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

Manzu Kua, Kay Jin Lim

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.

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

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 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.
Paper Structure (6 sections, 20 theorems, 48 equations, 1 figure)

This paper contains 6 sections, 20 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Heller Translates of the Simple Modules
  4. The stable Green ring of ${F}\mathfrak{S}_{p}$
  5. Stably Tensor-Semisimple Algebras
  6. Benson--Symonds invariants for ${F}\mathfrak{S}_{p}$

Key Result

Lemma 2.1

Figures (1)

  • Figure 1: $j$-diagram

Theorems & Definitions (41)

  • Lemma 2.1
  • proof
  • Theorem 2.2: bensonsymonds
  • proof
  • Theorem 2.3
  • Lemma 2.4
  • Corollary 2.5
  • proof
  • Lemma 3.1
  • proof
  • ...and 31 more