Séminaire Lotharingien de Combinatoire, B77c (2017), 40 pp.
Kronecker Coefficients For One Hook Shape
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
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
We prove that g\lambda
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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