Centers of quantum Schur superalgebras from Hecke algebras

Abstract

We study the center of the quantum Schur superalgebra $\mathcal{S}_v(m|n,r)$ associated with the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$. Using the super Schur--Weyl duality due to Mitsuhashi between the quantum supergroup $U_v(\mathfrak{gl}_{m|n})$ and the Hecke algebra $\mathcal{H}_v(\mathfrak{S}_r)$, we transfer two known bases of the center of $\mathcal{H}_v(\mathfrak{S}_r)$, namely the Geck--Rouquier basis and the Jones basis, to the center of $\mathcal{S}_v(m|n,r)$. This yields two distinct bases for $\mathscr{Z}(\mathcal{S}_v(m|n,r))$, indexed respectively by the symmetrized hook set $H^{\vee}(m|n,r):=H(\min(m,n)\mid \max(m,n),r)$ and by the full $(m|n)$-hook set $H(m|n,r)$. Our approach relies on a detailed analysis of the ring $\Lambda_{m|n}$ of doubly symmetric polynomials satisfying $f|_{x_m=t=-y_n}$ independent of $t$, and of its power-sum bases.