Schur Algebras and Quantum Symmetric Pairs With Unequal Parameters

Abstract

We study the (quantum) Schur algebras of type B/C corresponding to the Hecke algebras with unequal parameters. We prove that the Schur algebras afford a stabilization construction in the sense of Beilinson-Lusztig-MacPherson that constructs a multiparameter upgrade of the quantum symmetric pair coideal subalgebras of type AIII/AIV with no black nodes. We further obtain the canonical basis of the Schur/coideal subalgebras, at the specialization associated to any weight function. These bases are the counterparts of Lusztig's bar-invariant basis for Hecke algebras with unequal parameters. In the appendix we provide an algebraic version of a type D Beilinson-Lusztig-MacPherson construction which is first introduced by Fan-Li from a geometric viewpoint.

Global crystal bases for integrable modules over a quantum symmetric pair of type AIII

“…They are known as (generalized) q-Schur algebras. Multiparameter q-Schur algebra of type B is also studied in [LL19]. 7.1.…”

Multiplication formulas and isomorphism theorem of $\imath$Schur superalgebras

“…cellularity, quasi-hereditariness, semisimplicity and representation type) were derived in [LNX20]. We refer to [B17,FL15] for the type D analogue and refer to [LL21] for a 2-parameters generalization. Actually, all q-Schur algebras studied in [DJ89, Gr97, BKLW18, LL21] are special cases of the ıSchur algebras formulated in [SW21].…”

Equivariant K-theory approach to $\imath$-quantum groups

“…In [FLLLW2], a Hecke algebraic approach to ı-quantum groups and the corresponding affine q-Schur algebras was developed, which is the algebraic counterpart of the geometric approach studied in [FLLLW1]. The algebraic approach to other types has been studied in [LL15,LW17,LL18].…”