Decomposition numbers for the principal $\Phi_{2n}$-block of $\mathrm{Sp}_{4n}(q)$ and $\mathrm{SO}_{4n+1}(q)$

TL;DR

This paper calculates the decomposition numbers for unipotent characters in the principal -block of certain finite groups of Lie type, extending existing results on branching graphs for Harish-Chandra induction and restriction.

Contribution

It provides explicit decomposition numbers for the principal -block of  groups  and extends branching graph results to these groups.

Findings

Decomposition numbers for unipotent characters in the principal -block are computed.

Results extend the branching graph theory for Harish-Chandra induction to these finite groups.

Provides new tools for understanding modular representations of  groups.

Abstract

We compute the decomposition numbers of the unipotent characters lying in the principal $\ell$-block of a finite group of Lie type $B_{2n}(q)$ or $C_{2n}(q)$ when $q$ is an odd prime power and $\ell$ is an odd prime number such that the order of $q$ mod $\ell$ is $2n$. Along the way, we extend to these finite groups the results of \cite{DVV19} on the branching graph for Harish-Chandra induction and restriction.