Character correspondences for symmetric groups and wreath products

TL;DR

This paper proves a refined version of the Alperin--McKay conjecture for symmetric groups, establishes a canonical isometry between certain blocks, and generalizes character-theoretic results to arbitrary defect cases.

Contribution

It demonstrates the refined conjecture for all symmetric group blocks, identifies a canonical isometry, and extends character-theoretic results to arbitrary defect.

Findings

Refinement of Alperin--McKay conjecture holds for all symmetric group blocks.

Established a canonical isometry between principal blocks of symmetric groups and wreath products.

Generalized character-theoretic results to blocks with arbitrary defect.

Abstract

The Alperin--McKay conjecture relates irreducible characters of a block of an arbitrary finite group to those of its $p$-local subgroups. A refinement of this conjecture was stated by the author in a previous paper. We prove that this refinement holds for all blocks of symmetric groups. Along the way we identify a "canonical" isometry between the principal block of $S_{pw}$ and that of $S_p\wr S_w$. We also prove a general theorem on expressing virtual characters of wreath products in terms of certain induced characters. Much of the paper generalises character-theoretic results on blocks of symmetric groups with abelian defect and related wreath products to the case of arbitrary defect.