Representations of little $q$-Schur algebras

TL;DR

This paper investigates the representation theory of little $q$-Schur algebras, classifying simple modules, semisimple cases, and finite type algebras at odd roots of unity, expanding understanding of their structure.

Contribution

It provides a classification of simple modules, semisimple algebras, and finite representation type cases for little $q$-Schur algebras, especially at odd roots of unity.

Findings

Classified simple modules for little $q$-Schur algebras.

Identified semisimple little $q$-Schur algebras.

Determined finite representation type cases at odd roots of unity.

Abstract

In \cite{DFW} and \cite{Fu07}, little $q$-Schur algebras were introduced as homomorphic images of the infinitesimal quantum groups. In this paper, we will investigate representations of these algebras. We will classify simple modules for little $q$-Schur algebras and classify semisimple little $q$-Schur algebras. Moreover, through the classification of the blocks of little $q$-Schur algebras for $n=2$, we will determine little $q$-Schur algebras of finite representation type in the odd roots of unity case.