Universal characters twisted by roots of unity

TL;DR

This paper extends classical and recent factorisations of characters in symmetric and classical groups to universal characters, simplifying proofs and revealing clearer structural insights.

Contribution

It lifts known character factorisations to the universal level, unifying and simplifying the proofs while extending to new cases.

Findings

Unified factorisations for universal characters of classical groups

Simplified proofs of character factorisations

Extended factorisations to new classes of characters

Abstract

A classical result of Littlewood gives a factorisation for the Schur function at a set of variables "twisted" by a primitive $t$-th root of unity, characterised by the core and quotient of the indexing partition. While somewhat neglected, it has proved to be an important tool in the character theory of the symmetric group, the cyclic sieving phenomenon, plethysms of symmetric functions and more. Recently, similar factorisations for the characters of the groups $\mathrm{O}(2n,\mathbb{C})$, $\mathrm{Sp}(2n,\mathbb{C})$ and $\mathrm{SO}(2n+1,\mathbb{C})$ were obtained by Ayyer and Kumari. We lift these results to the level of universal characters, which has the benefit of making the proofs simpler and the structure of the factorisations more transparent. Our approach also allows for universal character extensions of some factorisations of a different nature originally discovered by Ciucu and Krattenthaler, and generalised by Ayyer and Behrend.