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

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.

29 pages