标题： 与循环相关的$S_3$-对称三对角代数的基本模 显示英文标题

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

摘要： 特里威利格最近引入了$S_3$对称的三对角代数，这是三对角代数的一个推广。 这个代数有六个生成元，自然与正六边形的顶点相关联：相邻的生成元满足三对角关系，而非相邻的则交换。 对于每个$Q$多项式距离正则图$\Gamma$，我们关联标量 $\beta, \gamma, \gamma^*, \varrho, \varrho^*$，并定义相应的 $S_3$对称的三对角代数$\mathbb{T} = \mathbb{T}(\beta, \gamma, \gamma^*, \varrho, \varrho^*)$。 设$V$表示$\Gamma$的标准模。 然后张量$V^{\otimes 3} := V \otimes V \otimes V$支持一个$\mathbb{T}$-模结构，在其中存在一个唯一的不可约$\mathbb{T}$-子模，称为基本$\mathbb{T}$-模，记为$\Lambda$。 在本文中，我们关注$\Gamma$是一个顶点集为$X$且直径为$D$的环的情况。 我们证明相关的标量满足： \begin{align*} \beta = \zeta + \zeta^{-1}, \quad \gamma = \gamma^* = 0, \quad \varrho = \varrho^* = -(\zeta-\zeta^{-1})^2, \end{align*}其中$\zeta$是一个固定的原始$|X|$次单位根。 我们证明了\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*}并为$\Lambda$构造了两个显式基，每个基都对角化了$\mathbb{T}$的一半生成元。最后，我们验证当$\Gamma$是一个环时，Terwilliger 的猜想成立。 显示英文摘要

摘要： 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.