Structure of the tensor product of two simple modules of quantum $GL_2$

TL;DR

This paper investigates the structure of tensor products of simple modules for quantum GL_2 over fields with non-zero characteristic, revealing they decompose into direct sums of indecomposable twisted tilting modules.

Contribution

It extends the understanding of tensor product decompositions from SL_2 and SL_3 to quantum GL_2, providing explicit structural descriptions.

Findings

Tensor products decompose into indecomposable twisted tilting modules

Generalizes previous results from SL_2 and SL_3 to quantum GL_2

Provides a framework for understanding module structures in quantum groups

Abstract

In this article, we consider the tensor product of two simple modules of quanum $GL_2$ over a field of characteristic $p\neq 0$. We show that it can be expressed as a direct sum of indecomposable twisted tilting modules. This problem has been studied by Henke and Doty [1] for $SL_2$ and also later on for $SL_3$ by S R Doty, Chris Bowman and Stuart Martin, ([7, 8]).