On Centres of 3-blocks of the Ree groups $^2G_2(q)$

TL;DR

This paper investigates the centers of principal blocks in Ree groups of type 2G2(q), revealing differences between the centers of blocks and their Brauer correspondents, supported by detailed subgroup analysis.

Contribution

It demonstrates that the centers of principal blocks and their Brauer correspondents are not isomorphic in Ree groups, providing explicit computations of conjugacy classes and character tables.

Findings

Centers of principal blocks and Brauer blocks differ in Ree groups.

Explicit conjugacy class and character table computations for maximal parabolic subgroups.

Shows non-isomorphism of centers in specific block cases.

Abstract

Let $G:={^2G_2}(q)$ be the simple Ree group with $q=3^{2k+1}$ and $k$ a positive integer. We show that the centre of the principal block $Z(kGe_0)$, where $k$ is an algebraically closed field of characteristic $3$, is not isomorphic to the centre of the Brauer corresponding block $Z(kN_G(P))$, where $N_G(P)$ is the normaliser in $G$ of a Sylow $3$-subgroup. As part of the proof, we compute the conjugacy classes of elements and the character tables of the maximal parabolic subgroups of $G$.