The berenstein-kirillov group and cactus groups

Michael S Chmutov, Max Glick, Pavlo Pylyavskyy

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


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 languageEnglish (US)
Pages (from-to)111-140
Number of pages30
JournalJournal of Combinatorial Algebra
Issue number2
StatePublished - 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.


  • Bender-Knuth involutions
  • Berenstein-Kirillov group
  • Cactus group


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