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

Terwilliger (in paper from 2025) recently introduced the S₃-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 S₃-symmetric tridiagonal algebra T = T(β, γ, γ*, \varrho, \varrho*). Let V denote the standard module of Γ. Then the tensor V^(⊗3) := V ⊗ V ⊗ V supports a T-module structure, and within it exists a unique irreducible T-submodule called the fundamental 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:β = ζ + ζ⁻¹, γ = γ* = 0, \varrho = \varrho* = −(ζ − ζ⁻¹)²,where ζ is a fixed primitive |X|th root of unity. We prove that dim(Λ) = 2D2+2 & if |X| is even,2D2 + 2D +1 & if |X| is odd, and construct two explicit bases for Λ, each of which diagonalizes half of the generators of T. Finally, we verify that Terwilliger’sconjectures (in paper from 2025) hold when Γ is a cycle.