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 $\mathcal{Z}({\mathcal{S}}_v(m|n,r))$, 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.

25 pages