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

TL;DR

This paper provides a concise proof demonstrating that the stretched Littlewood-Richardson coefficients are polynomial functions in the parameter t, building on previous work with a new approach using Steinberg's formula.

Contribution

It offers a short, alternative proof of polynomiality for stretched Littlewood-Richardson coefficients using Steinberg's formula and chamber complex arguments.

Findings

Confirmed polynomiality of $c^{t u}_{tla, la}$ using a simplified approach.

Utilized Steinberg's formula and chamber complex analysis.

Provided a more accessible proof compared to prior methods.

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.