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

2022/11/13 by Thawinrak, Warut
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2211.06810

Abstract

The stretched Littlewood-Richardson coefficient c^tν_tλ,tμ 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^ν_λ,μ is indeed a polynomial in variables ν, λ, μ provided they lie in certain polyhedral cones. Motivated by Rassart's approach, we give a short alternative proof of the polynomiality of c^tν_tλ,tμ using Steinberg's formula and a simple argument about the chamber complex of the Kostant partition function.

Related

• A polynomiality property for Littlewood-Richardson coefficients
• Positivity of stretched Littlewood-Richardson coefficients for partitions of length at most four
• Stretched Newell-Littlewood coefficients
• Geometric Proof of a Conjecture of King, Tollu, and Toumazet
• Quiver Generalization of a Conjecture of King, Tollu, and Toumazet