Character factorisations, $z$-asymmetric partitions and plethysm

TL;DR

This paper generalizes character factorizations related to symmetric functions, partitions, and group characters using a new framework involving $z$-asymmetric partitions, plethysm, and core-quotient combinatorics.

Contribution

It introduces a unified approach to character factorizations for various groups through $z$-asymmetric partitions and extends classical core-quotient theory.

Findings

New characterization of $t$-cores and $t$-quotients for $z$-asymmetric partitions

Generalization of character factorizations involving symmetric functions, plethysm, and group characters

Connections established between core-quotient combinatorics and symmetric group characters

Abstract

The Verschiebung operators $\varphi_t $ are a family of endomorphisms on the ring of symmetric functions, one for each integer $t\geq2$. Their action on the Schur basis has its origins in work of Littlewood and Richardson, and is intimately related with the decomposition of a partition into its $t$-core and $t$-quotient. Namely, they showed that the action on $s_\lambda$ is zero if the $t$-core of the indexing partition is nonempty, and otherwise it factors as a product of Schur functions indexed by the $t$-quotient. Much more recently, Lecouvey and, independently, Ayyer and Kumari have provided similar formulae for the characters of the symplectic and orthogonal groups, where again the combinatorics of cores and quotients plays a fundamental role. We embed all of these character factorisations in an infinite family involving an integer $z$ and parameter $q$ using a very general symmetric function defined by Hamel and King. The proof hinges on a new characterisation of the $t$-cores and $t$-quotients of $z$-asymmetric partitions which generalise the well-known classifications for self-conjugate and doubled distinct partitions. We also explain the connection between these results, plethysms of symmetric functions and characters of the symmetric group.