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Endotrivial complexes

Sam K. Miller

TL;DR

The paper develops a chain complex analogue of endotrivial modules by defining endotrivial chain complexes as invertible objects in the bounded homotopy category with $p$-permutation terms, yielding splendid Rickard autoequivalences. It introduces the h-mark framework to capture local Brauer-quotient data, and proves a canonical decomposition of the endotrivial complex Picard group $\\mathcal{E}_k(G)$ via Bouc's biset-functor theory, reducing questions to its faithful components. The authors provide complete descriptions of $\\mathcal{E}_k(G)$ for abelian groups and for $p$-groups of normal $p$-rank $1$, and establish surjectivity of the Lefschetz map $\\Lambda: \\mathcal{E}_k(G) \to O(T(kG))$ in these and related cases. They construct explicit faithful generators for several $2$-group families (dihedral, generalized quaternion, semidihedral) and determine the rank of $\\mathcal{E}_k(G)$ in these cases. Finally, they show that not all orthogonal units lift to endotrivial complexes by introducing a Frobenius-stability obstruction, providing concrete counterexamples.

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 $Λ: \mathcal{E}_k(G) \to O(T(kG))$, 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 $Λ$ must satisfy.

Endotrivial complexes

TL;DR

The paper develops a chain complex analogue of endotrivial modules by defining endotrivial chain complexes as invertible objects in the bounded homotopy category with -permutation terms, yielding splendid Rickard autoequivalences. It introduces the h-mark framework to capture local Brauer-quotient data, and proves a canonical decomposition of the endotrivial complex Picard group via Bouc's biset-functor theory, reducing questions to its faithful components. The authors provide complete descriptions of for abelian groups and for -groups of normal -rank , and establish surjectivity of the Lefschetz map in these and related cases. They construct explicit faithful generators for several -group families (dihedral, generalized quaternion, semidihedral) and determine the rank of in these cases. Finally, they show that not all orthogonal units lift to endotrivial complexes by introducing a Frobenius-stability obstruction, providing concrete counterexamples.

Abstract

Let be a finite group, a prime, and a field of characteristic . We introduce the notion of an endotrivial chain complex of -permutation -modules, which are the invertible objects in the bounded homotopy category of -permutation -modules, and study the corresponding Picard group of endotrivial complexes. Such complexes are shown to induce splendid Rickard autoequivalences of . The elements of are determined uniquely by integral invariants arising from the Brauer construction and a degree one character . Using ideas from Bouc's theory of biset functors, we provide a canonical decomposition of , and as an application, give complete descriptions of for abelian groups and -groups of normal -rank 1. Taking Lefschetz invariants of endotrivial complexes induces a group homomorphism , where 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 must satisfy.
Paper Structure (15 sections, 47 theorems, 94 equations, 9 figures)

This paper contains 15 sections, 47 theorems, 94 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. $p$-permutation modules and the Brauer construction
  4. Chain complexes
  5. Endotrivial modules
  6. The Burnside ring
  7. Endotrivial complexes and h-marks
  8. Basic properties of endotrivial complexes
  9. Relationships between h-marks and functorial constructions
  10. Endotrivial complexes induce splendid autoequivalences
  11. The faithful constituent of $\mathcal{E}_k(G)$
  12. Determining $\mathcal{E}_k(G)$ for some groups
  13. $\mathcal{E}_k(G)$ for abelian groups
  14. $\mathcal{E}_k(G)$ for $2$-groups with normal $2$-rank one
  15. Not all orthogonal units lift to endotrivial complexes

Key Result

Theorem 1.1

Let $C$ be an endotrivial complex of $p$-permutation $kG$-modules. Then $\operatorname{Ind}^{G\times G}_{\Delta G} C$, regarded as a complex of $(kG,kG)$-bimodules, induces a splendid Rickard autoequivalence of $kG$.

Figures (9)

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  • ...and 4 more figures

Theorems & Definitions (119)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • ...and 109 more