Slim cyclotomic q-Schur algebras

TL;DR

This paper introduces a new basis for slim cyclotomic q-Schur algebras using symmetric polynomials in Jucys--Murphy operators, revealing its relation to double coset bases and exploring cyclotomic Schur--Weyl duality.

Contribution

It constructs a novel basis for slim cyclotomic q-Schur algebras and analyzes its properties and relations to existing bases, advancing understanding of their structure.

Findings

New basis for $ ext{cys} ext{Sr}$ algebra constructed

Basis coincides with double coset basis at $q=1$

Discusses integral cyclotomic Schur--Weyl duality and category equivalence

Abstract

We construct a new basis for a slim cyclotomic $q$-Schur algebra $\cysSr$ via symmetric polynomials in Jucys--Murphy operators of the cyclotomic Hecke algebra $\cysHr$. We show that this basis, labelled by matrices, is not the double coset basis when $\cysHr$ is the Hecke algebra of a Coxeter group, but coincides with the double coset basis for the corresponding group algebra, the Hecke algebra at $q=1$. As further applications, we then discuss the cyclotomic Schur--Weyl duality at the integral level. This also includes a category equivalence and a classification of simple objects.