Leonard triples and hypercubes

Miklavič, Štefko

Journal of Algebraic Combinatorics, Tome 28 (2008) no. 3, pp. 397-424.

Résumé

Summary: Let $V$ denote a vector space over $\Bbb C$ with finite positive dimension. By a 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 $A$ Ĩ Mat $_{ X}(\mathbb C)$ A$\in {\rm $Mat_X(mathbbC) denote the adjacency matrix of $Q _{ D }$. Fix $x\in X$ and let $A ^{*}$ Ĩ Mat $_{ X}(\mathbb C)$ A^*$\in {\rm $Mat_X(mathbbC) denote the corresponding dual adjacency matrix. Let $T$ denote the subalgebra of Mat $_{ X}(\mathbb C) {\rm $Mat_X(mathbbC) generated by $A, A ^{*}$. We refer to $T$ as the Terwilliger algebra of $Q _{ D }$ with respect to $x$. The matrices $A$ and $A ^{*}$ are related by the fact that 2 i $A= A ^{*} A ^{ \epsilon } - A ^{ \epsilon } A ^{*}$ and 2 i $A ^{*}= A ^{ \epsilon } A - AA ^{ \epsilon }$, where 2 i $A ^{ \epsilon }= AA ^{*} - A ^{*} A$ and i $^{2}$= - 1.

Détail

Keywords: keywords leonard triple, distance-regular graph, hypercube, Terwilliger algebra