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

A combinatorial description of the crystal $\mathcal{B}(\mathrm{\infty })$ 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 $\mathcal{B}(\mathrm{\infty })$ 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.