On the semisimplicity and Schur elements of (super)symmetric superalgebras

TL;DR

This paper establishes semisimplicity criteria for (super)symmetric superalgebras using Schur elements, providing explicit formulas and applications to various cyclotomic superalgebras.

Contribution

It derives a closed formula for Schur elements of cyclotomic Hecke-Clifford superalgebras and applies this to establish semisimplicity criteria for multiple superalgebra classes.

Findings

Proved proportionality of two trace functions on Hecke-Clifford superalgebras.

Derived a semisimplicity criterion for cyclotomic superalgebras.

Provided explicit formulas for Schur elements of cyclotomic Hecke-Clifford superalgebras.

Abstract

In this paper, we use Schur elements to derive semisimplicity criteria for (super)symmetric superalgebras. We obtain a closed formula for the Schur elements of cyclotomic Hecke-Clifford superalgebras $\mathcal{H}^{f}_{\mathbb{K}}$. As applications, we prove that two trace functions $\gimel_n$ and $t_{1,n}$ on the Hecke-Clifford superalgebra, which are defined in different ways, are proportional. We give a semisimplicity criterion for $\mathcal{H}^{f}_{\mathbb{K}}$ when it is (super)symmetric. We also derive semisimplicity criteria for cyclotomic quiver Hecke superalgebras of types $A^{(1)}_{e}$, $C^{(1)}_{e}$, $A^{(2)}_{2e}$ and $D^{(2)}_{e+1}$.