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

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 $\Gamma$, we associate scalars $\beta, \gamma, \gamma^*, \varrho, \varrho^*$, and define the corresponding $S_3$-symmetric tridiagonal algebra $\mathbb{T} = \mathbb{T}(\beta, \gamma, \gamma^*, \varrho, \varrho^*)$. Let $V$ denote the standard module of $\Gamma$. 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 $\Lambda$. In this paper, we focus on the case where $\Gamma$ is a cycle with vertex set $X$ and diameter $D$. We show that the associated scalars satisfy: \begin{align*} \beta = \zeta + \zeta^{-1}, \quad \gamma = \gamma^* = 0, \quad \varrho = \varrho^* = -(\zeta-\zeta^{-1})^2, \end{align*} where $\zeta$ is a fixed primitive $|X|$th root of unity. We prove that \begin{align*} \operatorname{dim}(\Lambda) & = \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 $\Lambda$, each of which diagonalizes half of the generators of $\mathbb{T}$. Finally, we verify that Terwilliger's conjectures hold when $\Gamma$ is a cycle.