The Berenstein-Kirillov group and cactus groups

TL;DR

This paper explores the relationship between the Berenstein-Kirillov group and cactus groups, showing that the former is a quotient of the latter and deriving new relations and presentations connecting the two structures.

Contribution

It establishes a formal connection between the Berenstein-Kirillov group and cactus groups, providing new relations and a presentation for cactus groups via Bender-Knuth generators.

Findings

Berenstein-Kirillov group is a quotient of the cactus group

Derived new relations in the Berenstein-Kirillov group

Presented cactus groups in terms of Bender-Knuth generators

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.