Enumeration of Gelfand-Cetlin type reduced words

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 w0 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 = SLn+1(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 w0 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.