## Ernesto Vallejo - A diagrammatic approach to Kronecker squares

dmtcs:2416 - Discrete Mathematics & Theoretical Computer Science, January 1, 2014, DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014) - https://doi.org/10.46298/dmtcs.2416
A diagrammatic approach to Kronecker squares

Authors: Ernesto Vallejo 1

• 1 Centro de Ciencias Matematicas [Mexico]

In this paper we improve a method of Robinson and Taulbee for computing Kronecker coefficients and show that for any partition $\overline{ν}$ of $d$ there is a polynomial $k_{\overline{ν}}$ with rational coefficients in variables $x_C$, where $C$ runs over the set of isomorphism classes of connected skew diagrams of size at most $d$, such that for all partitions $\lambda$ of $n$, the Kronecker coefficient $\mathsf{g}(\lambda, \lambda, (n-d, \overline{ν}))$ is obtained from $k_{\overline{ν}}(x_C)$ substituting each $x_C$ by the number of $\lambda$-removable diagrams in $C$. We present two applications. First we show that for $\rho_{k} = (k, k-1,\ldots, 2, 1)$ and any partition $\overline{ν}$ of size $d$ there is a piecewise polynomial function $s_{\overline{ν}}$ such that $\mathsf{g}(\rho_k, \rho_k, (|\rho_k| - d, \overline{ν})) = s_{\overline{ν}} (k)$ for all $k$ and that there is an interval of the form $[c, \infty)$ in which $s_{\overline{ν}}$ is polynomial of degree $d$ with leading coefficient the number of standard Young tableaux of shape $\overline{ν}$. The second application is new stability property for Kronecker coefficients.

Volume: DMTCS Proceedings vol. AT, 26th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2014)
Section: Proceedings
Published on: January 1, 2014
Imported on: November 21, 2016
Keywords: Kronecker product,Young tableau,Schur function,Kostka number,Littlewood-Richardson rule,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]

## Linked publications - datasets - softwares

 Source : ScholeXplorer IsRelatedTo ARXIV 1304.0738 Source : ScholeXplorer IsRelatedTo DOI 10.1016/j.aim.2015.11.002 Source : ScholeXplorer IsRelatedTo DOI 10.48550/arxiv.1304.0738 10.1016/j.aim.2015.11.002 10.48550/arxiv.1304.0738 1304.0738 Kronecker products, characters, partitions, and the tensor square conjectures