On (super)symmetrizing forms and Schur elements of cyclotomic Hecke-Clifford algebras

TL;DR

This paper introduces Schur elements for supersymmetrizing superalgebras, characterizes when cyclotomic Hecke-Clifford algebras are symmetric or supersymmetric, and computes these elements in the semisimple case, with applications to new symmetrizing forms.

Contribution

It defines Schur elements for supersymmetrizing superalgebras and computes them for cyclotomic Hecke-Clifford and Sergeev algebras, establishing conditions for symmetry.

Findings

Cyclotomic Hecke-Clifford algebra is supersymmetric under certain conditions.

Schur elements are computed explicitly in the semisimple case.

New symmetrizing forms are constructed for related algebras.

Abstract

In this paper, we introduce Schur elements for supersymmetrizing superalgebras. We show that the cyclotomic Hecke-Clifford algebra $\mathcal{H}^f_{c}(n)$ is supersymmetric if $f=f^{(\mathtt{0})}_{\underline{Q}}$ and, symmetric if $f=f^{(\mathtt{s})}_{\underline{Q}}$ and an invertibility condition holds. In the semisimple case, we compute the Schur elements for both $\mathcal{H}^f_{c}(n)$ and the cyclotomic Sergeev algebra $\mathcal{h}^g_{c}(n)$. As applications, we define new symmetrizing forms on the Hecke-Clifford algebra $\mathcal{H}(n)$ and on the cyclotomic quiver Hecke algebras of types $A^{(1)}_{e-1}$ and $C^{(1)}_e$.