伦纳德三元组和超立方体
Leonard triples and hypercubes
摘要：
令$V$表示C上具有有限正维度的向量空间。$V$上的{ \em Leonard三元组}是指$V$上的线性算子的有序三元组，使得对于这些算子中的每一个都存在$V$的基，关于该基，表示该算子的矩阵是对角的，并且表示其他两个算子的矩阵是不可约三对角的。令$D$表示正整数，$Q_D$表示$D$维超立方体的图.令$X$表示$Q_D$的顶点集，令$A$表示$Q_D$的邻接矩阵。固定$x \in X$，让$A^*$表示相应的对偶邻接矩阵。令$T$表示由$A, A^*$生成的$Mat_X(C)$的子代数。我们将$T$称为{ \em Terwilliger代数} $Q_D$ { \em关于} $x$。矩阵$A$和$A^*$通过以下事实相关：$2 \im A = A^* A^e - A^e A^*$和$2 \im A^* = A^e A - A A^e$，其中$2 \im A^e = A A^* - A^* A$和$\im^2=-1$。我们证明了三元组$A$，$A^*$，$A^e$作为Leonard三元组作用在每个不可约$T$ -模上.我们给出了这些伦纳德三元组的详细描述。
英文摘要：
Let $V$ denote a vector space over C with finite positive dimension. By a {\em Leonard triple} on $V$ we mean an ordered triple of linear operators on $V$ such that for each of these operators there exists a basis of $V$ with respect to which the matrix representing that operator is diagonal and the matrices representing the other two operators are irreducible tridiagonal. Let $D$ denote a positive integer and let $Q_D$ denote the graph of the $D$-dimensional hypercube. Let $X$ denote the vertex set of $Q_D$ and let $A$ denote the adjacency matrix of $Q_D$. Fix $x \in X$ and let $A^*$ denote the corresponding dual adjacency matrix. Let $T$ denote the subalgebra of $Mat_X(C)$ generated by $A, A^*$. We refer to $T$ as the {\em Terwilliger algebra of} $Q_D$ {\em with respect to} $x$. The matrices $A$ and $A^*$ are related by the fact that $2 \im A = A^* A^e - A^e A^*$ and $2 \im A^* = A^e A - A A^e$, where $2 \im A^e = A A^* - A^* A$ and $\im^2=-1$. We show that the triple $A$, $A^*$, $A^e$ acts on each irreducible $T$-module as a Leonard triple. We give a detailed description of these Leonard triples.
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