Permutation dimensions of prime cyclic groups

TL;DR

The paper defines and analyzes the $p$-permutation dimension $\mathrm{ppdim}$ for $kG$-modules over a field $k$ of characteristic $p>0$, using finite $p$-permutation resolutions as a refinement of projective resolutions. Building on the Balmer–Gallauer framework, it proves that every $kG$-module admits such a finite resolution and introduces a computable invariant $\mathrm{dist_p}$ to measure module complexity. Focusing on the prime cyclic group $C_p$, it shows that $\mathrm{ppdim}(M)=\mathrm{dist_p}(M)$ and computes $\mathrm{ppdim}_{k}(C_p)=p-2$, with consequences for direct sums and tensor products. These results provide a finer, representation-theoretic gauge of complexity in the modular setting and connect to structural invariants of indecomposable modules.

Abstract

Based on recent successes concerning permutation resolutions of representations by Balmer and Gallauer we define a new invariant of finite groups: the p-permutation dimension. We define this analogously to the global dimension of a ring by replacing projective resolutions of ring modules with resolutions by p-permutation modules of modules over the group ring. We compute this invariant for cyclic groups of prime order.

Theorems & Definitions (24)

• Theorem
• Definition 2.1
• Remark 1
• Definition 2.2
• Example 1
• Proposition 2.3
• proof
• Definition 3.1
• Remark 2
• Lemma 3.2