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

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.