The fundamental module of $S_3$-symmetric tridiagonal algebra associated with cycles

Abstract

Terwilliger recently introduced the $S_3$-symmetric tridiagonal algebra, a generalization of the tridiagonal algebra. This algebra has six generators naturally associated with the vertices of a regular hexagon: adjacent generators satisfy the tridiagonal relations, while non-adjacent ones commute. To each $Q$-polynomial distance-regular graph $Γ$, we associate scalars $β, γ, γ^*, \varrho, \varrho^*$, and define the corresponding $S_3$-symmetric tridiagonal algebra $\mathbb{T} = \mathbb{T}(β, γ, γ^*, \varrho, \varrho^*)$. Let $V$ denote the standard module of $Γ$. Then the tensor $V^{\otimes 3} := V \otimes V \otimes V$ supports a $\mathbb{T}$-module structure, and within it exists a unique irreducible $\mathbb{T}$-submodule called the fundamental $\mathbb{T}$-module, denoted by $Λ$. In this paper, we focus on the case where $Γ$ is a cycle with vertex set $X$ and diameter $D$. We show that the associated scalars satisfy: \begin{align*} β= ζ+ ζ^{-1}, \quad γ= γ^* = 0, \quad \varrho = \varrho^* = -(ζ-ζ^{-1})^2, \end{align*} where $ζ$ is a fixed primitive $|X|$th root of unity. We prove that \begin{align*} \operatorname{dim}(Λ) & = \left\{\begin{array}{ll} \textstyle 2D^2+2 & \text{if } |X| \text{ is even},\\ \textstyle 2D^2 + 2D +1 & \text{if } |X| \text{ is odd}, \end{array} \right. \end{align*} and construct two explicit bases for $Λ$, each of which diagonalizes half of the generators of $\mathbb{T}$. Finally, we verify that Terwilliger's conjectures hold when $Γ$ is a cycle.