On (super)symmetrizing forms and Schur elements of cyclotomic Hecke-Clifford 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 $C^{(1)}_e$.