Stratifying q-Schur Algebras of Type D

TL;DR

This paper introduces two new families of q-Schur algebras of type D, proves their structural properties, and explores their relationships and potential for a unified theory in representation theory.

Contribution

It defines and analyzes the properties of q-Schur^{1.5} algebras, establishing their freeness, stability, and stratification, and connects these to existing families in type D.

Findings

q-Schur^{1.5} algebras are integrally free and stable under base change.

These algebras are standardly stratified in odd characteristic fields.

In linear prime cases, all three families are Morita equivalent.

Abstract

Two families of q-Schur algebras associated to Hecke algebras of type D are introduced, and related to a family used by Geck, Gruber and Hiss [10], [11]. We prove that the algebras in one family, called the q-Schur^{1.5} algebras, are integrally free, stable under base change, and are standardly stratified if the base field has odd characteristic. In the so-called linear prime case of [10], [11], all three families give rise to Morita equivalent algebras. A final section discusses a different example, and speculates on the direction of a general theory.