On the action of the cactus group on the set of Gelfand-Tsetlin patterns for orthogonal Lie algebras
Igor Svyatnyy
TL;DR
The paper constructs a natural cactus group action on the indexing sets of Gelfand-Tsetlin patterns for orthogonal Lie algebras by realizing $S^{\otimes N}$, with $S$ the spinor representation of $\frak{o}_{2n}$, as a module under $O_N\times \frak{o}_{2n}$ via Howe duality. Central to the approach are the regular cell diagrams $D^N_{\lambda}$, which biject with highest weights $\Delta^N$, and the bijection with short Young diagrams $\nu$ that encode the $O_N$-module structure of multiplicity spaces $U_{\lambda}$. Three complementary bases index these spaces: the principal basis $\mathcal{PR}(U_{\lambda})$, the GT-basis $GT(U_{\lambda})$, and the GT-patterns $\mathfrak{GT}(U_{\lambda})$, with natural bijections linking their indexing sets via regular cell tables $\mathfrak{Ctab}(D^N_{\lambda})$ and semi-standard short Young tableaux $SSSYT(\nu)$. By transporting the crystal commutor through these bijections, the cactus group $C_N$ acts on regular cell tables, hence on Gelfand-Tsetlin patterns and related tableaux, laying groundwork for explicit descriptions of the action and its combinatorial reflection. This framework unifies representation-theoretic multiplicities, combinatorial cell diagrams, and crystal-theoretic coboundary structures to produce a coherent, actionable picture of cactus-group symmetries in the orthogonal GT setting.
Abstract
The purpose of this work is to define a natural action of the cactus group on the set of Gelfand-Tsetlin patterns for orthogonal Lie algebras. These Gelfand-Tsetlin patterns are meant to index the Gelfand-Tsetlin basis in the irreducible representations of the orthogonal Lie algebra $\mathfrak{o}_N$ with respect to the chain of nested orthogonal Lie algebras $\mathfrak{o}_N \supset \mathfrak{o}_{N-1} \supset \ldots \supset \mathfrak{o}_3$. Using the Howe duality between $O_N$ and $\mathfrak{o}_{2n}$, we realize some representations of $\mathfrak{o}_N$ as multiplicity spaces inside the tensor power of the spinor representation $(Λ\mathbb{C}^{n})^{\otimes N}$. There is a natural choice of the basis inside the multiplicity space, which agrees with the decomposition of $(Λ\mathbb{C}^{n})^{\otimes N}$ into simple $\mathfrak{o}_{2n}$-modules. We call such basis principal. The action of the cactus group $C_N$ by the crystal commutors on the crystal arising from $(Λ\mathbb{C}^{n})^{\otimes N}$ induces the action of $C_N$ on the set indexing the principal basis inside the multiplicity space. We call this set regular cell tables. Regular cell tables are the analog of semi-standard Young tables. There is a natural bijection between a specific subset of semi-standard Young tables and regular cell tables. In this paper, we establish a natural bijection between the principal basis and the Gelfand-Tsetlin basis and, therefore, define an action of the cactus group on the set Gelfand-Tsetlin patterns.