A combinatorial description of the Gindikin-Karpelevich formula in type A

A combinatorial description of the crystal $\mathcal{B}(\infty)$ for finite-dimensional simple Lie algebras in terms of Young tableaux was developed by J. Hong and H. Lee. Using this description, we obtain a combinatorial rule for expressing the Gindikin-Karpelevich formula as a sum over $\mathcal{B}(\infty)$ when the underlying Lie algebra is of type A. We also interpret our description in terms of MV polytopes and irreducible components of quiver varieties.Comment: 17 pages. Edited using comments from referee

Young tableaux, canonical bases and the Gindikin-Karpelevich formula

A combinatorial description of the crystal B(infinity) for finite-dimensional simple Lie algebras in terms of certain Young tableaux was developed by J. Hong and H. Lee. We establish an explicit bijection between these Young tableaux and canonical bases indexed by Lusztig's parametrization, and obtain a combinatorial rule for expressing the Gindikin-Karpelevich formula as a sum over the set of Young tableaux.Comment: 19 page

The flush statistic on semistandard Young tableaux

In this note, a statistic on Young tableaux is defined which encodes data needed for the Casselman-Shalika formula.Comment: 6 page

Connecting marginally large tableaux and rigged configurations via crystals

We show that the bijection from rigged configurations to tensor products of Kirillov-Reshetikhin crystals extends to a crystal isomorphism between the $B(\infty)$ models given by rigged configurations and marginally large tableaux.Comment: 22 pages, 3 figure