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

Abstract

A combinatorial description of the crystal B(∞) 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 B(∞) 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.