Yokonuma-Schur algebras

TL;DR

This paper introduces Yokonuma-Schur algebras as endomorphism algebras of permutation modules for Yokonuma-Hecke algebras, proving their cellularity, quasi-hereditary structure, and defining tilting modules, with an extension to cyclotomic cases.

Contribution

It constructs Yokonuma-Schur algebras, proves their cellularity and quasi-hereditary properties, and introduces tilting modules, extending the framework to cyclotomic variants.

Findings

Yokonuma-Schur algebra is cellular.

Yokonuma-Schur algebra is a quasi-hereditary cover of the Yokonuma-Hecke algebra.

Introduction of tilting modules for Yokonuma-Schur algebra.

Abstract

In this paper, we define the Yokonuma-Schur algebra $\text{YS}_{q}(r,n)$ as the endomorphism algebra of a permutation module for the Yokonuma-Hecke algebra $\text{Y}_{r,n}(q).$ We prove that $\text{YS}_{q}(r,n)$ is cellular by constructing an explicit cellular basis following the approach in [DJM], and we further show that it is a quasi-hereditary cover of $\text{Y}_{r,n}(q)$ in the sense of Rouquier following [HM2]. We also introduce the tilting modules for $\text{YS}_{q}(r,n).$ In the appendix, we define and study the cyclotomic Yokonuma-Schur algebra in a similar way.