Linear characters and block algebra

TL;DR

This paper characterizes groups with blocks having only linear characters or Brauer characters, linking these properties to group structure and providing a formula relating block algebra dimension to irreducible Brauer characters.

Contribution

It establishes new criteria for blocks with linear characters and derives a dimension formula for block algebras in such groups.

Findings

Groups with only linear ordinary characters are p-nilpotent with abelian Sylow p-subgroups.

Groups with only linear Brauer characters satisfy specific subgroup inclusion conditions.

Dimension of block algebra relates to sum of squares of irreducible Brauer character degrees.

Abstract

This paper will prove that: 1. $G$ has a block only having linear ordinary characters if and only if $G$ is a $p$-nilpotent group with an abelian Sylow $p$-subgroup; 2. $G$ has a block only having linear Brauer characters if and only if $O_{p'}(G)\leq O_{p'p}(G)=HO_{p'}(G)= \textrm{Ker}(B_{0}^{*}) \leq O_{p'pp'}=G$, where $H=G^{'}O^{p'}(G), \textrm{Ker}(B_{0}^{*})=\bigcap_{\lambda \in \textrm{IBr}(B_{0})} \textrm{Ker}(V_{\lambda}), B_{0}$ is the principal block of $G$ and $V_{\lambda}$ is the $F[G]$-module affording the Brauer character $\lambda$; 3. if $G$ satisfies the conditions above, then for any block algebra $B$ of $G$, we have $$ \frac{\textrm{Dim}_{F}(B)}{|D|}= \sum_{\phi \in \textrm{IBr}(B)}\phi(1)^{2}$$ where $D$ is the defect group of $B$.