On the decomposition numbers of $\mathrm{SO}_8^+(2^f)$

TL;DR

This paper investigates the decomposition matrices of the special orthogonal group SO_8^+(2^f), describing conjugacy class fusion, proving unitriangularity of decomposition matrices for primes not equal to 2, and explicitly determining the matrices for certain primes dividing q+1.

Contribution

It provides a detailed analysis of the fusion of conjugacy classes and establishes the unitriangularity of decomposition matrices for all primes not equal to 2, including explicit calculations for specific cases.

Findings

Fusion of conjugacy classes of Sylow 2-subgroups in G described.

Proved unitriangularity of decomposition matrices for all  eq 2.

Explicitly determined -decomposition matrix for  \u2265 5 dividing q+1.

Abstract

Let $q=2^f$, and let $G=\mathrm{SO}_8^+(q)$ and $U$ be a Sylow $2$-subgroup of $G$. We first describe the fusion of the conjugacy classes of $U$ in $G$. We then use this information to prove the unitriangularity of the $\ell$-decomposition matrices of $G$ for all $\ell \ne 2$ by inducing certain irreducible characters of $U$ to $G$; the characters of $U$ of degree $q^3/2$ play here a major role. We then determine the $\ell$-decomposition matrix of $G$ in the case $\ell \mid q+1$, when $\ell \ge 5$ and $(q+1)_\ell>5$, up to two non-negative indeterminates in one column.