The big cactus group, invariant tensors, and rectangular tableaux
Abstract
The cactus group acts on tensor products of crystals and, through Bender--Knuth involutions, on semistandard Young tableaux. We show that the big cactus group J'n=π1Sn(M̅0,n(ℝ)), n≥3, is obtained from the cactus group by imposing n−1 Dynkin-reversal relations, which say that conjugation by the full reversal acts on the Bender--Knuth generators by the Dynkin diagram automorphism; the quotient map is induced by forgetting the distinguished marked point. We give a second presentation, in which these relations are replaced by the requirement that every reduced word for the longest element of Sn adapted to an orientation of An−1 represents the full reversal. These relations also hold on rectangular tableaux, where every adapted reduced word therefore acts as evacuation, and on invariant tensors: for finite normal crystals of any complex semisimple Lie algebra, the cactus action on invariants factors through J'n. We prove the latter combinatorially in type A and in general via Gaudin eigenline coverings. Haiman's promotion relation follows from a single adapted-word relation, but for n≥4 it is strictly weaker than the whole family.
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BibTeX
@misc{frieden2026cactus,
title = {{The big cactus group, invariant tensors, and rectangular tableaux}},
author = {Gabriel Frieden and Pavlo Pylyavskyy and Leonid Rybnikov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24842},
eprint = {2610.24842},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24842},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The big cactus group, invariant tensors, and rectangular tableaux
AU - Gabriel Frieden
AU - Pavlo Pylyavskyy
AU - Leonid Rybnikov
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24842
UR - https://arxiv.org/abs/2610.24842
AB - The cactus group acts on tensor products of crystals and, through Bender--Knuth involutions, on semistandard Young tableaux. We show that the big cactus group J'_n=π_1^S_n(M̅_0,n(ℝ)), n≥3, is obtained from the cactus group by imposing n−1 Dynkin-reversal relations, which say that conjugation by the full reversal acts on the Bender--Knuth generators by the Dynkin diagram automorphism; the quotient map is induced by forgetting the distinguished marked point. We give a second presentation, in which these relations are replaced by the requirement that every reduced word for the longest element of S_n adapted to an orientation of A_n−1 represents the full reversal. These relations also hold on rectangular tableaux, where every adapted reduced word therefore acts as evacuation, and on invariant tensors: for finite normal crystals of any complex semisimple Lie algebra, the cactus action on invariants factors through J'_n. We prove the latter combinatorially in type A and in general via Gaudin eigenline coverings. Haiman's promotion relation follows from a single adapted-word relation, but for n≥4 it is strictly weaker than the whole family.
ER -