Séminaire Lotharingien de Combinatoire, B77c (2017), 40 pp.

Jonah Blasiak

Kronecker Coefficients For One Hook Shape

Abstract. We give a positive combinatorial formula for the Kronecker coefficient g\lambda \mu(d) \nu for any partitions \lambda, \nu of n and hook shape \mu(d) := (n-d,1d). 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 \lambda and total color d (CYT\lambda,d) to be the set of mixed insertion tableaux of colored words w with exactly d barred letters and such that wblft is a Yamanouchi word of content \lambda, where wblft is the ordinary word formed from w by shuffling its barred letters to the left and then removing their bars. We prove that g\lambda \mu(d) \nu is equal to the number of CYT\lambda,d of shape \nu with unbarred southwest corner.

Received: December 1, 2016. Accepted: December 2, 2016. Final version: December 6, 2016.

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