The permutation dimension of the Klein 4-group

TL;DR

This paper investigates the $p$-permutation dimension of the Klein 4-group over fields of characteristic 2, providing explicit calculations for the group and its indecomposable modules, inspired by recent advances in modular representation theory.

Contribution

It introduces and computes the $p$-permutation dimension for the Klein 4-group, extending the understanding of this invariant beyond cyclic groups.

Findings

The global $p$-permutation dimension of the Klein 4-group in characteristic 2 is explicitly calculated.

Dimensions for each indecomposable module of the Klein 4-group are determined.

The results build on recent theoretical developments in $p$-permutation resolutions.

Abstract

Let $G$ be a finite group and $k$ a field of characteristic $p > 0$. Balmer and Gallauer's recent result on finite $p$-permutation resolutions of $kG$-modules motivates the study of an intriguing new invariant; the $p$-permutation dimension. Following Walsh's success with cyclic groups of prime order, we compute the (global) $p$-permutation dimension of the Klein 4-group in characteristic $p := 2$, along with the dimensions for each of its indecomposable modules.