Endotrivial complexes

TL;DR

This paper introduces and studies endotrivial chain complexes of p-permutation modules over finite groups, establishing their structure, invariants, and autoequivalences, with explicit classifications for certain groups.

Contribution

It defines endotrivial complexes in the homotopy category, characterizes their invariants, and provides a canonical decomposition and classifications for specific group classes.

Findings

Endotrivial complexes induce splendid Rickard autoequivalences.

Complete descriptions of endotrivial complexes for abelian and p-groups of rank 1.

A homomorphism to the orthogonal unit group with Frobenius stability conditions.

Abstract

Let $G$ be a finite group, $p$ a prime, and $k$ a field of characteristic $p$. We introduce the notion of an endotrivial chain complex of $p$-permutation $kG$-modules, which are the invertible objects in the bounded homotopy category of $p$-permutation $kG$-modules, and study the corresponding Picard group $\mathcal{E}_k(G)$ of endotrivial complexes. Such complexes are shown to induce splendid Rickard autoequivalences of $kG$. The elements of $\mathcal{E}_k(G)$ are determined uniquely by integral invariants arising from the Brauer construction and a degree one character $G \to k^\times$. Using ideas from Bouc's theory of biset functors, we provide a canonical decomposition of $\mathcal{E}_k(G)$, and as an application, give complete descriptions of $\mathcal{E}_k(G)$ for abelian groups and $p$-groups of normal $p$-rank 1. Taking Lefschetz invariants of endotrivial complexes induces a group homomorphism $\Lambda: \mathcal{E}_k(G) \to O(T(kG))$, where $O(T(kG))$ is the orthogonal unit group of the trivial source ring. Using recent results of Boltje and Carman, we give a Frobenius stability condition elements in the image of $\Lambda$ must satisfy.