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.