On the semisimplicity and Schur elements of (super)symmetric 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 ${\mathcal{H}}_{\mathbb{K}}^f$. 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 ${\mathcal{H}}_{\mathbb{K}}^f$ when it is (super)symmetric. We also derive semisimplicity criteria for cyclotomic quiver Hecke superalgebras of types $A_e^{(1)}$, $C_e^{(1)}$, $A_{2e}^{(2)}$ and $D_{e+1}^{(2)}$.

42 pages. Comments welcome!