Hypercubes, Leonard triples and the anticommutator spin algebra
Hypercubes, Leonard triples and the anticommutator spin algebra
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发表时间：
2013-01
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通讯作者：
G. Brown
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作者：
G. Brown
本文涉及三类对象：伦纳德三元组、距离正则图和反换子自旋代数模块。令 $\K$ 表示特征为零的代数闭域。令 $V$ 表示 $\K$ 上具有有限正维数的向量空间。 $V$ 上的伦纳德三元组是 $\mathrm{End}(V)$ 中线性变换的有序三元组，使得对于每个变换都存在 $V$ 的基，表示该变换的矩阵是对角的，表示其他两个变换的矩阵是不可约三对角的。我们感兴趣的伦纳德三元组据说完全是 B/AB 类型，属于坂内/伊藤类型。 Bannai/Ito 类型的完全 B/AB 伦纳德三元组与反换子自旋代数 $\mathcal{A}$ 一起出现，单位结合 $\K$ 代数由生成元 $x,y,z$ 和关系 \[xy+yx=2z,\qquad yz+zy=2x,\qquad zx+xz=2y.\] 定义。\] 让 $D\geq0$ 表示一个整数，设 $Q_{D}$ 表示直径为 $D$ 的超立方体，并设 $\tilde{Q}_{D}$ 表示对映商。令$T$（分别为$\tilde{T}$）表示$Q_{D}$（分别为$\tilde{Q}_{D}$）的Terwilliger 代数。我们得到以下内容。当 $D$ 为偶数（或奇数）时，我们表明在 $Q_{D}$（或 $\tilde{Q}_{D}$）上存在唯一的 $\mathcal{A}$ 模块结构，使得 $x,y$ 分别充当邻接矩阵和对偶邻接矩阵。我们将所得的不可约 $\mathcal{A}$ 模块分类为同构。我们引入$Q_{D}$、$\tilde{Q}_{D}$的加权邻接矩阵。当$D$为偶数（或奇数）时，我们证明$Q_{D}$（分别为$\tilde{Q}_{D}$）的邻接矩阵、对偶邻接矩阵和加权邻接矩阵在任何不可约$T$-模块（分别为$\tilde{T}$-模块）上的动作形成Bannai/Ito类型的完全二分（或几乎二分）伦纳德三元组，并对伦纳德三倍同构。
This paper is about three classes of objects: Leonard triples, distance-regular graphs and the modules for the anticommutator spin algebra. Let $\K$ denote an algebraically closed field of characteristic zero. Let $V$ denote a vector space over $\K$ with finite positive dimension. A Leonard triple on $V$ is an ordered triple of linear transformations in $\mathrm{End}(V)$ such that for each of these transformations there exists a basis for $V$ with respect to which the matrix representing that transformation is diagonal and the matrices representing the other two transformations are irreducible tridiagonal. The Leonard triples of interest to us are said to be totally B/AB and of Bannai/Ito type. Totally B/AB Leonard triples of Bannai/Ito type arise in conjunction with the anticommutator spin algebra $\mathcal{A}$, the unital associative $\K$-algebra defined by generators $x,y,z$ and relations\[xy+yx=2z,\qquad yz+zy=2x,\qquad zx+xz=2y.\] Let $D\geq0$ denote an integer, let $Q_{D}$ denote the hypercube of diameter $D$ and let $\tilde{Q}_{D}$ denote the antipodal quotient. Let $T$ (resp. $\tilde{T}$) denote the Terwilliger algebra for $Q_{D}$ (resp. $\tilde{Q}_{D}$). We obtain the following. When $D$ is even (resp. odd), we show that there exists a unique $\mathcal{A}$-module structure on $Q_{D}$ (resp. $\tilde{Q}_{D}$) such that $x,y$ act as the adjacency and dual adjacency matrices respectively. We classify the resulting irreducible $\mathcal{A}$-modules up to isomorphism. We introduce weighted adjacency matrices for $Q_{D}$, $\tilde{Q}_{D}$. When $D$ is even (resp. odd) we show that actions of the adjacency, dual adjacency and weighted adjacency matrices for $Q_{D}$ (resp. $\tilde{Q}_{D}$) on any irreducible $T$-module (resp. $\tilde{T}$-module) form a totally bipartite (resp. almost bipartite) Leonard triple of Bannai/Ito type and classify the Leonard triple up to isomorphism.