On a Conjecture about an analogue of Tokuyama's theorem for $G_2$

TL;DR

This paper proves a conjecture related to sums over Littelmann patterns for the $G_2$ root system, extending Tokuyama's theorem from type $A_r$ to $G_2$ using elementary polynomial identities.

Contribution

It establishes the conjecture of Friedlander et al. for the $G_2$ root system, providing a new analogue of Tokuyama's theorem with elementary proof techniques.

Findings

Confirmed the conjecture for $G_2$ root system

Reduced the proof to finite polynomial identities

Extended the scope of Tokuyama's theorem to $G_2$

Abstract

We prove the conjecture of Friedlander et al. about sums over Littelmann patterns for the the root system of type $G_2$, which is an analogue of Tokuyama's theorem for root systems of type $A_r$. We use elementary means to show that the conjecture is implied by a finite set of polynomial identities.