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arXiv · 2211.06810

A Short Proof for the Polynomiality of the Stretched Littlewood-Richardson Coefficients

Warut Thawinrak
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Abstract

The stretched Littlewood-Richardson coefficient $c^{t\nu}_{t\lambda,t\mu}$ was conjectured by King, Tollu, and Toumazet to be a polynomial function in $t.$ It was shown to be true by Derksen and Weyman using semi-invariants of quivers. Later, Rassart used Steinberg's formula, the hive conditions, and the Kostant partition function to show a stronger result that $c^{\nu}_{\lambda,\mu}$ is indeed a polynomial in variables $\nu, \lambda, \mu$ provided they lie in certain polyhedral cones. Motivated by Rassart's approach, we give a short alternative proof of the polynomiality of $c^{t\nu}_{t\lambda,t\mu}$ using Steinberg's formula and a simple argument about the chamber complex of the Kostant partition function.

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Warut Thawinrak. 2022-11-13. A Short Proof for the Polynomiality of the Stretched Littlewood-Richardson Coefficients. https://arxiv.org/abs/2211.06810

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