Centers of q-Schur algebras

Abstract

We give a direct BLM-coordinate description of the centers of generic $q$-Schur algebras for type $A$ and type $B$. For the $q$-Schur algebras ${\mathcal{S}}_q(n,r)$ of type $A$, we show that any central element is supported on some matrices satisfying certain balance properties, and that its coefficients are precisely the solutions of an explicit finite linear system derived from the left and right multiplication formulas for the Chevalley generators of ${\mathcal{S}}_q(n,r)$. This yields an effective reduced row echelon form construction of a $\mathbb{Q}(\upsilon )$-basis of the center, a basis that we then compare with the central bases obtained by Fu via quantum Schur-Weyl duality. For the $q$-Schur algebras ${\mathcal{S}}^{ȷ}(n,r)$ of type $B$, we first apply a method similar to that for type $A$ to provide a basis for the center. In addition, we provide another structural description of the center by decomposing ${\mathcal{S}}^{ȷ}(n,r)$ into the $q$-Schur algebras of type $A$, and these two descriptions are compatible. The center of a variant ${\mathcal{S}}^{ı}(n,r)$ of ${\mathcal{S}}^{ȷ}(n,r)$ is also discussed.

21 pages