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arXiv · 1403.5327

Kronecker Coefficients For Some Near-Rectangular Partitions

Vasu V. Tewari
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Abstract

We give formulae for computing Kronecker coefficients occurring in the expansion of $s_μ*s_ν$, where both $μ$ and $ν$ are nearly rectangular, and have smallest parts equal to either 1 or 2. In particular, we study $s_{(n,n-1,1)}*s_{(n,n)}$, $s_{(n-1,n-1,1)}*s_{(n,n-1)}$, $s_{(n-1,n-1,2)}*s_{(n,n)}$, $s_{(n-1,n-1,1,1)}*s_{(n,n)}$ and $s_{(n,n,1)}*s_{(n,n,1)}$. Our approach relies on the interplay between manipulation of symmetric functions and the representation theory of the symmetric group, mainly employing the Pieri rule and a useful identity of Littlewood. As a consequence of these formulae, we also derive an expression enumerating certain standard Young tableaux of bounded height, in terms of the Motzkin and Catalan numbers.

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Vasu V. Tewari. 2014-03-21. Kronecker Coefficients For Some Near-Rectangular Partitions. https://arxiv.org/abs/1403.5327

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