Generic direct summands of tensor products for simple algebraic groups and quantum groups

TL;DR

This paper introduces new classes of indecomposable modules called generic direct summands in tensor products of simple modules for algebraic and quantum groups, extending understanding of module decompositions.

Contribution

It defines generic direct summands, establishes a Steinberg-Lusztig tensor product theorem for them, and provides explicit examples for types A1 and A2.

Findings

New classes of indecomposable modules identified.

Steinberg-Lusztig tensor product theorem extended to generic summands.

Explicit examples for types A1 and A2 provided.

Abstract

Let $\mathbf{G}$ be either a simple linear algebraic group over an algebraically closed field of positive characteristic or a quantum group at a root of unity. We define new classes of indecomposable $\mathbf{G}$-modules, which we call generic direct summands of tensor products because they appear generically in Krull-Schmidt decompositions of tensor products of simple $\mathbf{G}$-modules and of Weyl modules. We establish a Steinberg-Lusztig tensor product theorem for generic direct summands of tensor products of simple $\mathbf{G}$-modules and provide examples of generic direct summands for $\mathbf{G}$ of type $\mathrm{A}_1$ and $\mathrm{A}_2$.