Critical classes, Kronecker products of spin characters, and the Saxl conjecture

Bessenrodt, Christine

Algebraic Combinatorics, Tome 1 (2018) no. 3, pp. 353-369.

Using critical conjugacy classes, we find a new criterion for constituents in Kronecker products of spin characters of the double covers of the symmetric and alternating groups. This is applied together with earlier results on spin characters to obtain constituents in Kronecker products of characters of the symmetric groups. Via this tool, we make progress on the Saxl conjecture; this claims that for a triangular number $n$, the square of the irreducible character of the symmetric group $S_n$ labelled by the staircase contains all irreducible characters of $S_n$ as constituents. With the new criterion we deduce a large number of constituents in this square which were not detected by other methods, notably all double-hooks. The investigation of Kronecker products of spin characters also inspires a spin variant of Saxl’s conjecture.

Classification : 20C30, 05E15
Mots clés : symmetric groups, double cover groups, characters, hook character, spin characters, Kronecker products, Saxl conjecture, unimodal sequences

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