Richardson coefficient”. Littlewood made further important contributions to the

theory of symmetric functions, for instance in the study of certain “Hall-Littlewood polynomi-

als” (originally introduced by Hall, but now bearing both their names because of Littlewood’s

contributions) and also through the introduction of “plethystic substitutions”. He also con-

tributed greatly to the mathematics community in Wales, being head of department in Bangor

for many years. For a comprehensive review of Littlewood’s contributions to mathematics, see

the obituary of Morris-Baker [18].

We should also mention the work of A. Morris, who did important work on developing further

the theory of Hall-Littlewood symmetric polynomials (with applications to the representation

theory of the general linear group), and also on the projective representation theory of the

symmetric group.

2. Part II: Quotient singularities

We have seen already that one can quite easily describe (at least on a coarse level) the

irreducible representations of a particular finite group. For a given representation V of a group

6

G, one can start to ask more detailed questions. For instance, what do the orbits of G in V

look like?

Since there will be infinitely many orbits, it is not possible to simply list them all. Instead,

we can try to study the space of G-orbits. This space has a natural algebraic structure, or more

precisely, is an example of an affine variety1. Namely, we define

V/G := SpecC[V ]G,

so that the closed points of V/G are precisely the G-orbits in V . Here C[V ]G is the subalgebra of

C[V ] consisting of all polynomials invariant under the action of G. The power of this definition

stems from the fact that the structure of V/G as a variety reflects (in a rather complicated way)

the action of G on V . One can then use the tools of algebraic geometry to study the space V/G.

This is illustrated by the classical Chevalley-Shephard-Todd Theorem. First, we may assume

that G ⊂ GL(V ) acts faithfully on V . Then s ∈ G is said to be a reflection if rk(1− s) = 1. In

other words, s has exactly one non-trivial eigenvalue. If S ⊂ G is the set of reflections, then we

say that G is a complex reflection group if S generates G.

Theorem 2.1. The space V/G is smooth if and only if G is a complex reflection group.

Notice that being a complex reflection group is really a property of the action of G on V . In

the case where V/G is smooth, it is known that V/G ' An where n = dimV i.e. C[V ]G is a

polynomial ring.

2.1. Symplectic actions. In most cases, the vector space V will be endowed with some ad-

ditional structure, and we will assume that G preserves the additional structure. In our case,

we will assume that V is a symplectic vector space i.e. there is a non-degenerate bilinear form

ω : V × V → C that is anti-symmetric:

ω(v, w) = −ω(w, v), ∀ v, w ∈ V.

In particular, this implies that V is even dimensional. We will now assume that V is symplectic

and that G preserves ω i.e.

ω(g · v, g · w) = ω(v, w), ∀ v, w ∈ V, g ∈ G.

In this case, every g ∈ G has determinant one, so G cannot contain any reflections. In particular,

this implies that V/G is always singular when G 6= {1} is non-trivial.

To better understand these singularities, and the ways they encode information about the

action of G on V , we take the standard approach in algebraic geometry and consider resolutions

of singularities. For us, a resolution of singularities will always mean a projective, birational

morphism π : Y → V/G from a smooth variety Y .

As illustrated by the case of Kleinian singularities below, an arbitrary resolution of singu-

larities is too far removed from V/G to remember meaningful information about the action of

G on V . Therefore we hunt for resolutions which are “minimal”. In dimension two, minimal

resolutions are exactly what you would expect - a resolution of singularities through which all

other resolutions factor. Unfortunately, in higher dimensions, minimal resolutions in this strong

sense do not exist. A suitable alternative definition was proposed by Reid. He introduced the

notion of crepant resolutions - those resolutions with zero discrepancy; see [21] for the precise

1We will not recall here the basics of algebraic geometry, but refer the reader to the standard text [13]

7

definition. In dimension two, a resolution is crepant if and only if it is minimal. However, even

this alternative definition has a significant defect; in general crepant resolutions need not exist !

This motivates the key problem:

Classify those finite groups G ⊂ Sp(V ) such that V/G admits a crepant resolution.

By analogy with the Chevalley-Shephard-Todd Theorem, the key result that makes the above

problem tractable is:

Theorem 2.2 (Verbitsky [24]). If the space V/G admits a crepant resolution then G is a

symplectic reflection group.

Here, we say that s ∈ G is a symplectic reflection if rk(1 − s) = 2 i.e. s has exactly

two non-trivial eigenvalues (the smallest number possible in Sp(V ), without forcing s = 1).

Then G is a symplectic reflection group if G = 〈S〉, where S ⊂ G is the set of all symplectic

reflections. It is easily shown that every symplectic reflection group can be expressed as a

product of “symplectically irreducible symplectic reflection groups”. That is, there is no proper

decomposition of V into a direct sum V1 ⊕ V2 of G-submodules with each Vi ⊂ V a symplectic

subspace. The latter were classified by Cohen in [7]. The rank of a symplectic reflection group

is defined to be the dimension of the space V .

Unfortunately, unlike the Chevalley-Shephard-Todd Theorem, Verbistky’s Theorem is not an

if and only if statement. There are many examples of symplectic reflection groups for which

V/G does not admit a crepant resolution.

2.2. Kleinian singularities. As noted above, the dimension of a symplectic vector space is al-

ways even. Therefore the smallest possible dimension for V is 2. In this case, Sp(V ) = SL(2,C),

and we are considering quotient varieties corresponding to finite subgroups of SL(2,C). The

classification of finite subgroups of SL(2,C), up to conjugation, is a classical result going back

at least to the work of Felix Klein. We recall that the groups are classified by the corresponding

simply laced Dynkin diagram (if you are unfamiliar with this notion see e.g. [14]). Moreover, as

algebraic varieties, the corresponding quotient singularity C2/G is a hypersurface V (f) ⊂ C3.

We list the groups and corresponding defining equation f = 0 in table (2.2.1).

Diagram Group Equation

An, n ≥ 1 Cyclic Zn+1 xy − zn+1 = 0

Dn, n ≥ 3 Binary dihedral BD4(n−2) x2 + y2z + zn−1 = 0

E6 Binary tetraherdral T x2 + y3 + z4 = 0

E7 Binary octahedral O x2 + y3 + yz3 = 0

E8 Binary icosahedral I x2 + y3 + z5 = 0

(2.2.1)

The binary dihedral group BD4m, of order 4m, is the subgroup of SL(2,C) generated by

g =

(ε 0

0 ε−1

), h =

(0 1

−1 0

),

where ε is a primitive (2m)th root of unity. The binary tetraherdral T, binary octahedral O,

and binary icosahedral I groups have order 24, 48 and 120 respectively. They are double covers

of the rotational symmetry groups of the corresponding 3-dimensional Platonic solids. If we

take G = BD4 of type D3, then a real slice of the Kleinian singularity C2/G is shown in Figure

2.1.

8

Figure 2.1. A real picture of the D3 Kleinian singularity.

As explained previously, since we are in dimension two, there is a unique minimal resolution

of the singularity V/G. This can be obtained by repeated blowup of the (reduced) singular

locus. One need not stop there - you can continue to blowup points as much as you like to

get infinitely many different resolutions. However, it is much harder to extract meaningful

information about the original singularity from these new resolutions. For an introduction to

the concept of “blowing up”, see [13, Section II.7] and in the context of surfaces, [13, Section

V.5].

2.3. Poisson deformations. The goal of the remainder of part II is to explain how represen-

tation theory can be used to classify those groups G for which V/G admits a crepant resolution.

First, we must explain how crepant resolutions are related to Poisson deformations of V/G.

The space V/G has two important features. First, the scaling action of C× on V commutes

with the action of G, so it descends to an action of C× on V/G. Secondly, the fact that V

is a symplectic vector space implies that V has the structure of a Poisson variety (see [6] and

references therein for this important notion). Again, the fact that G preserves the symplectic

structure implies that the Poisson bracket on V descends to a Poisson bracket on V/G. Alge-

braically, this is simply saying that C[V ]G is a Poisson subalgebra of C[V ]. Remarkably, it was

shown by Namikawa [20] and Ginzburg-Kaledin [11] that there is a close relationship between

9

crepant resolutions and deformations of V/G that preserve both the Poisson structure and the

C×-action.

Firstly, we say that a pair f : X → S is a graded Poisson deformation of V/G if:

(a) f is a flat morphism between Poisson varieties, where the Poisson structure on S is

trivial.

(b) C× acts on X and S, making f equivariant.

(c) There is a point s0 ∈ S, fixed by C×, and a C×-equivariant Poisson isomorphism

f−1(s0) ' V/G.

(d) The action of C× on X is compatible2 with the Poisson structure.

We note that (a) implies that every fibre of f is a Poisson variety.

Theorem 2.3 (Namikawa). There exists a universal graded Poisson deformation ρ : X→ H/W

of V/G.

By universal, we mean that if f : X → S is any graded Poisson deformation of V/G then

there exists a unique morphism S → H/W such that X ' S ×H/W X. Here H is a certain

vector space and W is a finite group acting as a Weyl group on H; the quotient H/W is thus

smooth.

The relation to crepant resolutions is given by:

Theorem 2.4 (Namkiawa,Ginzburg-Kaledin). There exists a crepant resolution of V/G if and

only if there exists a graded Poisson deformation f : X → S such that the generic fibre f−1(s)

is smooth.

The existence of the universal graded Poisson deformation implies that it suffices to check

that generic fibres of the universal deformation are smooth.

Corollary 2.5. There exists a crepant resolution of V/G if and only if the generic fibre ρ−1(h)

of ρ is smooth.

2.4. Calogero-Moser spaces. Equipped with the above results, the focus of our problem

turns to constructing graded Poisson deformations of V/G. The power of this point of view

is that we can now begin to use representation theory and non-commutative algebras to try

and construct the deformations. This was one of the motivations behind the seminal work of

Etingof-Ginzburg [10], where they introduced the wonderful class of non-commutative algebras

called symplectic reflection algebras. We won’t give the definition of these algebras here, since

it is not essential for the story, but we explain how the algebras are parametrized. First we note

that the set S of symplectic reflections is a union of G-conjugacy classes. Therefore, we fix

c = {c : S → C | c is conjugate invariant},

the space of all conjugate invariant functions from S ⊂ G to C. For each c ∈ c, Etingof and

Ginzburg constructed the symplectic reflection algebra Hc(G). They showed that the algebra

Hc(G) is a finite module over its centre Zc(G), and the family

Xc(G) := {Xc(G) | c ∈ c} → c

where Xc(G) := SpecZc(G), is a graded Poisson deformation of X0(G) = V/G.

2See [2, §3.3] for precisely what this means.

10

T ∗P1

C2/Z2

X1(Z2)

P1

Figure 2.2. The resolution and the deformation of the Z2 quotient singularity.

This implies that there is a unique morphism ν : c → H/W such that Xc(G) = c ×H/W X.

Ginzburg-Kaledin [11] showed that the map ν is generically etale. More precisely, we show in

[2] that:

Theorem 2.6 (Bellamy). There is a W -equivariant isomorphism c ∼→ H such that the diagram

c H

H/W

∼

ν q

commutes. Here q : H → H/W is the quotient map.

Corollary 2.7. The quotient V/G admits a crepant resolution if and only if the Calogero-Moser

space Xc(G) is smooth for generic c.

Miraculously, there is a very easy representation theoretic criterion for deciding whether the

space Xc(G) is smooth.

Theorem 2.8 (Etingof-Ginzburg). For all c ∈ c:

(a) If L is an irreducible Hc(G)-module then dimL ≤ |G|.(b) Xc(G) is smooth if and only if dimL = |G| for all irreducible Hc(G)-modules L.

Thus, to decide if V/G admits a crepant resolution, it suffices to compute the dimension of

all irreducible Hc(G)-modules when c is generic. Unfortunately, this is a bit harder that it

might first seem. None the less, we have been able to do this now for most symplectic reflection

groups.

Example 2.9. When G = Z2, acting on C2, is of type A1, the centre of the symplectic reflection

algebra Hc(Z2) has a presentation

Zc(Z2) 'C[x, y, z]

(xy − (z + c)(z − c)).

Hence the space Xc(Z2) equals V (xy − (z + c)(z − c)) ⊂ C3. As c varies over A1, this gives

a flat Poisson deformation of V (xy − z2). We note that this space is smooth for all c 6= 0.

The minimal resolution is given by T ∗P1, which can be constructed by blowing up the origin in

V (xy − z2) once. This setup is illustrated in Figure 2.4.

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2.5. The classification. The classification of quotients V/G admitting crepant resolutions is

almost complete. However, a number of examples are still to be dealt with. The symplectic

reflection groups were classified by A. Cohen [7]. In order to be able to effectively use his

classification, we partition the groups into four distinct classes.

First, we introduce some terminology. A symplectic reflection group G is said to be proper

if there is no G-stable Lagrangian subspace h such that V ' h × h∗ as a symplectic G-

representation. Next, the group G is said to be imprimitive if there is a direct sum decom-

position V = V1 ⊕ · · · ⊕ Vr into proper subspaces such that for each g ∈ G and 1 ≤ i ≤ r, there

exists a j such that g(Vi) ⊂ Vj . We say that G is symplectically imprimitive if, in the above

decomposition, each Vi is a symplectic subspace of V . It is important to note that there are

symplectic reflection groups that are imprimitive but not symplectically imprimitive. Recal that

every symplectic reflection group is a (unique) produce of symplectically irreducible symplectic

reflection groups. We assume now that all groups are symplectically irreducible. We can divide

the groups into four classes as follows:

(i) symplectic reflection groups that are not proper,

(ii) proper symplectic reflection groups that are symplectically imprimitive,

(iii) proper symplectic reflection groups that are imprimitive but not symplectically imprim-

itive; and

(iv) proper symplectic reflection groups that are not imprimitive.

Since there are very few, relatively speaking, symplectic reflection groups that admit crepant

resolutions, in summarizing the classification programme to date, we will first list those groups

that are known to admit a crepant resolution. Then we will describe, for each of the four above

classes, those groups for which we still do not know if they admit a crepant resolution or not.

Theorem 2.10. The following symplectic reflection groups admit a crepant resolution:

(a) The wreath product Sn o Γ acting on (C2)n = C2n, where Γ ⊂ SL(2,C) is a finite group.

(b) The complex reflection group G4 acting on the four-dimensional representation h× h∗.

(c) The rank four symplectic reflection group Q8 ×Z2 D8, where Q8 is the quaternion group

of order 8 and D8 is the dihedral group of order 8.

The case (a) is classical. In this case, if Y → C2/Γ is the minimal resolution of the Kleinian

singularity C2/Γ, then the Hilbert scheme HilbnY is a crepant resolution of C2n/G; see [19]. As

we have noted in section 2.2, the finite subgroups Γ of SL(2,C) are classified up to conjugation

by the corresponding simply laced Dynkin diagram. If Γ = Z` is of type A then Sn o Γ is not a

proper symplectic reflection group (it belongs to class (i)). If Γ is of type D or E then, at least

for n ≥ 2, the group Sn o Γ is of type (ii).

The group G4 appearing in (b) is also an example of a non-proper symplectic reflection group;

it is the first of the exceptional complex reflection groups, and is abstractly isomorphic to the

binary tetrahedral group. The fact that it admits a crepant resolution was first discovered by

the author in [1]. An explicit crepant resolution was constructed in [17]. The group listed

in (c) belongs to class (ii). The fact that it admits a crepant resolution was first discovered

by the author in joint work with T. Schedler [3]. Using the Cox ring construction, Donten-

Bury and Wisniewski [9] constructed all possible crepant resolutions of this quotient singularity

(remarkably, it admits 81 different crepant resolutions).

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Finally, we explain the current state of affairs, by listing fo each of the above classes those

groups in the class for which we do not know if the corresponding quotient singularity admits

a crepant resolution.

(i) In this case the classification is complete; see [1].

(ii) In this case the classification is also complete, except when G has rank 4. In the notation

of Cohen [7], it is know when the quotient admits a crepant resolution except if G belongs

to one of the families (G),(K),(P),(Q),(U) or (V). See [4] for more details.

(iii) It has been shown by Cohen [7] that all groups in this class have rank 4. It is not known

whether or not any of these groups admit symplectic resolutions or not.

(iv) Up to conjugation, there are only 13 proper symplectic reflection groups that are not

imprimitive. They are listed in Table III of [7]. We do not know whether or not any of

these groups admit crepant resolutions.

In addition to the work of Namikawa and Ginzburg-Kaledin referenced above, the classifica-

tion is a summary of results from [12], [1], [3], and [4].

References

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School of Mathematics and Statistics, University Gardens, University of Glasgow, Glasgow,

G12 8QW, UK.

Email address: gwyn.bellamy@glasgow.ac.uk

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