Enumeration of Gelfand–⁠Cetlin Type Reduced Words

Abstract

The combinatorics of reduced words and their commutation classes plays an important role in geometric representation theory. For a semisimple complex Lie group $G$, a string polytope is a convex polytope associated with each reduced word of the longest element $w_0$ in the Weyl group of $G$ encoding the character of a certain irreducible representation of $G$. In this paper, we deal with the case of type $A$, i.e., $G = \mathrm{SL}_{n+1}(\mathbb{C})$. A Gelfand–⁠Cetlin polytope is one of the most famous examples of string polytopes of type $A$. We provide a recursive formula enumerating reduced words of $w_0$ such that the corresponding string polytopes are combinatorially equivalent to a Gelfand–⁠Cetlin polytope. The recursive formula involves the number of standard Young tableaux of shifted shape. We also show that each commutation class is completely determined by a list of quantities called indices.