Kronecker coefficients for one hook shape

2012/09/10 by Jonah Blasiak, Blasiak, Jonah · 3 citations
Mathematics · #05E10 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1209.2018

openalex publication_date 2012/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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,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 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 w^blft is a Yamanouchi word of content lambda, 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_{lambda mu(d) nu} is equal to the number of CYT_{lambda, d} of shape nu with unbarred southwest corner.

Cited by

• The problem of deciding the positivity of Kronecker coefficients and Saxl conjecture
• Permanent versus determinant, obstructions, and Kronecker coefficients
• Kronecker Coefficients, Crystals, and Bitableaux

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