Tensor Product and the Stable Green Ring of the Symmetric Group Algebra $F\mathfrak{S}_p$

TL;DR

This paper provides explicit formulas for tensor product decompositions of indecomposable modules over the symmetric group algebra, revealing semisimplicity modulo projectives and computing Benson--Symonds invariants.

Contribution

It introduces a detailed formula for tensor product decompositions of indecomposable modules over $F\mathfrak{S}_p$, advancing understanding of their module structure.

Findings

Tensor product of two simple modules is semisimple modulo projectives.

Explicit decomposition formulas for all indecomposable non-projective modules.

Computed Benson--Symonds invariants for these modules.

Abstract

We give an explicit formula for the decomposition of the tensor product of any two indecomposable non-projective modules for the symmetric group algebra $F \mathfrak{S}_p$ modulo projective modules. In particular, we show that the tensor product of two simple modules is semisimple modulo projectives. We also compute the Benson--Symonds invariants for all such indecomposable non-projective modules.