Seamus Albion: Cores, quotients and symmetric functions

Title: Cores, quotients and symmetric functions

Seamus Albion

Abstract: The Littlewood decomposition is a family of bijections on integer partitions, one for each $t\geq 2$, decomposing a partition into its $t$-core and $t$-quotient. This relatively simple construction, which can be thought of as a generalisation of integer division for partitions, has deep applications in modular representation theory, number theory, algebraic geometry and combinatorics. In (algebraic) combinatorics, it describes the factorisation and vanishing of Schur polynomials with variables twisted by a primitive $t$-th root of unity, a result which is due to Littlewood and Richardson. Much more recently, Lecouvey and, independently, Ayyer and Kumari, considered characters of the symplectic and orthogonal groups with similar twists by roots of unity, and proved vanishing and factorisation results. I will explain how, using new properties of the Littlewood decomposition for so-called $z$-asymmetric partitions, one can provide a uniform proof and extension of their results. Along the way I will discuss connections with plethysm and character values of the symmetric group, which have not yet been fully explored.