Enumeration of Gelfand-Cetlin type reduced words (Q2121754)

Summary: 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.