The number of independent Traces and Supertraces on the Symplectic Reflection Algebra $H_{1,\nu}(\Gamma \wr S_N)$

TL;DR

This paper explicitly calculates the dimensions of the spaces of traces and supertraces for symplectic reflection algebras associated with wreath product groups, revealing their independence from the deformation parameter.

Contribution

It provides explicit formulas for the number of traces and supertraces on symplectic reflection algebras for wreath product groups, extending understanding of their algebraic structure.

Findings

Calculated $T(G)$ and $S(G)$ for $G= ext{finite subgroup of } Sp(2, \, \mathbb{C}) \wr S_N$

Showed $T(G)$ and $S(G)$ depend only on the group $G$, not on the parameter $\nu$

Enhanced understanding of trace structures in symplectic reflection algebras

Abstract

Symplectic reflection algebra $ H_{1, \,\nu}(G)$ has a $T(G)$-dimensional space of traces whereas, when considered as a superalgebra with a natural parity, it has an $S(G)$-dimensional space of supertraces. The values of $T(G)$ and $S(G)$ depend on the symplectic reflection group $G$ and do not depend on the parameter $\nu$. In this paper, the values $T(G)$ and $S(G)$ are explicitly calculated for the groups $G= \Gamma \wr S_N$, where $\Gamma$ is a finite subgroup of $Sp(2,\mathbb C)$.