Character correspondences above fully ramified sections and Schur indices
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 IrrA(N) of irreducible complex characters invariant under the action of A, and the irreducible characters of the centralizer of A in N, Irr(CN(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= NG(A). Let ∈ IrrA(N) and * ∈ Irr(CN(A)) correspond. Then there is a canonical bijection between Irr(G | ) and Irr(U | *) preserving Schur indices. We also give simplified and more conceptual proofs of (known) character correspondences above fully ramified sections.
Turn this paper into a lesson
ArcXiv compiles a structured reading guide from this paper's metadata: plain-English importance, contributions, prerequisite concepts, which sections to read first, flashcards, and a quiz. Grounded in the abstract, never invented.