Schur Index and Extensions of Witt-Berman's Theorems

Abstract

Let $G$ be a finite group, and $F$ a field of characteristic $0$ or prime to the order of $G$. In $1952$, Witt and in $1956$, Berman independently proved that the number of inequivalent irreducible $F$-representations of $G$ is equal to the number of $F$-conjugacy classes of the elements of $G$, where "$F$-conjugacy" was defined in a certain way. In this paper, we define $F$-conjugacy on $G$ in a natural way and give a proof of the above Witt-Berman theorem. In addition, we give an explicit formula for computing a primitive central idempotent (pci) of the group algebra $F[G]$ corresponding to an irreducible $F$-representation of $G$, which can be obtained from the "$F$-character table" of $G$. Let $G$ be a finite group with a normal subgroup $H$ of index $p$, a prime. In $1955$, in case $F$ is algebraically closed, Berman computed the primitive central idempotent (pci) of $F[G]$ corresponding to an irreducible $F$-representation of $G$, in terms of pci's of $F[H]$. In this paper, we give a complete proof of this Berman's theorem, and extend this result when $F$ is not necessarily algebraically closed. Also, using classical Schur's theory and Wedderburn's theory, we work out decomposition of induced representation of an irreducible $F$-representation of $H$, into irreducible components.