Relatively endotrivial complexes

2024/02/12 by Sam K. Miller, Miller, Sam K. · 1 citation
Materials Science · #20C05 #20C20 #20J05 #FOS: Mathematics #Group Theory (math.GR) #Lanthanide and Transition Metal Complexes #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2402.08042

openalex publication_date 2024/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a finite group and k be a field of characteristic p > 0. In prior work, we studied endotrivial complexes, the invertible objects of the bounded homotopy category K^b(_kGtriv) of p-permutation kG-modules. Using the notion of projectivity relative to a kG-module, we expand on this study by defining notions of "relatively" endotrivial chain complexes, analogous to Lassueur's construction of relatively endotrivial kG-modules. We obtain equivalent characterizations of relative endotriviality and find corresponding local homological data which almost completely determine the isomorphism class of a relatively endotrivial complex. We show this local data must partially satisfy the Borel-Smith conditions, and consider the behavior of restriction to subgroups containing Sylow p-subgroups S of G.

Cited by

• The Euler characteristic of an endotrivial complex

Related

• Endotrivial complexes
• The classification of endotrivial complexes
• On the orbit category on nontrivial p-subgroups and endotrivial modules
• The Euler characteristic of an endotrivial complex
• Relatively Projectivity and the Green correspondence for complexes