Schur Index and Extensions of Witt-Berman's Theorems

TL;DR

This paper extends classical theorems on the representation theory of finite groups over fields, providing new proofs, explicit formulas, and generalizations for primitive central idempotents and induced representations.

Contribution

It offers a natural definition of $F$-conjugacy, an explicit formula for primitive central idempotents, and extends Berman's theorem to non-algebraically closed fields.

Findings

Provided a proof of Witt-Berman theorem with a natural $F$-conjugacy definition.

Derived an explicit formula for primitive central idempotents from the $F$-character table.

Extended Berman's theorem to fields not necessarily algebraically closed.

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.