[Paper Review] Finite dimensional representations of symplectic reflection algebras associated to wreath products II

Silvia Montarani|ArXiv.org|Jan 11, 2005

Advanced Algebra and Geometry5 references4 citations

TL;DR

This paper constructs finite-dimensional irreducible representations of symplectic reflection algebras associated to wreath products of the form $S_N \ltimes \Gamma^N$, where $\Gamma \subset SL(2,\mathbb{C})$ is a finite subgroup. By inducing representations from a parabolic subalgebra and analyzing trace conditions, it proves that such representations can be uniquely deformed along a linear subspace of codimension $r$ in the parameter space $C(\mathcal{S})$, corresponding to $r$ non-isomorphic irreducible modules of the rank-one algebra $H_{1,c}(\Gamma)$.

ABSTRACT

This note extends some results of a previous paper (math.RT/0403250) about finite dimensional representations of the wreath product symplectic reflection algebra H(k,c,N,G) of rank N attached to a finite subgroup G of SL(2,C) (here k is a number and c a class function on the set of nontrivial elements of G). Specifically, let N'=(N_1,...,N_r) be a partition of N. Consider W=W_1\otimes >...\otimes W_r an irreducible representation of S_N'=S_{N_1} imes ... imes S_{N_r}\subset S_N s.t. W_i has rectangular Young diagram for any i and let {Y_i}, i=1,...,r be a collection of irreducible non isomorphic representation of the rank 1 algebra B=H(c_0,1,G) s.t. Ext^1_B(Y_i, Y_j)=0 for any i eq j. Consider the module M'=W\otimes Y, where Y=Y_1^{N_1}\otimes ...\otimes Y_r^{N_r}, over the subalgebra S_N'#B^N \subset S_N#B^N= H(0,c_0,N,G) and form the induced module M over H(0,c_0,N,G). We show that M can be uniquely deformed along a linear subspace of codimension r in the space of the parameters (k,c) passing through c_0. This result implies the main result of math.RT/0403250 as a particular case, the case of the trivial partition N'=(N).

Motivation & Objective

• To extend previous results on finite-dimensional representations of symplectic reflection algebras associated to wreath products $S_N \ltimes \Gamma^N$.
• To characterize the parameter space $C(\mathcal{S})$ for which induced representations from parabolic subalgebras admit unique deformations.
• To establish conditions under which irreducible representations of $H_{1,k,c}(\mathbf{\Gamma}_N)$ can be lifted from rank-one building blocks $H_{1,c}(\Gamma)$.
• To determine the precise linear subspace in $C(\mathcal{S})$ along which such deformations exist, using homological and trace-based techniques.

Proposed method

• Constructs an induced module $M = \mathrm{Ind}_{S_{\vec{N}} \sharp B^{\otimes N}}^{S_N \sharp B^{\otimes N}} (W \otimes Y)$ from a tensor product of irreducible representations $W$ of $S_{\vec{N}}$ and $Y = \bigotimes_{i=1}^r Y_i^{\otimes N_i}$ of the rank-one algebra $B = H_{1,c}(\Gamma)$.
• Uses the decomposition of $M$ into isotypic components to compute traces of operators derived from the defining relations of $H_{1,k,c}(\mathbf{\Gamma}_N)$.
• Applies trace conditions to the relation $[x_1, y_1] = 1 + \frac{k}{2} \sum_{j \neq 1} \sum_{\gamma \in \Gamma} s_{1j} \gamma_1 \gamma_j^{-1} + \sum_{\gamma \in \Gamma \setminus \{1\}} c_\gamma \gamma_1$ to derive linear constraints on parameters.
• Leverages character theory and Young diagram content to compute $\mathrm{Tr}_{W_1}(s_{1j})$ and $\mathrm{Tr}_{Y_1^{\otimes N_1}}(s_{1j} \gamma_1 \gamma_j^{-1})$, leading to explicit trace formulas.
• Derives a system of $r$ independent linear equations in the parameters $k$ and $c_\gamma$, each corresponding to a hyperplane $\mathcal{H}_{Y_i, m_i, l_i}$, and shows the deformation space is the formal neighborhood of the intersection of these hyperplanes.
• Uses the fact that the dimension vectors of non-isomorphic $B$-modules are linearly independent to ensure the $r$ conditions are independent.

Experimental results

Research questions

• RQ1 Under what conditions on the parameters $k$ and $c$ does an induced representation $M$ of $H_{1,k,c}(\mathbf{\Gamma}_N)$ admit a unique deformation?
• RQ2 How does the structure of the Young diagrams of $W_i$ influence the trace computation and the resulting deformation conditions?
• RQ3 What is the precise codimension of the subspace in $C(\mathcal{S})$ where finite-dimensional irreducible representations exist?
• RQ4 How do the Ext-vanishing and non-isomorphism conditions on the $Y_i$ ensure independence of the derived linear constraints?
• RQ5 Can the deformation space of $M$ be described as the formal neighborhood of a linear subspace defined by $r$ hyperplanes, each associated with a distinct irreducible $B$-module?

Key findings

• The induced module $M = \mathrm{Ind}_{S_{\vec{N}} \sharp B^{\otimes N}}^{S_N \sharp B^{\otimes N}} (W \otimes Y)$ admits a unique deformation along a linear subspace of codimension $r$ in $C(\mathcal{S})$.
• For each $i = 1, \dots, r$, the deformation condition yields a hyperplane $\mathcal{H}_{Y_i, m_i, l_i}$ defined by the equation $\mathrm{dim}\,Y_i + \frac{k}{2}|\Gamma|(m_i - l_i) + \sum_{\gamma \in \Gamma \setminus \{1\}} c_\gamma \chi_{Y_i}(\gamma) = 0$, where $m_i$ and $l_i$ are the dimensions of the rectangular Young diagram of $W_i$.
• The $r$ hyperplanes are independent, so their intersection has codimension $r$, and the deformation space is the formal neighborhood of this intersection at the origin.
• The parameter $k$ and the structure of the $Y_i$'s (specifically their dimension vectors and characters) directly determine the deformation locus.
• The proof relies on trace computations using the character of $S_{N_i}$ on $W_i$ and the trace of transpositions, which depends on the content of the rectangular Young diagram of $W_i$.
• The result confirms that the deformation space is isomorphic to the formal neighborhood of the intersection $\bigcap_{i=1}^r \mathcal{H}_{Y_i, m_i, l_i}$, with the subspace $S$ being the formal neighborhood of $\bigcap_{i=1}^r \mathcal{H}_{Y_i, m_i, l_i} - (0, c_0)$.