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

The stretched Littlewood-Richardson coefficient $c_{t\lambda ,t\mu }^{t\nu }$ was conjectured by King, Tollu, and Toumazet to be a polynomial function in $t\mathrm{.}$ 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 $c_{\lambda ,\mu }^{\nu }$ 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 $c_{t\lambda ,t\mu }^{t\nu }$ using Steinberg’s formula and a simple argument about the chamber complex of the Kostant partition function.