Maximal transitivity of the cactus group on standard Young tableaux

TL;DR

This paper investigates the transitivity properties of the cactus group acting on standard Young tableaux, revealing conditions for 2-transitivity and the nature of the image group, with implications for Galois groups in integrable systems.

Contribution

It characterizes when the cactus group action is 2-transitive on Young tableaux and determines the possible image groups, extending understanding of Galois groups in related mathematical models.

Findings

The action is 2-transitive iff the shape is not self-transpose and not a single hook.

The image of the cactus group is either the full symmetric or the alternating group.

Both cases occur infinitely often for different shapes.

Abstract

The action of the cactus group $C_n$ on Young tableaux of a given shape $\lambda$ goes back to Berenstein and Kirillov and arises naturally in the study of crystal bases and quantum integrable systems. We show that this action is $2$-transitive on standard Young tableaux of the shape $\lambda$ if and only if $\lambda$ is not self-transpose and not a single hook. Moreover, we show that in these cases, the image of the cactus group in the permutation group of standard Young tableaux is either the whole permutation group or the alternating group, and prove that both cases are possible for infinitely many $\lambda$ (though the alternating group is more frequent). As an application, this implies that the Galois group of solutions to the Bethe ansatz in the Gaudin model attached to the Lie group $GL_d$ is, in many cases, at least the alternating group. This also extends the results of Sottile and White on the multiple transitivity of the Galois group of Schubert calculus problems in Grassmannians to many new cases.