1Department of Mathematics Education, Sungkyunkwan University 2Department of Mathematics, Sungkyunkwan University 3Center for Geometry and Physics, Institute for Basic Science (IBS)
The electronic journal of combinatorics, Tome 29 (2022) no. 1
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.
Classification :
05A15, 05E14, 05E10, 06A07
Mots-clés :
Young tableaux of shifted shape, commutation classes, string polytopes: Gelfand-Cetlin polytope
Affiliations des auteurs :
Yunhyung Cho
1
;
Jang Soo Kim
2
;
Eunjeong Lee
3
1
Department of Mathematics Education, Sungkyunkwan University
2
Department of Mathematics, Sungkyunkwan University
3
Center for Geometry and Physics, Institute for Basic Science (IBS)
@article{10_37236_10071,
author = {Yunhyung Cho and Jang Soo Kim and Eunjeong Lee},
title = {Enumeration of {Gelfand-Cetlin} type reduced words},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {1},
doi = {10.37236/10071},
zbl = {1486.05013},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10071/}
}
TY - JOUR
AU - Yunhyung Cho
AU - Jang Soo Kim
AU - Eunjeong Lee
TI - Enumeration of Gelfand-Cetlin type reduced words
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/10071/
DO - 10.37236/10071
ID - 10_37236_10071
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%0 Journal Article
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%A Jang Soo Kim
%A Eunjeong Lee
%T Enumeration of Gelfand-Cetlin type reduced words
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/10071/
%R 10.37236/10071
%F 10_37236_10071
Yunhyung Cho; Jang Soo Kim; Eunjeong Lee. Enumeration of Gelfand-Cetlin type reduced words. The electronic journal of combinatorics, Tome 29 (2022) no. 1. doi: 10.37236/10071
