Kronecker coefficients for one hook shape

TL;DR

This paper provides a positive combinatorial formula for specific Kronecker coefficients involving hook shapes, using a generalized insertion algorithm and colored tableaux, advancing understanding in algebraic combinatorics.

Contribution

It introduces a combinatorial formula for Kronecker coefficients with hook shapes using mixed insertion and colored tableaux, extending previous methods.

Findings

Provides a combinatorial interpretation for Kronecker coefficients g_{lambda mu(d) nu}

Defines colored Yamanouchi tableaux and relates their count to the coefficients

Uses Haiman's mixed insertion to generalize Schensted insertion for this purpose

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 \emph{mixed insertion}. This is a generalization of Schensted insertion to \emph{colored words}, words in the alphabet of barred letters \bar{1},\bar{2},... and unbarred letters 1,2,.... We define the set of \emph{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.