Maximal transitivity of the cactus group on standard Young tableaux

TL;DR

The paper identifies maximal transitivity levels for the cactus group $C_n$ acting on standard Young tableaux of shape $oldsymbol{λ}$, proving transitivity in all cases and 2-transitivity except for hook-shaped or self-transpose shapes. It shows that for non-hook, non-self-transpose shapes the action is so large that the image in the symmetric group on $ ext{SYT}(oldsymbol{λ})$ is either $S_N$ or $A_N$, with both occurring for infinitely many shapes. These results yield substantial Galois-group consequences: in many Gaudin-model Bethe ansatz problems and Schubert calculus settings, the monodromy/Galois group is at least alternating, extending prior work on multiple transitivity. The work connects crystal-theoretic cactus actions to monodromy, providing a unified framework for transitivity results and explicit exclusions of exceptional finite groups, with numerical evidence suggesting $A_N$-dominance in large regimes.

Abstract

The action of the cactus group $C_n$ on Young tableaux of a given shape $λ$ 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 $λ$ if and only if $λ$ 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 $λ$ (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.

Figures (2)

• Figure 1: The number of partitions $\lambda$ such that the image of $\text{SYT}(\lambda)$ under the cactus group is $S_N$ or $A_N$. The first graph considers all possible partitions, whereas the second only considers generic partitions (those that fit into a $2\sqrt2\cdot \sqrt{|\lambda|}$ by $2\sqrt2\cdot \sqrt{|\lambda|}$ grid).
• Figure 2: Describes how even $|\text{SYT}(\lambda)|$ is for two parameters of $\lambda$: the number of boxes and the maximum of its height and width. Each square on the plot represents all partitions of a given size ($x$-axis) and a given $k$ where $\lambda\subseteq k\times k$ but $\lambda\not\subseteq (k-1)\times(k-1)$ ($y$-axis). The square is assigned a color based on the maximum power of 2 which divides $|\text{SYT}(\lambda)|$ for all such $\lambda$ considered by these two criteria. For example, every partition of 32 boxes which fit into a $16\times 16$ grid have an even number of standard Young tableaux, but there is at least one 32-box partition whose height or width is 17 that has an odd number of standard Young tableaux.

Theorems & Definitions (54)

• Proposition 2.1
• Theorem 2.1
• proof
• Definition
• Example 3.1
• Definition
• Lemma 3.1
• proof
• Lemma 3.2
• proof