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,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.Comment: 37 pages, 3 figure

Promotion and cyclic sieving on families of SSYT

We examine a few families of semistandard Young tableaux, for which we observe the cyclic sieving phenomenon under promotion. The first family we consider consists of stretched hook shapes, where we use the cocharge generating polynomial as CSP-polynomial. The second family we consider consists of skew shapes, consisting of rectangles. Again, the charge generating polynomial together with promotion exhibits the cyclic sieving phenomenon. This generalizes earlier result by B. Rhoades and later B. Fontaine and J. Kamnitzer. Finally, we consider certain skew ribbons, where promotion behaves in a predictable manner. This result is stated in form of a bicyclic sieving phenomenon. One of the tools we use is a novel method for computing charge of skew semistandard tableaux, in the case when every number in the tableau occur with the same frequency