The orthogonal unit group of the trivial source ring

TL;DR

This paper characterizes the structure of the trivial source ring of a finite group over a p-modular system, describing its orthogonal units via coherent tuples and connections to the Burnside ring and fusion system automorphisms.

Contribution

It provides a new isomorphism for the trivial source ring and describes its orthogonal units in terms of fusion system data and character tuples.

Findings

Trivial source ring is isomorphic to coherent G-stable tuples of virtual characters.

Orthogonal units form a group isomorphic to a product involving the Burnside ring and homomorphisms.

Results relate orthogonal units to p-permutation autoequivalences.

Abstract

Let $G$ be a finite group, $p$ a prime, and $(K,\mathcal{O},F)$ a $p$-modular system. We prove that the trivial source ring of $\mathcal{O} G$ is isomorphic to the ring of {\em coherent} $G$-stable tuples $(\chi_P)$, where $\chi_P$ is a virtual character of $K[N_G(P)/P]$, $P$ runs through all $p$-subgroups of $G$, and the coherence condition is the equality of certain character values. We use this result to describe the group of orthogonal units of the trivial source ring as the product of the unit group of the Burnside ring of the fusion system of $G$ with the group of coherent $G$-stable tuples $(\varphi_P)$ of homomorphisms $N_G(P)/P\to F^\times$. The orthogonal unit group of the trivial source ring of $\mathcal{O} G$ is of interest, since it embeds into the group of $p$-permutation autoequivalences of $\mathcal{O} G$.