Character correspondences above fully ramified sections and Schur indices

TL;DR

This paper proves that Isaacs' correspondence between certain invariant irreducible characters and characters of centralizers preserves Schur indices over the rationals, with applications to character correspondences in group theory.

Contribution

It demonstrates that Isaacs' character correspondence preserves Schur indices over the rationals and provides simplified proofs for known character correspondences above fully ramified sections.

Findings

Isaacs' correspondence preserves Schur indices over Q.

Established a canonical bijection between character sets of G and U preserving Schur indices.

Provided simplified, conceptual proofs of existing character correspondences.

Abstract

Let N be a finite group of odd order and A a finite group that acts on N such that the orders of N and A are coprime. Isaacs constructed a natural correspondence between the set Irr_A(N) of irreducible complex characters invariant under the action of A, and the irreducible characters of the centralizer of A in N, Irr(C_N(A)). We show that this correspondence preserves Schur indices over the rational numbers. Moreover, suppose that the semidirect product AN is a normal subgroup of the finite group G and set U= N_G(A). Let \chi \in Irr_A(N) and \chi* \in Irr(C_N(A)) correspond. Then there is a canonical bijection between Irr(G | \chi) and Irr(U | \chi*) preserving Schur indices. We also give simplified and more conceptual proofs of (known) character correspondences above fully ramified sections.