On relative height zero Brauer characters

Abstract

Let $N \triangleleft G$ where $G$ is a finite group and let $B$ be a $p$-block of $G$, where $p$ is a prime. A Brauer character $\psi \in \mathop{\mathrm{IBr}}_{p}(B)$ is said to be of relative height zero with respect to $N$ provided that the height of $\psi$ is equal to that of an irreducible constituent of $\psi_{N}$. Now assume $G$ is $p$-solvable. In this paper, we count the number of relative height zero irreducible Brauer characters of $B$ with respect to $N$ that lie over any given $\varphi \in \mathop{\mathrm{IBr}}_{p}(N)$. As a consequence, we show that if $D$ is a defect group of $B$ and $\hat{B}$ is the unique $p$-block of $NN_{G}(D)$ with defect group $D$ such that $\hat{B}^{G} = B$, then $B$ and $\hat{B}$ have equal numbers of relative height zero irreducible Brauer characters with respect to $N$.