Séminaire Lotharingien de Combinatoire 92 (2025), Article B92a

COMBINATORICS OF THE IRREDUCIBLE COMPONENTS OF H_{n}^{Γ} IN TYPE D AND E

RAPHAËL PAEGELOW

Abstract. We give a combinatorial model (in terms of symmetric cores) of the indexing set of the irreducible components of H_{n}^{Γ} (the Γ-fixed points of the Hilbert scheme of n points in the plane) containing a monomial ideal, whenever Γ is a finite subgroup of SL_{2}( ) isomorphic to the binary dihedral group. Moreover, we show that, if Γ is a subgroup of SL_{2}( ) isomorphic to the binary tetrahedral group, to the binary octahedral group or to the binary icosahedral group, then the Γ-fixed points of H_{n} which are also fixed under the maximal diagonal torus of SL_{2}( ) are in fact SL_{2}( )-fixed points. Finally, we prove that in this case the irreducible components of H_{n}^{Γ} containing a monomial ideal are zero-dimensional.

1. Introduction

Let Γ be a finite subgroup of SL_{2}( ) and, for n ∈ _{≥0} , let H_{n} be the Hilbert scheme of n points in ^{2}. The natural action of Γ on ^{2} induces a Γ-action on [x, y] and thus on H_{n}. In this article, we are interested in the combinatorics of the parametrization set of the irreducible components of H_{n}^{Γ}. When Γ is equal to the cyclic subgroup of the maximal diagonal torus of SL_{2}( ), a combinatorial model using partitions has already been constructed by Iain Gordon [Gor08, Lemma 7.8] and by Cédric Bonnafé and Ruslan Maksimau [BM21, Lemma 4.9]. We will therefore only consider the groups of type D and E. Type D corresponds to the class of finite subgroups of SL_{2}( ) that are isomorphic to the binary dihedral subgroups. In the second section, we introduce important notation concerning affine root systems and partitions of integers. In the third section, we then define the binary dihedral group and give its character table and its McKay graph. We then present a folding of that Dynkin diagram which will be of use in the subsequent section. In section four, we define and give the main properties of a generalization of the residue to type D. In the fifth section, we prove the first theorem, which can be stated as follows.

Theorem 1. Let ℓ be an integer greater than or equal to 2, and let Γ be a binary dihedral subgroup of SL_{2}( ) of order 4ℓ. Then the set of all irreducible components of H_{n}^{Γ} containing a monomial ideal is in bijection with the set of all symmetric 2ℓ-cores λ such that |λ| ≡ n (mod 2ℓ) and |λ| ≤ n. Moreover, for all symmetric partitions μ_{1}, μ_{2} of n, the monomial ideals attached to μ_{1} and μ_{2} are in the same irreducible component of H_{n}^{Γ} if and only if the 2ℓ-cores of μ_{1} and μ_{2} are equal.

In section six, we start by giving a presentation of the binary tetrahedral group, its character table and its McKay graph. Moreover, we prove that, if Γ is isomorphic to the binary tetrahedral group, then the points in H_{n} that are fixed under Γ and the maximal diagonal torus of SL_{2}( ) are exactly the SL_{2}( )-fixed points. Since the binary octahedral group and the binary icosahedral group contain a subgroup isomorphic to the binary tetrahedral group, the previous result generalizes to these two isomorphism classes of finite subgroups of SL_{2}( ). Finally, in section seven, we prove the following theorem.

Theorem 2. If Γ is a finite subgroup of SL_{2}( ) of type E, then, for I ∈ H_{n}^{SL_{2}( )} , the irreducible component of H_{n}^{Γ} containing I is zero-dimensional.

Acknowledgement. The author would like to thank Cédric Bonnafé for suggesting the study of the combinatorics of the irreducible components of the Γ-fixed points in the punctual Hilbert scheme of ^{2}, when Γ is of type D and E, and the referee for valuable comments, suggestions and improvements.

2. Starting point

Fix a finite subgroup Γ of SL_{2}( ). In this subsection we recall the general description of the indexing set of the irreducible components of H_{n}^{Γ} in terms of roots that has been obtained in [Pae1]. Denote by I_{Γ} the set of all irreducible characters of Γ and let χ_{0} ∈ I_{Γ} denote the trivial character of Γ. Let Δ_{Γ}^{+} (⊂ Δ_{Γ}) be the free monoid (respectively free abelian group) associated with I_{Γ}. Let T̃_{Γ} be the type of the McKay graph seen as an affine Dynkin diagram. One can then associate with Γ a realization (cf. [Kac, §1.1])

( 𝔥_{Γ}, Π_{Γ} := {α_{χ} | χ ∈ I_{Γ}} , Π_{Γ}^{∨} := {α_{χ}^{∨} | χ ∈ I_{Γ}} )

of the generalized Cartan matrix of type T̃_{Γ}. Denote by Q(T̃_{Γ}) and W(T̃_{Γ}) the root lattice and Weyl group, respectively, associated with the previously mentioned realization. From now on, we will identify Q(T̃_{Γ}) with Δ_{Γ}. Let δ^{Γ} denote the null root.

For d ∈ Δ_{Γ}, let (d_{χ}) ∈ ^{|I_{Γ}|} be such that d = ∑ _{χ∈I_{Γ}} d_{χ}χ. For d ∈ Δ_{Γ}^{+} , let |d|_{Γ} := ∑ _{χ∈I_{Γ}} d_{χ}δ_{χ}^{Γ} ∈ _{≥0} . Finally, a new statistic on Δ_{Γ} has been defined in [Pae1, Definition 4.8]. The group W(T̃_{Γ}) naturally acts by reflections on 𝔥_{Γ}^{∗}. This action will be denoted by ∗. Define a new action of W(T̃_{Γ}) on Δ_{Γ} denoted by . such that

ω ∗ (Λ_{χ_{0}} − d) = Λ_{χ_{0}} − ω.d, for all (ω, d) ∈ W(T̃_{Γ}) × Δ_{Γ},

where Λ_{χ_{0}} is the fundamental weight associated with χ_{0}, the trivial character of Γ.

One can then prove that, for d ∈ Δ_{Γ}, there exists a unique integer r such that d and rδ^{Γ} are in the W(T̃_{Γ})-orbit for the . action, see [Pae1, Lemma 4.7]. We denote this integer r by wt(d).

Recall the result of [Pae1, Theorem 4.10], which will be our starting point. For a finite subgroup Γ of SL_{2}( ) we have indexed the irreducible components of H_{n}^{Γ} by the set

A_{Γ}^{n} := {d ∈ Δ_{Γ}^{+} | |d|_{Γ} = n and wt(d) ≥ 0} .

Before diving into the type D study, let us introduce a bit more notation. A partition λ of n is a tuple (λ_{1} ≥ λ_{2} ≥ ⋯≥ λ_{r} ≥ 0) of integers, such that |λ| := ∑ _{i=1}^{r} λ_{i} is equal to n. Denote by P_{n} the set of all partitions of n and by P the set of all partitions of integers. For λ = (λ_{1}, …, λ_{r}) ∈ P , denote by Y(λ) := {(i, j) ∈ _{≥0}^{2} | i < λ_{1}, j < r} its associated Young diagram. The conjugate partition of a partition λ of n, denoted by λ^{∗}, is the partition associated with the reflection of Y(λ) in the diagonal (which is again a Young diagram of a partition of n). We will draw Young diagrams upright and the box that is lowest and furthest to the left will have index (0, 0). Let i and j denote the row and column indices, respectively. For example, consider λ = (2, 2, 1). Its associated Young diagram is

.

In that case λ^{∗} = (3, 2). A partition λ will be called symmetric if it is equal to its conjugate. We denote the set of all symmetric partitions by P^{s}, and we define P_{n}^{s} := P^{s} ∩ P_{n}.

COMBINATORICS OF THE IRREDUCIBLE COMPONENTS OF H_{n}^{Γ} IN TYPE D AND E 3

A hook of a partition λ in position (i, j) ∈ Y(λ), denoted by H_{(i,j)}(λ), is

{(a, b) ∈ Y(λ) | (a = i and b ≥ j) or (a > i and b = j)} .

Define the length of a hook H_{(i,j)}(λ) to be its cardinality.

Definition 2.1. For a given integer r ≥ 1, a partition λ is said to be an r-core if Y(λ) does not contain any hook of length r. The set of all r-cores is denoted by ℭ_{r}, and we define ℭ_{r}^{s} := ℭ_{r} ∩ P^{s}.

3. From type D to type C

Fix ℓ ≥ 2. Let μ_{ℓ} denote the cyclic subgroup of SL_{2}( ) generated by the diagonal matrix diag(ζ_{ℓ}, ζ_{ℓ}^{−1} ), where ζ_{ℓ} = e^{\frac{2iπ}{ℓ}} . We will work with the following model of the binary dihedral group in SL_{2}( ). Let BD_{2ℓ} := ⟨ω_{2ℓ}, s⟩, where

ω_{2ℓ} := ( \begin{matrix}ζ_{2ℓ} & 0 \\ 0 & ζ_{2ℓ}^{−1} \end{matrix} ) ,  s := ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) .

The group BD_{2ℓ} is of order 4ℓ. Note that BD_{4} is isomorphic to the quaternion group (cf. [CM13, §1.7]). Let τ_{2ℓ} be the character of μ_{2ℓ} that maps ω_{2ℓ} to ζ_{2ℓ}. For i ∈ , let

χ_{i} := Ind_{μ_{2ℓ}}^{BD_{2ℓ}}( τ_{2ℓ}^{i}).

Note that χ_{i} is irreducible if and only if i is not congruent to 0 or ℓ modulo 2ℓ. If ℓ is even, the character table of BD_{2ℓ} is

cardinality  1  1  2  ℓ  ℓ
classes ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) ω_{2ℓ} ^{p}(0 < p < ℓ)  s sω_{2ℓ}
χ_{0^{+}}  1  1  1  1  1
χ_{0^{−}}  1  1  1  −1 −1
χ_{ℓ^{+}}  1  1  (−1)^{p}  −1 1
χ_{ℓ^{−}}  1  1  (−1)^{p}  1 −1
\begin{matrix}χ_{k} \\ (0 < k < ℓ) \end{matrix}  2  (−1)^{k}2  2cos ( \frac{kpπ}{ℓ} )  0  0

and if ℓ is odd, the character table of BD_{2ℓ} is

cardinality  1  1  2  ℓ  ℓ classes ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) ( \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ) ω_{2ℓ} ^{p}(0 < p < ℓ)  s sω_{2ℓ} χ_{0^{+}}  1  1  1  1  1 χ_{0^{−}}  1  1  1  −1 −1 χ_{ℓ^{+}}  1  −1  (−1)^{p}  ζ_{4} −ζ_{4} χ_{ℓ^{−}}  1  −1  (−1)^{p}  −ζ_{4} ζ_{4} \begin{matrix}χ_{k} \\ (0 < k < ℓ) \end{matrix}  2  (−1)^{k}2  2cos ( \frac{kpπ}{ℓ} )  0  0 .

The McKay graph of BD_{2ℓ} is a Dynkin diagram of affine type D̃_{ℓ+2}

.

The irreducible characters of BD_{2ℓ} are labeled by their index in the McKay graph.

4 RAPHAËL PAEGELOW

We want to give a combinatorial description of A_{BD_{2ℓ}}^{n}. Let _{1} denote the maximal diagonal torus of SL_{2}( ). In what follows, using symmetric partitions, we will give a combinatorial description of the irreducible components of H_{n}^{BD_{2ℓ}} containing a monomial ideal. To do so, restrict A_{BD_{2ℓ}}^{n} to the irreducible components of H_{n}^{BD_{2ℓ}} containing a _{1}-fixed point. Let us denote this subset of A_{BD_{2ℓ}}^{n} by A_{BD_{2ℓ}}^{n, _{1}}. Note also that, in this context, the coefficients of the null root in the basis of simple roots are

δ_{χ_{i}}^{BD_{2ℓ}} := \{ \begin{matrix}1, & if i = 0^{+} , 0^{−} , ℓ^{+} , ℓ^{−} , \\ 2, & otherwise. \end{matrix}

The central object of study will be the affine root lattice of type D̃_{ℓ+2} (which is the same object as the coroot lattice of type D̃_{ℓ+2} since it is a simply laced type) Q(D̃_{ℓ+2} ) ⊂ 𝔥_{BD_{2ℓ}}^{∗}. Let τ_{ℓ} := α_{χ_{0^{−}}} + α_{χ_{ℓ^{+}}} + α_{χ_{ℓ^{−}}} + ∑ _{i=1}^{ℓ−1} 2α_{χ_{i}} be the highest root of the finite root system of type D_{ℓ+2} . Definition 3.1. Define a bijection from the set I_{BD_{2ℓ}} to itself by

σ_{0^{−}} : \begin{matrix}I_{BD_{2ℓ}} & → & I_{BD_{2ℓ}}, \\ χ & ↦ & χ_{0^{−}} .χ, \end{matrix}

and define also an automorphism of the Dynkin diagram of type D̃_{ℓ+2} by

σ : \begin{matrix}Π_{BD_{2ℓ}} & → & Π_{BD_{2ℓ}}, \\ α_{χ} & ↦ & α_{σ_{0^{−}} (χ)} . \end{matrix}

This automorphism swaps the first two vertices (the ones with labels 0^{+} and 0^{−} ) and the last two (with the labels ℓ^{+} and ℓ^{−} ) and fixes all the others. We can apply Stembridge’s construction [Stem] to the root system of type D̃_{ℓ+2} and to the automorphism σ. Denote the simple roots by (β_{i})_{i∈⟦0,ℓ⟧} and the simple coroots associated with the root system Φ(D̃_{ℓ+2}^{σ} ) by (β_{i}^{∨})_{i∈⟦0,ℓ⟧}. By construction, we have

• β_{0} = α_{χ_{0^{+}}} + α_{χ_{0^{−}}} , ; • for all i ∈ ⟦1, ℓ − 1⟧, β_{i} = α_{χ_{i}}, ; • β_{ℓ} = α_{χ_{ℓ^{+}}} + α_{χ_{ℓ^{−}}} ,  • β_{0}^{∨} = ^{\frac{α_{χ_{0^{+}}}^{∨} +α_{χ_{0^{−}}}^{∨}}{2}} , ; • for all i ∈ ⟦1, ℓ − 1⟧, β_{i}^{∨} = α_{χ_{i}}^{∨}, ; • β_{ℓ}^{∨} = ^{\frac{α_{χ_{ℓ^{+}}}^{∨} +α_{χ_{ℓ^{−}}}^{∨}}{2}} .

Let A = (a_{ij}) be a generalized Cartan matrix. Recall that, if two vertices (i, j) are connected by more than one edge in the associated Dynkin diagram, then these edges are equipped with an arrow pointing towards i if |a_{ij}| > 1. With these conventions, the root system Φ(D̃_{ℓ+2}^{σ} ) has the following Dynkin diagram:

.

Proposition 3.2. The set Φ(D̃_{ℓ+2}^{σ} ) is a crystallographic root system of type C̃_{ℓ}. Definition 3.3. Let Q(D̃_{ℓ+2}^{σ} )[0^{+} ] := Q(D̃_{ℓ+2}^{σ} ) ⊔ (Q(D̃_{ℓ+2}^{σ} ) + α_{χ_{0^{+}}} ) ⊂ Q(D̃_{ℓ+2} ). Written more explicitly,

Q(D̃_{ℓ+2}^{σ} )[0^{+} ] = { ∑_{χ∈I_{_{BD}_{2ℓ}}} a_{χ}α_{χ} ∈ Q(D̃_{ℓ+2} ) | 0 ≤ a_{χ_{0}^{+}} − a_{χ_{0}^{−}} ≤ 1 and a_{χ_{ℓ^{+}}} = a_{χ_{ℓ^{−}}}  } .

Definition 3.4. Define the map

T : Q(D̃_{ℓ+2}^{σ} )[0^{+} ] → Q^{∨} (D̃_{ℓ+2}^{σ} ) = Q^{∨} (C̃_{ℓ})

given by

∑_{i=0}^{ℓ} a_{i}β_{i} + qα_{χ_{0^{+}}} ↦ (2a_{0} + q)β_{0}^{∨} + ∑_{i=1}^{ℓ−1} a_{i}β_{i}^{∨} + 2a_{ℓ}β_{ℓ}^{∨},

with q ∈ {0, 1}.

In type C̃_{ℓ}, the null root is δ(C̃_{ℓ}) := β_{0} + ∑ _{i=1}^{ℓ−1} 2β_{i} + β_{ℓ} ∈ Q(C̃_{ℓ}) , and the null coroot is δ^{∨} (C̃_{ℓ}) := ∑ _{i=0}^{ℓ} β_{i}^{∨} . For χ ∈ I_{BD_{2ℓ}}, let s_{χ} ∈ W(D̃_{ℓ+2} ) denote the simple reflection associated with α_{χ}.

Definition 3.5. For χ ∈ I_{BD_{2ℓ}}, let σ.s_{χ} := s_{σ_{0^{−}} (χ)} and extend this action to W(D̃_{ℓ+2} ), the Weyl group of type D̃_{ℓ+2} . Let W(D̃_{ℓ+2} )^{σ} := {ω ∈ W(D̃_{ℓ+2} ) | σ.ω = ω}, which is a subgroup of W(D̃_{ℓ+2} ).

Remark 3.6. The set {s_{0} := s_{χ_{0^{+}}} s_{χ_{0^{−}}} , s_{1} := s_{χ_{1}}, …, s_{ℓ−1} := s_{χ_{ℓ−1}} , s_{ℓ} := s_{χ_{ℓ^{+}}} s_{χ_{ℓ^{−}}} } is a set of generators of W(D̃_{ℓ+2} )^{σ}. Applying [Stem, Claim 3] to our situation, we obtain a group isomorphism from W(D̃_{ℓ+2}^{σ} ) to W(D̃_{ℓ+2} )^{σ}. Let us, from now on, identify these two groups and refer to them as W(C̃_{ℓ}). This group acts naturally by reflections on Q(D̃_{ℓ+2}^{σ} )[0^{+} ] and Q^{∨} (C̃_{ℓ}). Denote this action by ∗.

Definition 3.7. Define a W(C̃_{ℓ})-action on Q(D̃_{ℓ+2}^{σ} )[0^{+} ] by

s_{i} α := s_{i} ∗ α + δ_{i}^{0}α_{χ_{0^{+}}} , for all i ∈ ⟦0, ℓ⟧ and α ∈ Q(D̃_{ℓ+2}^{σ} )[0^{+} ].

Similarly, define a W(C̃_{ℓ})-action on the coroot lattice Q^{∨} (D̃_{ℓ+2}^{σ} ) by

s_{i} β^{∨} := s_{i} ∗ β^{∨} + δ_{i}^{0}β_{0}^{∨}, for all i ∈ ⟦0, ℓ⟧ and β^{∨} ∈ Q^{∨} (C̃_{ℓ}).

A simple computation shows the equivariance of T with respect to the former defined actions.

Proposition 3.8. The map T is W(C̃_{ℓ})-equivariant.

Remark 3.9. Note also that T preserves sizes,

|α|D̃_{ℓ+2} = |T (α)|C̃_{ℓ}, for all α ∈ Q(D̃_{ℓ+2}^{σ} )[0^{+} ].

Let G be an abstract group acting on a set X. For x ∈ X, we denote the orbit of x under the action of G by \overline{x}^{G}. The following lemma will be used later on when proving the first theorem.

Lemma 3.10. If β^{∨} ∈ \overline{0}^{W(C̃_{ℓ})} ⊂ Q^{∨} (C̃_{ℓ}) and k ∈ , then (β^{∨} + kδ^{∨} (C̃_{ℓ})) ∈ \overline{kδ^{∨} (C̃_{ℓ})}^{W(C̃_{ℓ})}.

Proof.It is enough to check this on the set of generators {s_{i} | i ∈ ⟦0, ℓ⟧}. If i ∈ ⟦1, ℓ⟧, the action is by reflections. It is then linear and s_{i} stabilizes kδ^{∨} (C̃_{ℓ}). For i = 0, we can combine the fact

s_{0} (β_{1}^{∨} + β_{2}^{∨}) = s_{0} β_{1}^{∨} + s_{0} β_{2}^{∨} − β_{0}^{∨}, for all β_{1}^{∨}, β_{2}^{∨} ∈ Q^{∨} (C̃_{ℓ})

with the fact that s_{0} δ^{∨} (C̃_{ℓ}) = δ^{∨} (C̃_{ℓ}) + β_{0}^{∨} to conclude that

s_{0} (β^{∨} + kδ^{∨} (C̃_{ℓ})) = s_{0} β^{∨} + kδ^{∨} (C̃_{ℓ}).  □

Finally, let us say a few words about the dual root system of Φ(D̃_{ℓ+2}^{σ} ). It can be obtained as a folding of type A. This will simplify proofs in the next section. Recall that μ_{2ℓ} denotes the cyclic subgroup of order 2ℓ contained in the maximal diagonal torus of SL_{2}( ) and that τ_{2ℓ} denotes the irreducible character of μ_{2ℓ} mapping the generator ω_{2ℓ} to ζ_{2ℓ}. The McKay graph of μ_{2ℓ} is a Dynkin diagram of affine type Ã_{2ℓ} with 2ℓ vertices (since μ_{2ℓ} is abelian). Consider the automorphism of the Dynkin diagram of type Ã_{2ℓ} given by

ς: \begin{matrix}Π_{μ_{2ℓ}} & → & Π_{μ_{2ℓ}}, \\ α_{τ^{i}} & ↦ & α_{τ^{−i}} . \end{matrix}

It fixes α_{τ^{0}} and α_{τ^{ℓ}}. Applying Stembridge’s construction to (Ã_{2ℓ}, ς) and identifying it with the dual root system (cf. [Kac, §3.1]) of Φ(D̃_{ℓ+2}^{σ} ), we obtain the following result.

Proposition 3.11. The set Φ(Ã_{2ℓ}^{ς}) is the dual root system of Φ(D̃_{ℓ+2}^{σ} ).

4. BD_{2ℓ}-Residue

The _{1}-fixed points in H_{n} are the ideals I_{λ} generated by {x^{i}y^{j} | (i, j) ∈ ^{2} \ Y(λ)} for λ a partition of n. These ideals are called monomial ideals. Among these ideals, the ideals fixed by s ∈ BD_{2ℓ} are exactly the monomial ideals parametrized by symmetric partitions of n. This implies that [x, y]/I_{λ} is a BD_{2ℓ}-module whenever λ is symmetric. In this section, our goal is to generalize the residue “of type A”, i.e., the usual residue of partitions, to a residue of type D. Recall that we identify the root lattice constructed out of Γ with the Grothendieck ring of Γ. The property from the residue that we want to generalize is that the residue of a partition λ is equal to the character of the representation [x, y]/I_{λ}. Thus, we want to construct a map Res_{D} from P_{n}^{s} to Q(D̃_{ℓ+2} ). To do so, let us first define the functions d_{k} : P_{n}^{s} → _{≥0} , for k ∈ ⟦0, ℓ⟧.

Let Y(λ)_{k} := {(i, j) ∈ Y(λ) | i − j ≡ k (mod 2ℓ)} for k ∈ ⟦0, 2ℓ − 1⟧.

Definition 4.1. For k ∈ ⟦1, ℓ⟧ define d_{k}(λ) := #(Y(λ)_{k} ∪ Y(λ)_{2ℓ−k} ). For k = 0, we put d̃_{0}(λ) := #{(i, j) ∈ Y(λ) | i = j} and d_{0}(λ) := #Y(λ)_{0} − d̃_{0}(λ).

Furthermore, we introduce the notations

• d′ 0(λ) := ^{\frac{d_{0}(λ)}{2}} + d̃_{0}(λ) − ⌊^{\frac{d̃_{0}(λ)}{2}} ⌋,  • d{′′ \atop 0}(λ) := ^{\frac{d_{0}(λ)}{2}} + ⌊^{\frac{d̃_{0}(λ)}{2}} ⌋.

We are now able to define the residue in type D.

Definition 4.2. The residue of type D is

Res_{D} : \begin{matrix}P_{n}^{s} & → & Q(D̃_{ℓ+2} ), \\ λ & ↦ & d′_{0}(λ)α_{χ_{0^{+}}} + d′′_{0}(λ)α_{χ_{0^{−}}} + ∑ _{i=1}^{ℓ−1} \frac{d_{i}(λ)}{2} α_{χ_{i}} + \frac{d_{ℓ}(λ)}{2} (α_{χ_{ℓ^{+}}} + α_{χ_{ℓ^{−}}} ). \end{matrix}

Remark 4.3. Using the fact that the partition is symmetric, it is easy to see that the image of Res_{D} is indeed in the -span of the roots {α_{χ} | χ ∈ I_{BD_{2ℓ}}}. Note moreover, that

for all λ ∈ P_{n}^{s}, |Res_{D}(λ)|D̃_{ℓ+2} = |λ| = n.

Example 4.4. Take ℓ = 2 and consider λ = (4, 4, 3, 2), which is symmetric and has Young diagram

\begin{matrix}1 & 2 \\ 2 & 1 & 0^{+} \\ 1 & 0^{−} & 1 & 2 \\ 0^{+} & 1 & 2 & 1 \end{matrix} .

Hence, Res_{D}(λ) = 2α_{χ_{0^{+}}} + α_{χ_{0^{−}}} + 3α_{χ_{1}} + 2α_{χ_{2^{+}}} + 2α_{χ_{2^{−}}} .

Proposition 4.5. For λ ∈ P_{n}^{s}, Res_{D}(λ) is the character of the BD_{2ℓ}-representation [x, y]/I_{λ}.

Proof.Consider a basis (\overline{x^{i}y^{j}})_{(i,j)∈Y(λ)} of the representation [x, y]/I_{λ}. Since λ is symmetric, restrict the attention to Y^{−} (λ) := {(i, j) ∈ Y(λ) | i > j} and to the diagonal \{(i, j) ∈ Y(λ) | i = j\} . Take first (i, j) ∈ Y^{−} (λ) and consider V_{(i,j)} = Vect(\overline{x^{i}y^{j}}, \overline{x^{j}y^{i}}) , a subspace of [x, y]/I_{λ}. Let k be an element of ⟦1, ℓ − 1⟧. For (i, j) ∈ Y^{−} (λ) such that i − j ≡ k (mod 2ℓ), we have V_{(i,j)} ≃_{BD_{2ℓ}} X_{χ_{k}} (recall that X_{χ_{k}} is an irreducible representation of BD_{2ℓ} with character equal to χ_{k}). Moreover when i − j ≡ 2ℓ − k (mod 2ℓ), we have V_{(i,j)} ≃_{BD_{2ℓ}} X_{χ_{k}}. If k = ℓ, then, for each pair (i, j) ∈ Y^{−} (λ) such that i − j ≡ ℓ (mod 2ℓ), we have V_{(i,j)} ≃_{BD_{2ℓ}} X_{χ_{ℓ^{+}}} ⊕ X_{χ_{ℓ^{−}}} . In the same way, if (i, j) ∈ Y^{−} (λ) such that i ≡ j (mod 2ℓ), we have V_{(i,j)} ≃_{BD_{2ℓ}} X_{χ_{0^{+}}} ⊕ X_{χ_{0^{−}}} . It remains to understand the action of BD_{2ℓ} on the diagonal. For i ∈ _{≥0} , we have ω_{2ℓ}.\overline{x^{i}y^{i}} = \overline{x^{i}y^{i}} and s.\overline{x^{i}y^{i}} = (−1)^{i} \overline{x^{i}y^{i}}. These two computations show that, if i ≡ 0 (mod 2), then V_{i} := V_{(i,i)} ≃_{BD_{2ℓ}} X_{χ_{0^{+}}} and that, if i ≡ 1 (mod 2), then V_{i} ≃_{BD_{2ℓ}} X_{χ_{0^{−}}} . To sum it all up, the character of [x, y]/I_{λ} is Res_{D}(λ). □

By construction Res_{D} factors through Q(D̃_{ℓ+2}^{σ} )[0^{+} ]. For a, b ∈ , let rem(a, b) ∈ ⟦0, b − 1⟧ denote the remainder of the Euclidean division of a by b. By the work of Christopher R. H. Hanusa and Brant C. Jones [HJ12, Theoreom 5.8], we can endow the set ℭ_{2ℓ}^{s} of symmetric 2ℓ-cores with a W(C̃_{ℓ})-action. Let us quickly recall how this action is constructed.

Definition 4.6. For a symmetric 2ℓ-core λ define the C-residue of a box positioned in row i and column j in the Young diagram of λ as

\{ \begin{matrix}rem(j − i, 2ℓ), & if 0 ≤ rem(j − i, 2ℓ) ≤ ℓ, \\ 2ℓ − rem(j − i, 2ℓ), & if l < rem(j − i, 2ℓ) < 2ℓ. \end{matrix}

Example 4.7. Take ℓ = 2 and the same symmetric 4-core (4, 4, 3, 2). The Young diagram filled with the C-residue of each box is

1 2 2 1 0 1 0 1 2 0 1 2 1

Remark 4.8. Note that for each symmetric 2ℓ-core λ, the C-residue of each box of λ is always an integer between 0 and ℓ.

Definition 4.9. The action of W(C̃_{ℓ}) on ℭ_{2ℓ}^{s} is defined on generators. Take s_{i} ∈ W(C̃_{ℓ}) and λ ∈ ℭ_{2ℓ}^{s}. There are three disjoint cases. Either we can add boxes with C-residue i, or we can remove such boxes, or there are no such boxes. Define s_{i}.λ as the partition obtained from λ in either adding all boxes of Y(λ) with C-residue i so that s_{i}.λ remains a partition or removing all boxes of Y(λ) with C-residue i so that s_{i}.λ remains a partition.

Definition 4.10. The C-region of index k ∈ of a symmetric 2ℓ-core is the following subset of Y(λ):

R_{k} := {(i, j) ∈ Y(λ) | (i − j) ∈ {2kℓ, …, 2(k + 1)ℓ − 1}}.

More generally, we can define a shifted C-region. Let (k, h) ∈ ^{2} and define the hshifted C-region of index k by

R_{k,h}:= {(i, j) ∈ Y(λ) | (i − j) ∈ {2kℓ + h, …, 2(k + 1)l − 1 + h}}.

Proposition 4.11. Res_{D} : ℭ_{2ℓ}^{s} → Q(D̃_{ℓ+2}^{σ} )[0^{+} ] is W(C̃_{ℓ})-equivariant.

Proof.By Proposition 3.11, we have Q(D̃_{ℓ+2}^{σ} )[0^{+} ] ⊂ Q^{∨} (Ã_{2ℓ}). Moreover, the type Ã_{2ℓ} is simply laced. We can thus identify Q^{∨} (Ã_{2ℓ}) with Q(Ã_{2ℓ}). Using Proposition 3.11, we can also identify W(Ã_{2ℓ}^{ς}) with W(C̃_{ℓ}). Now, by [BJV09, Proposition 3.2.5], the usual residue map, Res : ℭ_{2ℓ} → Q(Ã_{2ℓ}), is W(Ã_{2ℓ})-equivariant. Finally, the restriction of this map to ℭ_{2ℓ}^{s} gives Res_{D} : ℭ_{2ℓ}^{s} → Q(D̃_{ℓ+2}^{σ} )[0^{+} ]. Indeed, using Proposition 4.5and the definition of the irreducible characters of BD_{2ℓ} we see that it is already true that Res_{D} is the restriction of the usual residue map to symmetric partitions. We can thus conclude that Res_{D} : ℭ_{2ℓ}^{s} → Q(D̃_{ℓ+2}^{σ} )[0^{+} ] is W(Ã_{2ℓ}^{ς})-equivariant. □

Proposition 4.12. T ∘ Res_{D} : ℭ_{2ℓ}^{s} → \overline{0}^{W(C̃_{ℓ})} ⊂ Q^{∨} (C̃_{ℓ}) is a bijection.

Proof.By definition, we have T (Res_{D}(∅)) = 0 and the stabilizer of ∅ ∈ ℭ_{2ℓ}^{s} in W(C̃_{ℓ}) is equal to W(C_{ℓ}), the Weyl group of the finite type C_{ℓ}, which is equal to the stabilizer of

0 ∈ \overline{0}^{W(C̃_{ℓ})}in W(C̃_{ℓ}). Moreover, using Propositions 3.8and 4.11, we know that T ∘ Res_{D} is W(C̃_{ℓ})-equivariant. To conclude, it is enough to show that the W(C̃_{ℓ})-action defined on ℭ_{2ℓ}^{s} (Definition 4.9) is transitive. This has been proven in [HJ12, Proposition 6.2]. □

Remark 4.13. Note that Proposition 4.12 can also be deduced from [BM21, Proposition 4.4] and Proposition 3.11.

Proposition 4.14. The composition of maps

φ : ℭ_{2ℓ}^{s}  Q^{∨} (C̃_{ℓ})  Q^{∨} (C̃_{ℓ})/ δ^{∨} (C̃_{ℓ}) T ∘Res_{D} π

is a bijection.

Proof.Consider the bijection \overline{0}^{W(C̃_{ℓ})} ⟶^{∼} Q^{∨} (C_{ℓ}) which is the composition of the two bijections

\overline{0}^{W(C̃_{ℓ})} ⟶^{∼} W(C̃_{ℓ})/W(C_{ℓ}) ⟶^{∼} Q^{∨} (C_{ℓ}).

The second bijection boils down to the choice of a representative with coordinate 0 along β_{0}^{∨}. Moreover, consider the bijection

\begin{matrix}Q^{∨} (C̃_{ℓ})/ δ^{∨} (C̃_{ℓ}) & ⟶^{∼} & Q^{∨} (C_{ℓ}), \\ β^{∨} & ↦ & β^{∨} − β_{0}^{∨}δ^{∨} (C̃_{ℓ}). \end{matrix}

We then have the following commutative diagram:

\begin{aligned}\overline{0}^{W(C̃_{ℓ})} & & Q^{∨} (C̃_{ℓ})/ δ^{∨} (C̃_{ℓ}) \\ & Q^{∨} (C_{ℓ}) \end{aligned} ; ∼ ; π ; ∼ .

From there, we can use Proposition 4.12 to prove that φ is a bijection. □

COMBINATORICS OF THE IRREDUCIBLE COMPONENTS OF H_{n}^{Γ} IN TYPE D AND E 9

5. Combinatorial description in type D

We now have everything needed to give a combinatorial description of the set A_{BD_{2ℓ}}^{n, _{1}}. Note that, by Proposition 4.5, A_{BD_{2ℓ}}^{n, _{1}} ⊂ A_{BD_{2ℓ}}^{n} ∩ Q(D̃_{ℓ+2}^{σ} )[0^{+} ].

Consider the map

\begin{matrix}ϵ : Q(D̃_{ℓ+2}^{σ} )[0^{+} ] & → & ℭ_{2ℓ}^{s}, \\ d & ↦ & (φ^{−1} ∘ π ∘ T )(d). \end{matrix}

Theorem 5.1. The map ϵ defines a bijection between A_{BD_{2ℓ}}^{n, _{1}} and the symmetric 2ℓ-cores λ, with the property that |λ| ≡ n (mod 2ℓ) and |λ| ≤ n. Moreover, for μ_{1}, μ_{2} ∈ P_{n}^{s}, I_{μ_{1}} and I_{μ_{2}} are in the same irreducible component of H_{n}^{BD_{2ℓ}} if and only if the 2ℓ-cores of μ_{1} and μ_{2} are equal.

Proof.We show first that, if d ∈ A_{BD_{2ℓ}}^{n, _{1}}, then |ϵ(d)| ≡ n (mod 2ℓ). Put λ := ϵ(d). Then

there exists k ∈ such that T (d) = T (Res_{D}(λ)) + kδ^{∨} (C̃_{ℓ}).

In particular |T (d)|C̃_{ℓ} = |T (Res_{D}(λ))|C̃_{ℓ} + 2kℓ. Now, since d ∈ A_{BD_{2ℓ}}^{n, _{1}}, we have |d|D̃_{ℓ+2} = n. Furthermore, using Remark 3.9, we have n = |λ| + 2kℓ. Moreover, we claim that if d ∈ A_{BD_{2ℓ}}^{n, _{1}} , then |ϵ(d)| ≤ n. Indeed, by Lemma 3.10 and the fact

that T (Res_{D}(λ)) ∈ \overline{0}^{W(C̃_{ℓ})}, we have T (d) ∈ \overline{kδ^{∨} (C̃_{ℓ})}^{W(C̃_{ℓ})}. Since wt(d) ≥ 0, there exists k′∈ _{≥0} such that d ∈ \overline{k′δ(D̃_{ℓ+2} )}^{W(D̃_{ℓ+2} )}. In fact d ∈ \overline{k′δ(D̃_{ℓ+2} )}^{W(C̃_{ℓ})} since d ∈ Q(D̃_{ℓ+2}^{σ} )[0^{+} ]. The map T sends δ(D̃_{ℓ+2} ) to 2δ^{∨} (C̃_{ℓ}), which then implies that

T (d) ∈ \overline{2k′δ^{∨} (C̃_{ℓ})}^{W(C̃_{ℓ})}. Hence, we obtain k = 2k′by construction of wt(d) (cf. [Pae1, Lemma 4.7]). Since n = |λ| + 2kℓ, we have that k ≥ 0 if and only if |λ| ≤ n. The map ϵ : A_{BD_{2ℓ}}^{n, _{1}} → {λ ∈ ℭ_{2ℓ}^{s} | |λ| ≡ n (mod 2ℓ), |λ| ≤ n} has now been proven to be well defined. By construction, ϵ is the inverse map of Res_{D} and establishes a bijection between A_{BD_{2ℓ}}^{n, _{1}} and {λ ∈ ℭ_{2ℓ}^{s} | |λ| ≡ n (mod 2ℓ), |λ| ≤ n} .

Concerning the second assertion, we observe that I_{μ_{1}} and I_{μ_{2}} are in the same irreducible component of H_{n}^{BD_{2ℓ}} if and only if the character of [x, y]/I_{μ_{1}} is equal to the character of [x, y]/I_{μ_{2}}, by [Pae1, Corollary 4.3]. Now using Proposition 4.5, we know that this is the case if and only if Res_{D}(μ_{1}) = Res_{D}(μ_{2}). By construction, for i ∈ {1, 2} , ϵ(Res_{D}(μ_{i})) is the 2ℓ-core of μ_{i} which then yields the result. □

Remark 5.2. Take d ∈ A_{BD_{2ℓ}}^{n, _{1}} and λ ∈ P_{n}^{s} such that I_{λ} is in the irreducible component parametrized by d. Let γ_{2ℓ} denote the 2ℓ-core of λ. We have, as a by-product of the proof of Theorem 5.1, that \frac{n−|γ_{2ℓ}|}{2ℓ} , which is the number of 2ℓ-hooks that we need to remove from λ to obtain its 2ℓ-core, is equal to 2wt(d). Example 5.3. The set A_{BD_{2ℓ}}^{n, _{1}} is a proper subset of A_{BD_{2ℓ}}^{n}. If ℓ = 2, for r ∈ _{>0} we can find an irreducible component of H_{8r+4}^{BD_{4}} of dimension 2r that is parametrized by an element of A_{BD_{4}}^{8r+4} \ A_{BD_{4}}^{8r+4, _{1}} . Let ω = s_{χ_{2^{+}}} s_{χ_{1}}s_{χ_{0^{+}}} ∈ W(T̃_{BD_{4}}), and consider ω.rδ^{BD^{4}}. We have (ω.rδ^{BD^{4}}) _{χ_{2^{+}}} = (ω.rδ^{BD^{4}}) _{χ_{2^{−}}} + 1, which implies that this element is not in A_{BD_{4}}^{8r+4, _{1}} due to Proposition 4.5.

6. Absence of combinatorics in type E

The binary tetrahedral group Ã_{4} is a central extension of A_{4}, the alternating group on 4 elements [CM13, §6.5]. It has order 24 and has the following presentation

⟨a, b, c | a^{2} = b^{3} = c^{3} = abc⟩.

We set z := abc, which is a central element of Ã_{4}. Note that z has order 2. The group Ã_{4} has the following character table:

cardinality  1  1 6 4 4 4  4
classes ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) z a b c b^{2} c^{2}
χ_{0}  1  1 1 1 1 1  1
ψ  1  1 1 ζ_{3} ζ_{3}^{2} ζ_{3}^{2} ζ_{3}
ψ^{2}  1  1 1 ζ_{3}^{2} ζ_{3} ζ_{3} ζ_{3}^{2}
X  3  3 −1 0 0 0  0
χ_{std}  2  −2 0 1 1 −1 −1
ψχ_{std}  2  −2 0 ζ_{3} ζ_{3}^{2} −ζ_{3}^{2} −ζ_{3}
ψ^{2}χ_{std}  2  −2 0 ζ_{3}^{2} ζ_{3} −ζ_{3} −ζ_{3}^{2}

The McKay graph of any finite subgroup of SL_{2}( ) isomorphic to Ã_{4} is of affine type Ẽ_{6}

.

The goal here is to study the combinatorics of the irreducible components of H_{n}^{Γ} when Γ is of type Ẽ_{6} (meaning that Γ is isomorphic to Ã_{4}). We claim that the irreducible components containing a monomial ideal are fixed under SL_{2}( ). Indeed, let X_{std} denote the standard representation of SL_{2}( ) with its canonical basis (e_{1}, e_{2}), and denote by B_{1} and B_{2} the stabilizers of e_{1} and e_{2} in SL_{2}( ), respectively. The subgroups B_{1} and B_{2} are the two Borel subgroups of SL_{2}( ) containing _{1}. Let us fix Γ a finite subgroup of type Ẽ_{6} in SL_{2}( ).

Lemma 6.1. The group Γ is not conjugate to any subgroup of the normalizer of _{1} in SL_{2}( ), which we denote by N_{SL_{2}( )} ( _{1}). Furthermore, the group Γ is neither conjugate to a subgroup of B_{1} nor of B_{2}.

Proof.The representation X_{std} ⊗ X_{std}^{∗} is isomorphic to the direct sum of the trivial representation (generated by e_{1} ⊗ e_{1}^{∗} + e_{2} ⊗ e_{2}^{∗}) and the adjoint representation of SL_{2}( ). On the one hand, note that for the character χ_{std} of Γ, we have ⟨(χ_{std})^{2}, (χ_{std})^{2}⟩ = 2, which implies that the restriction of the adjoint representation to Γ is irreducible. On the other hand, the restriction of the adjoint representation to N_{SL_{2}( )} ( _{1}) is not irreducible since the one-dimensional subspace of X_{std}⊗ X_{std}^{∗} generated by e_{1}⊗ e_{1}^{∗} − e_{2}⊗ e_{2}^{∗} is N_{SL_{2}( )} ( _{1})-stable. Moreover, the one-dimensional subspace of X_{std} generated by e_{1} is B_{1}-stable and the one-dimensional subspace of X_{std} generated by e_{2} is B_{2}-stable. □

Proposition 6.2. The subgroup G of SL_{2}( ) generated by _{1} and Γ is SL_{2}( ).

Proof.By Lemma 6.1, there exists x ∈ Γ such that _{1} ≠ x _{1}x^{−1} . The two subgroups _{1} and x _{1}x^{−1} are both irreducible and connected subgroups of SL_{2}( ). Let us denote the subgroup of SL_{2}( ) generated by these two one-dimensional tori by H. From [Hump , Section 7.5], we know that H is a closed connected subgroup of SL_{2}( ). Since H is not equal to SL_{2}( ), and is of dimension at least two, H is of dimension 2. Using [Bor12, Corollary 11.6] we know that H is solvable. The algebraic group H is then a Borel subgroup of SL_{2}( ) containing _{1} and contained in G. Moreover, by the Bruhat decomposition [Bor12, Theorem 14.12], we know that SL_{2}( ) = B_{1}sB_{1} ⊔ B_{1} = B_{2}sB_{2} ⊔ B_{2}. Combining the Bruhat decomposition with Lemma 6.1, we see that s ∈ G. By [Bor12, Proposition 11.19], we know that all Borel subgroups containing _{1} are conjugated by the Weyl group of _{1}, which we denote by W( _{1}). It is by construction the group generated by s̄∈ W( _{1}). We infer that all Borel subgroups containing _{1} are in G. Finally, using [Bor12, Proposition 13.7], we have G = SL_{2}( ). □

Definition 6.3. A partition is called a staircase partition if there exists a certain integer m such that the partition is equal to λ_{m} := (m, m − 1, …, 1) ⊢ \frac{m(m+1)}{2} . Note that ℭ_{2} is equal to the set of all staircase partitions.

Proposition 6.4. The only SL_{2}( ) fixed points of H_{n} are the monomial ideals associated with staircase partitions of size n.

Proof.We already know that _{1}-fixed points are exactly monomial ideals. Moreover, by [KT, Lemma 12], the fixed points under the subgroup _{2} of GL_{2}( ) consisting of all upper triangular matrices are parametrized by staircase partitions. Let _{2} be the maximal diagonal torus of GL_{2}( ). Since _{2} = _{2}B_{1}, we get that B_{1}-fixed points of H_{n} are also parametrized by staircase partitions, and the result follows. □

Finally, the binary octahedral group (type Ẽ_{7}) and the binary icosahedral group (type Ẽ_{8}) both contain a subgroup isomorphic to Ã_{4}, which then implies that the combinatorics of fixed points which are also _{1}-fixed is the same as the one of SL_{2}( ). We thus have proved the following result.

Proposition 6.5. If Γ is a finite subgroup of SL_{2}( ) of type Ẽ_{6}, Ẽ_{7} or Ẽ_{8}, then, for n ∈ _{≥1} , there is at most one irreducible component of H_{n}^{Γ} containing a _{1}-fixed point, and it is indexed by the staircase partition of size n (when it exists).

7. Dimension of the irreducible components containing a _{1}-fixed point

In this section we show that each irreducible component of H_{n}^{Γ} containing a _{1}-fixed point is zero-dimensional whenever Γ is of type Ẽ_{6} in SL_{2}( ). By Proposition 6.5, we know that it is enough to compute the dimensions of the irreducible components of H_{n}^{Γ} which contain a _{1}-fixed point indexed by a staircase partition. The results of this section will not depend on the choice of Γ but only on the McKay graph. Since we need to make explicit computations, we choose to work with the following model of the binary tetrahedral group. Let t ∈ SL_{2}( ) be the matrix

\frac{1}{\sqrt{2}} ( \begin{matrix}ζ_{8} & ζ_{8} \\ ζ_{8}^{3} & ζ_{8}^{−1} \end{matrix} ) .

Consider the subgroup of SL_{2}( ) generated by ω_{4}, s and t. Let us denote this group by BT. By setting a = sω_{4}, b = t and c = st^{2}, one can show that BThas the desired presentation (namely the one of Ã_{4}). Note also that BT= BD_{4} ⋊ ⟨t^{2}⟩.

7.1. Ẽ_{6}-Residue. The irreducible components of H_{n}^{Γ} are isomorphic to quiver varieties over the doubled, framed McKay quiver (cf. [Pae1, Proposition 3.19]). We are interested in the irreducible components containing a _{1}-fixed point. Take m ∈ _{≥1} . We know that the dimension parameter of this quiver variety is equal to the character of BT of the representation [x, y]/I_{λ_{m}}. In this subsection, we will then construct a map ResẼ_{6} : ℭ_{2} → Q(Ẽ_{6}) which computes the decomposition into irreducible characters of the character of the representations [x, y]/I_{λ_{m}} for λ_{m} ∈ ℭ_{2}. Before doing that, we need to introduce some notation. If m = 2k ∈ _{≥0} , define

d_{m}^{0} := \{ \begin{matrix}1 + ⌊ \frac{k−2}{3} ⌋, & if m ≡ 0 (mod 3), \\ ⌊ \frac{k}{3}⌋, & if m ≡ 1 (mod 3), \\ 1 + ⌊ \frac{k−1}{3} ⌋, & if m ≡ 2 (mod 3). \end{matrix}

Let d_{m} := \frac{m−2d_{m}^{0}}{4} . The fact that d_{m} ∈ _{≥0} results from the definition of d_{m}^{0}. If now m = 2k + 1 ∈ _{≥0} , define

a_{m} := 1 + ⌊ \frac{k − 1}{2} ⌋ .

Let b_{m} := m − 3a_{m}. Moreover define

e_{m}^{0} := \{ \begin{matrix}⌊ \frac{b_{m}+1}{3} ⌋, & if m ≡ 0 (mod 3), \\ 1 + ⌊ \frac{b_{m}}{3}⌋, & if m ≡ 1 (mod 3), \\ ⌊ \frac{b_{m}}{3}⌋, & if m ≡ 2 (mod 3). \end{matrix}

Let e_{m} := \frac{b_{m}−e_{m}^{0}}{2} . The fact that e_{m} ∈ _{≥0} results from the definition of e_{m}^{0}. For the sake of clarity, we introduce

β_{m} := \{ \begin{matrix}d_{m}^{0}α_{χ_{std}} + d_{m}α_{ψχ_{std}} + d_{m}α_{ψ^{2}χ_{std}} , & if m is even, \\ a_{m}α_{X}+ e_{m}^{0}α_{χ_{0}} + e_{m}α_{ψ} + e_{m}α_{ψ^{2}}, & otherwise, \end{matrix} for all m ∈ _{≥0} .

We define ResẼ_{6} such that the difference between ResẼ_{6}(λ_{m}) and ResẼ_{6}(λ_{m−1} ) is exactly the element β_{m} of the Ẽ_{6}-root lattice.

Definition 7.1. Define the map Ẽ_{6}-Residue in the following way

ResẼ_{6} : \begin{matrix}ℭ_{2} & → & Q(Ẽ_{6}), \\ λ_{0} & ↦ & 0, \\ λ_{m} & ↦ & ResẼ_{6}(λ_{m−1} ) + β_{m}. \end{matrix}

Example 7.2. Take λ_{3} = (3, 2, 1). Its Young diagram is filled as follows:

X ; ^{χ_{std} X} ; ^{χ_{0} χ_{std} X} .

This gives ResẼ_{6}(λ_{3}) = α_{χ_{0}} + α_{χ} + α_{X}.

The following proposition assures us that ResẼ_{6} fulfills its purpose.

Proposition 7.3. For λ_{m} ∈ ℭ_{2}, ResẼ_{6}(λ_{m}) is the character of the BT-representation [x, y]/I_{λ_{m}}.

Proof.To decompose the character of the SL_{2}( )-representation [x, y]/I_{λ_{m}} along the irreducible characters of BT, we will use the fact that BT= BD_{4}⋊ ⟨t^{2}⟩. The group ⟨t^{2}⟩ is conjugate to μ_{3} in SL_{2}( ). Moreover, we deduce from the character tables and Clifford theory (see [I11, Theorem 6.2]) that X = Ind_{BD_{4}}^{BT} (χ_{2^{+}} ). By Proposition 4.5, we infer that the recursive Definition 7.1 is the character of the BT-representation [x, y]/I_{λ_{m}}. □

Now that we have computed the decomposition of the character of [x, y]/I_{λ_{m}} for λ_{m} ∈ ℭ_{2}, we need to define the Euler form to compute the dimension of the irreducible component of HBT_{\frac{m(m+1)}{2}} containing I_{λ_{m}}. To define this form, one needs to choose an

orientation of the McKay quiver. Let us work with the following orientation:

Let EẼ_{6} be the set of oriented arrows of the McKay quiver Ẽ_{6}. For an arrow h ∈ EẼ_{6}, we denote by h′and h′′the source and target of h, respectively.

7.2. Zero-dimensional irreducible components.

Definition 7.4. The Euler form is a bilinear form defined on the root lattice (which is identified with the lattice of dimension parameters) in the following way:

⟨v, w⟩ := ∑_{χ∈IẼ_{6}} v_{χ}w_{χ} − ∑_{h∈EẼ_{6}} v_{h′}w_{h′′}, for all (v, w) ∈ Q(Ẽ_{6})^{2}.

Remark 7.5. Our results will only involve the Tits form (which is the associated quadratic form). Thus they will not depend on the choice of an orientation. Only the intermediate computations will use the Euler form.

Theorem 7.6. For m ∈ _{≥0} , the irreducible component of HBT_{\frac{m(m+1)}{2}} containing I_{λ_{m}} is of di-

mension 0.

Proof.Combining [Pae1, Proposition 3.19] and Proposition 7.3, we see that the irreducible component of HBT_{\frac{m(m+1)}{2}} containing I_{λ_{m}} is isomorphic to the quiver variety on

the McKay quiver with dimension parameter ResẼ_{6}(λ_{m}). By [Nak98, Corollary 3.12], the dimension of this quiver variety is equal to

2 ( ResẼ_{6}(λ_{m})_{χ_{0}} − ⟨ResẼ_{6}(λ_{m}), ResẼ_{6}(λ_{m})⟩) .

It remains to prove that this integer is equal to zero. To improve readability, we prove the remaining equality in Proposition 7.9. □

Before being able to finish the proof of Theorem 7.6, we need to prove a technical lemma.

Lemma 7.7. For k ∈ _{≥1} , we have

d_{2(k+1)}^{0} + d_{2k}^{0} = e_{2k+1}^{0} + a_{2k+1} ,  (1)
e_{2k−1}^{0} + e_{2k+1}^{0} = d_{2k}^{0},  (2)
k + 1 = a_{2k+1} + a_{2k+3} .  (3)

Proof.To prove relation (1), we consider the following cases:

• If 2k − 1 ≡ 0 (mod 3) , then d_{2(k+1)}^{0} + d_{2k}^{0} = 1 + ⌊ \frac{k−1}{3} ⌋ + ⌊ \frac{k}{3}⌋ and e_{2k+1}^{0} + a_{2k+1} =

⌊ \frac{2k+1}{3} ⌋. In that case d_{2(k+1)}^{0} + d_{2k}^{0} = e_{2k+1}^{0} + a_{2k+1} .

• If 2k − 1 ≡ 1 (mod 3) , then d_{2(k+1)}^{0} + d_{2k}^{0} = ⌊ \frac{k+1}{3} ⌋ + 1 + ⌊ \frac{k−1}{3} ⌋ and e_{2k+1}^{0} +

a_{2k+1} = ⌊ \frac{2k+2}{3} ⌋. In that case d_{2(k+1)}^{0} + d_{2k}^{0} = e_{2k+1}^{0} + a_{2k+1} .

• If 2k − 1 ≡ 2 (mod 3) , then d_{2(k+1)}^{0} + d_{2k}^{0} = 1 + ⌊ \frac{k}{3}⌋ + 1 + ⌊ \frac{k−1}{3} ⌋ and e_{2k+1}^{0} +

a_{2k+1} = 1 + ⌊ \frac{2k+1}{3} ⌋. In that case d_{2(k+1)}^{0} + d_{2k}^{0} = e_{2k+1}^{0} + a_{2k+1} .

The same can be done to prove relation (2). Relation (3) is a direct consequence of the definition of a_{2k+1} . □

Definition 7.8. Take m ∈ _{≥1} . For k ∈ ⟦1, m⟧, define the slice k of λ_{m} to be the subset of the Young diagram of λ_{m} given by {(i, j) ∈ Y(λ_{m}) | i + j = k − 1} . Note that, if we remove the slice m from λ_{m}, we obtain λ_{m−1} .

We are now able to finish the proof of Theorem 7.6.

Proposition 7.9. For m ∈ _{≥0} , we have

⟨ResẼ_{6}(λ_{m}), ResẼ_{6}(λ_{m})⟩ = ResẼ_{6}(λ_{m})_{χ_{0}}.

Proof.We proceed by induction on m. The proposition is obviously true for m = 0. Due to the definition of ResẼ_{6}, we need to distinguish two cases depending on the parity of m.

• If m = 2k, then by Definition 7.1 we have

⟨ResẼ_{6}(λ_{m}), ResẼ_{6}(λ_{m})⟩ = ⟨ResẼ_{6}(λ_{m−1} ), ResẼ_{6}(λ_{m−1} )⟩
+ ⟨ResẼ_{6}(λ_{m−1} ), β_{m}⟩
+ ⟨β_{m}, ResẼ_{6}(λ_{m−1} )⟩
+ (d_{m}^{0})^{2} + 2(d_{m})^{2}.

In the interest of readability, we abbreviate ResẼ_{6}(λ_{m}) by R(m). By the induction hypothesis, we have ⟨R(m − 1), R(m − 1)⟩ = R(m − 1) _{χ_{0}}. By Definition 7.1, we know that R(m − 1) _{χ_{0}} = R(m)_{χ_{0}}, since m is even. It is then enough to prove that

⟨R(m − 1), β_{m}⟩ + ⟨β_{m}, R(m − 1)⟩ + (d_{m}^{0})^{2} + 2(d_{m})^{2} = 0 .  (4)

By Definition 7.1, we have

R(m − 1)_{χ_{0}} = ∑_{j=0}^{k−1} e_{2j+1}^{0} ,

R(m − 1)_{X} = ∑_{j=0}^{k−1} a_{2j+1} ,

R(m − 1)_{χ_{std}} = ∑_{j=0}^{k−1} d_{2j}^{0}.

By (1), we obtain

∑_{j=0}^{k−1} d_{2(j+1)}^{0} + ∑_{j=0}^{k−1} d_{2j}^{0} = ∑_{j=0}^{k−1} e_{2j+1}^{0} + ∑_{j=0}^{k−1} a_{2j+1} .

Since d_{j}^{0} = 0 for j = 0, the previous equality yields

d_{m}^{0} = R(m − 1)_{χ_{0}} + R(m − 1)_{X}− 2R(m − 1)_{χ_{std}}.  (5)

Moreover, by Proposition 7.3, the number of boxes that lie in the odd slices between slice 1 and slice m − 1 is equal to

R(m − 1)_{χ_{0}} + 2R(m − 1)_{ψ} + 3R(m − 1)_{X}.

Thus we have

R(m − 1)_{χ_{0}} + 2R(m − 1)_{ψ} + 3R(m − 1)_{X} = k^{2}.

In the same way, the number of boxes that lie in the even slices between slice 1 and slice m − 1 is equal to 2R(m − 1)_{χ_{std}} + 4R(m − 1)_{ψχ_{std}} . Hence we have

2R(m − 1)_{χ_{std}} + 4R(m − 1)_{ψχ_{std}} = k(k − 1).

From there we have the following two relations:

d_{m}^{0} = R(m − 1)_{χ_{0}} + R(m − 1)_{X}− 2R(m − 1)_{χ_{std}},
k = R(m − 1)_{χ_{0}} + 2R(m − 1)_{ψ} + 3R(m − 1)_{X}− (2R(m − 1)_{χ_{std}} + 4R(m − 1)_{ψχ_{std}} ) .

They imply that

k − d_{m}^{0} = 2 (R(m − 1)_{ψ} + R(m − 1)_{X}− 2R(m − 1)_{ψχ_{std}} ) .

Now, since m = 2k and d_{m} = \frac{m−2d_{m}^{0}}{4} , we obtain

d_{m} = R(m − 1)_{ψ} + R(m − 1)_{X}− 2R(m − 1)_{ψχ_{std}} .  (6)

Recall that, since m is even, β_{m} = d_{m}^{0}α_{χ_{std}} + d_{m}α_{ψχ_{std}} + d_{m}α_{ψ^{2}χ_{std}} . By construction of the Euler form, we have

⟨R(m − 1), β_{m}⟩ + ⟨β_{m}, R(m − 1)⟩ = (R(m − 1), β_{m}),

where ( , ) denotes the nondegenerate bilinear form on 𝔥_{Γ}^{∗} (cf. [Kac, §2.1]). Using the McKay graph of type Ẽ_{6}, we deduce that

(R(m − 1), β_{m}) = 2d_{m}^{0}R(m − 1)_{χ_{std}} + 4d_{m}R(m − 1)_{ψχ_{std}}

− d_{m}^{0}(R(m − 1)_{χ_{0}} + R(m − 1)_{X}) − 2d_{m}(R(m − 1)_{X}+ R(m − 1)_{ψ}) .

Rearranging the right-hand side of the previous equation, we may rewrite this righthand side as

d_{m}^{0}( 2R(m − 1)_{χ_{std}} − R(m − 1)_{χ_{0}} − R(m − 1)_{X})

+ 2d_{m}( 2R(m − 1)_{ψχ_{std}} − R(m − 1)_{X}− R(m − 1)_{ψ}).

We recognize the expressions of d_{m}^{0} and d_{m} obtained in (5) and (6). This leads to the equation

(R(m − 1), β_{m}) = −(d_{m}^{0})^{2} − 2(d_{m})^{2}.

This gives the desired equality (4) and concludes the proof when m is even.

• Let us suppose now that m = 2k + 1 is odd. It is then enough to prove that

⟨R(m − 1), β_{m}⟩ + ⟨β_{m}, R(m − 1)⟩ + e_{m}^{0}^{2} + 2e_{m}^{2} + a_{m}^{2} = e_{m}^{0}.

Due to relations (2) and (3), we first have

2R(m − 1)_{χ_{0}} + e_{m}^{0} − R(m − 1)_{ψχ_{std}} = 1.  (7)

Secondly, we have

4R(m − 1)_{ψ} + 2e_{m} − 2R(m − 1)_{ψχ_{std}} = 0.  (8)

Thirdly, we have

2R(m − 1)_{X}+ a_{m} − R(m − 1)_{χ_{std}} − 2R(m − 1)_{ψχ_{std}} = 0.  (9)

Equations (7), (8) and (9) give the desired result when m is odd.

This concludes the proof of the proposition and also of Theorem 7.6. □

Remark 7.10. Theorem 7.6 implies that, if Γ is a finite subgroup of SL_{2}( ) isomorphic to the binary octahedral group (of type Ẽ_{7}), or if Γ is a finite subgroup of SL_{2}( ) isomorphic to the binary icosahedral group (of type Ẽ_{8}), then all irreducible components of H_{n}^{Γ} containing a _{1}-fixed point are of dimension 0 since these two finite groups contain a subgroup of type Ẽ_{6}.

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Institut Montpelliérain Alexander Grothendieck, Université Montpellier, 2, Place Eugène Bataillon, 34095 Montpellier Cedex, France

Email address: raphael.paegelow@umontpellier.fr