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

TL;DR

This paper studies the structure of the fundamental module of an $S_3$-symmetric tridiagonal algebra linked to cycles, providing explicit bases, dimension formulas, and confirming conjectures for cycle graphs.

Contribution

It explicitly characterizes the fundamental module of the algebra for cycle graphs, including dimension, bases, and validation of Terwilliger's conjectures.

Findings

Dimension formulas depend on cycle length parity.

Constructed explicit bases diagonalizing half of the generators.

Confirmed Terwilliger's conjectures for cycle graphs.

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 $\Gamma$, 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.