Blocks of the Grothendieck ring of equivariant bundles on a finite group

TL;DR

This paper investigates the block structure of the Grothendieck ring of $G$-equivariant vector bundles on a finite group, providing a detailed description of its decomposition over certain integral extensions.

Contribution

It offers a new description of the ${ m O}$-blocks of the algebra ${ m O} {f K}_G(G)$, enhancing understanding of its semisimple decomposition.

Findings

The algebra ${ m O} {f K}_G(G)$ is split semisimple over a large extension of ${f Q}_p$.

The paper characterizes the block decomposition of ${ m O} {f K}_G(G)$.

Provides explicit descriptions of the blocks in the Grothendieck ring.

Abstract

If $G$ is a finite group, the Grothendieck group ${\mathbf{K}}\_G(G)$ of the category of $G$-equivariant ${\mathbb{C}}$-vector bundles on $G$ (for the action of $G$ on itself by conjugation) is endowed with a structure of (commutative) ring. If $K$ is a sufficiently large extension of ${\mathbb{Q}}\_{\! p}$ and ${\mathcal{O}}$ denotes the integral closure of ${\mathcal{Z}}\_{\! p}$ in $K$, the $K$-algebra $K{\mathbf{K}}\_G(G)=K \otimes\_{\mathbb{Z}} {\mathbf{K}}\_G(G)$ is split semisimple. The aim of this paper is to describe the ${\mathcal{O}}$-blocks of the ${\mathcal{O}}$-algebra ${\mathcal{O}} {\mathbf{K}}\_G(G)$.