Enumeration of Gelfand–Cetlin Type Reduced Words
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Published:2022-02-11
Issue:1
Volume:29
Page:
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ISSN:1077-8926
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Container-title:The Electronic Journal of Combinatorics
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language:
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Short-container-title:Electron. J. Combin.
Author:
Cho Yunhyung,Kim Jang Soo,Lee Eunjeong
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.
Publisher
The Electronic Journal of Combinatorics
Subject
Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics