The berenstein-kirillov group and cactus groups

Michael S Chmutov; Max Glick; Pavlo Pylyavskyy

doi:10.4171/JCA/36

The berenstein-kirillov group and cactus groups

Michael S Chmutov, Max Glick, Pavlo Pylyavskyy

School of Mathematics

Research output: Contribution to journal › Article › peer-review

7 Scopus citations

Abstract

Berenstein and Kirillov have studied the action of Bender-Knuth moves on semistandard tableaux. Losev has studied a cactus group action in Kazhdan-Lusztig theory; in type A this action can also be identified in the work of Henriques and Kamnitzer. We establish the relationship between the two actions. We show that the Berenstein-Kirillov group is a quotient of the cactus group. We use this to derive previously unknown relations in the Berenstein-Kirillov group. We also determine precise implications between subsets of relations in the two groups, which yields a presentation for cactus groups in terms of Bender-Knuth generators.

Original language	English (US)
Pages (from-to)	111-140
Number of pages	30
Journal	Journal of Combinatorial Algebra
Volume	4
Issue number	2
DOIs	https://doi.org/10.4171/JCA/36
State	Published - 2020

Bibliographical note

Funding Information:
M.C. was partially supported by NSF grant DMS-1503119. M. G. was partially supported by NSF grant DMS-1303482. P. P. was partially supported by NSF grants DMS-1148634, DMS-1351590, and Sloan Fellowship.

Publisher Copyright:
© 2020 European Mathematical Society.

Keywords

Bender-Knuth involutions
Berenstein-Kirillov group
Cactus group

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10.4171/JCA/36

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title = "The berenstein-kirillov group and cactus groups",

abstract = "Berenstein and Kirillov have studied the action of Bender-Knuth moves on semistandard tableaux. Losev has studied a cactus group action in Kazhdan-Lusztig theory; in type A this action can also be identified in the work of Henriques and Kamnitzer. We establish the relationship between the two actions. We show that the Berenstein-Kirillov group is a quotient of the cactus group. We use this to derive previously unknown relations in the Berenstein-Kirillov group. We also determine precise implications between subsets of relations in the two groups, which yields a presentation for cactus groups in terms of Bender-Knuth generators.",

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AU - Chmutov, Michael S

AU - Glick, Max

AU - Pylyavskyy, Pavlo

N1 - Funding Information: M.C. was partially supported by NSF grant DMS-1503119. M. G. was partially supported by NSF grant DMS-1303482. P. P. was partially supported by NSF grants DMS-1148634, DMS-1351590, and Sloan Fellowship. Publisher Copyright: © 2020 European Mathematical Society.

PY - 2020

Y1 - 2020

N2 - Berenstein and Kirillov have studied the action of Bender-Knuth moves on semistandard tableaux. Losev has studied a cactus group action in Kazhdan-Lusztig theory; in type A this action can also be identified in the work of Henriques and Kamnitzer. We establish the relationship between the two actions. We show that the Berenstein-Kirillov group is a quotient of the cactus group. We use this to derive previously unknown relations in the Berenstein-Kirillov group. We also determine precise implications between subsets of relations in the two groups, which yields a presentation for cactus groups in terms of Bender-Knuth generators.

AB - Berenstein and Kirillov have studied the action of Bender-Knuth moves on semistandard tableaux. Losev has studied a cactus group action in Kazhdan-Lusztig theory; in type A this action can also be identified in the work of Henriques and Kamnitzer. We establish the relationship between the two actions. We show that the Berenstein-Kirillov group is a quotient of the cactus group. We use this to derive previously unknown relations in the Berenstein-Kirillov group. We also determine precise implications between subsets of relations in the two groups, which yields a presentation for cactus groups in terms of Bender-Knuth generators.

KW - Bender-Knuth involutions

KW - Berenstein-Kirillov group

KW - Cactus group

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The berenstein-kirillov group and cactus groups

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