Brauer's height zero conjecture for central quotients of special linear and special unitary groups
Let \(G\) be a finite group and \(B\) be any \(r\)-block of \(G\), where \(r\) is a prime number. By the height of an irreducible character \(\chi\) belonging to \(B\) we mean \(\nu(\chi(1))-\nu([G:D])\) where \(D\) is a defect group of \(B\) and \(\nu\) is the discrete valuation associated to \(r\). Brauer's height zero conjecture states that all ordinary irreducible characters of \(G\) belonging to any \(r\)-block \(B\) have height zero if and only if \(B\) has an Abelian defect group. This conjecture has been verified by \textit{P. Fong} and \textit{B. Srinivasan} [Invent. Math. 69, 109-153 (1982; Zbl 0507.20007)] for all \(r\)-blocks of all the general linear and unitary groups over finite fields where \(r>2\) and is different from the characteristic of the ground field. The purpose of the paper under review is to prove Brauer's height zero conjecture for all \(r\)-blocks of all central factor groups of the special linear and special unitary groups.
scientific article; zbMATH DE number 1271822