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Foulkes characters: deflations, twists and algorithms

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Conference: Representations of Symmetric Groups, Hecke Algebras and KLR Algebras

Let φ(mn) be the permutation character of the symmetric group Smn acting on the set partitions of {1,...,mn} into n sets each of size m. A fundamental problem in algebraic combinatorics asks for the decomposition of φ(mn) into irreducible characters. Closely related is Foulkes’ Conjecture, which states that φ(mn) contains φ(nm) whenever m_≤_n.

My talk will begin with a deflation map that sends characters of Smn to characters of Sn, via characters of the wreath product of Sm by Sn. This deflation map leads to a new algorithm for decomposing Foulkes characters that has been used to verify Foulkes’ Conjecture when m+n≤19. I will then state a combinatorial rule giving the values of the deflations of irreducible characters of Smn and show that this rule generalises the Murnaghan–Nakayama rule and Young’s rule. This is joint work with Anton Evseev and Rowena Paget.

For each partition ν of n there is a twisted analogue of the Foulkes character φ(mn) corresponding to the plethysm of the Schur functors Δν and Symm. These characters may also be decomposed using the deflation maps. I will end with a recent result, obtained in joint work with Rowena Paget, that uses the colexicographic order on sets to determine all minimal irreducible constituents of twisted Foulkes characters.

This talk is part of the Algebra seminar series.

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