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

TL;DR

This paper provides a combinatorial rule for the Gindikin-Karpelevich formula in type A Lie algebras using Young tableaux, MV polytopes, and quiver varieties, linking algebraic and geometric perspectives.

Contribution

It introduces a new combinatorial description of the Gindikin-Karpelevich formula in type A, connecting crystal bases with Young tableaux, MV polytopes, and quiver varieties.

Findings

Derived a combinatorial rule for the Gindikin-Karpelevich formula in type A.

Connected crystal bases with Young tableaux, MV polytopes, and quiver varieties.

Provided a unified combinatorial and geometric interpretation.

Abstract

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.