Kronecker Coefficients For One Hook Shape

Jonah Blasiak

Séminaire lotharingien de combinatoire, Tome 77 (2017-2018)

Résumé

We give a positive combinatorial formula for the Kronecker coefficient g_{λ μ(d) ν} for any partitions λ, ν of n and hook shape μ(d) := (n-d,1^d). Our main tool is Haiman's mixed insertion. This is a generalization of Schensted insertion to colored words, words in the alphabet of barred letters 1^-, 2^-, ... and unbarred letters 1, 2, ... We define the set of colored Yamanouchi tableaux of content λ and total color d (CYT_λ,d) to be the set of mixed insertion tableaux of colored words w with exactly d barred letters and such that w^blft is a Yamanouchi word of content λ, where w^blft is the ordinary word formed from w by shuffling its barred letters to the left and then removing their bars. We prove that g_{λ μ(d) ν} is equal to the number of CYT_λ,d of shape ν with unbarred southwest corner.