Séminaire Lotharingien de Combinatoire, 93B.93 (2025), 12 pp.
Ming Yean Lim
The Number of Irreducibles in the Plethysm sλ[sm]
Abstract.
We give a formula for the number of irreducibles (with multiplicity) in the decomposition of the plethysm sλ[sm] of Schur functions in terms of the number of lattice points in certain rational polytopes. In the case where λ = n consists of a single part, we will give a combinatorial interpretation of this number as the cardinality of a set of matrices modulo permutation equivalence. This is also the setting of Foulkes' conjecture, and our results allow us to state a weaker version that only involves comparing the cardinalities of such sets, rather than the multiplicities of irreducible representations.
Received: November 15, 2024.
Accepted: February 15, 2025.
Final version: April 1, 2025.