On the action of the cactus group on the set of Gelfand-Tsetlin patterns for orthogonal Lie algebras

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.