Crystal graphs, Tokuyama’s theorem, and the Gindikin–Karpelevič formula for $$G_{2}$$ G 2

Abstract

We conjecture a deformation of the Weyl character formula for type $$G_{2}$$G2 in the spirit of Tokuyama's formula for type $$A$$A. Using our conjecture, we prove a combinatorial version of the Gindikin---Karpelevia? formula for $$G_{2}$$G2, in the spirit of Bump---Nakasuji's formula for type $$A$$A.