The classification of endotrivial complexes

2024/03/06 by Sam K. Miller, Miller, Sam K. · 2 citations
Medicine · Mathematics · #Medical Imaging Techniques and Applications #Advanced Topics in Algebra #Advanced Operator Algebra Research

paper · pdf · doi:10.48550/arxiv.2403.04088

Abstract

Let G be a finite group and k a field of prime characteristic p. We give a complete classification of endotrivial complexes, i.e. determine the Picard group E_k(G) of the tensor-triangulated category K^b(_kGtriv), the bounded homotopy category of p-permutation modules, which Balmer and Gallauer recently considered. For p-groups, we identify E_k(-) with the rational p-biset functor CF_b(-) of Borel-Smith functions and recover a short exact sequence of rational p-biset functors constructed by Bouc and Yalçin. As a consequence, we prove that every p-permutation autoequivalence of a p-group arises from a splendid Rickard autoequivalence. Additionally, we give a positive answer to a question of Gelvin and Yalçin, showing the kernel of the Bouc homomorphism for an arbitrary finite group G is described by superclass functions f: s_p(G) → ℤ satisfying the oriented Artin-Borel-Smith conditions.

Cited by

• The Euler characteristic of an endotrivial complex
• The Balmer spectrum of integral permutation modules

Related

• Endotrivial complexes
• Relatively endotrivial complexes
• The Euler characteristic of an endotrivial complex
• On the orbit category on nontrivial p-subgroups and endotrivial modules
• Diagonal p-permutation functors