Alcove paths and Gelfand-Tsetlin patterns

TL;DR

This paper establishes an explicit formula linking alcove paths and Gelfand-Tsetlin patterns, providing a combinatorial understanding of crystal isomorphisms in type A Lie groups.

Contribution

It introduces a simple, explicit formula for the crystal isomorphism between alcove paths and Gelfand-Tsetlin patterns in type A, extending combinatorial models for Lie theory.

Findings

Derived an explicit formula for crystal isomorphism

Connected alcove paths with Gelfand-Tsetlin patterns

Enhanced combinatorial understanding of type A crystals

Abstract

In their study of the equivariant K-theory of the generalized flag varieties $G/P$, where $G$ is a complex semisimple Lie group, and $P$ is a parabolic subgroup of $G$, Lenart and Postnikov introduced a combinatorial tool, called the alcove paths model. It provides a model for the highest weight crystals with dominant integral highest weights, generalizing the model by semistandard Young tableaux. In this paper, we prove a simple and explicit formula describing the crystal isomorphism between the alcove paths model and the Gelfand-Tsetlin patterns model for type $A$.