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

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