An inequality involving the local eigenvalues of a distance-regular graph

Terwilliger, Paul

Journal of Algebraic Combinatorics, Tome 19 (2004) no. 2, pp. 143-172.

Résumé

Summary: Let $\mathbb R$ mathbbR [ [$( h)$tilde] $\tilde \eta = -1 - b _{1}(1+ \mathbb C$ mathbbC ) generated by $A, E* _{0}, E* _{1}, \dots , E* _{ D }$, where $A$ denotes the adjacency matrix of $Gamma$ and $E* _{ i }$ denotes the projection onto the $i$th subconstituent of $Gamma$ with respect to $X. T$ is called the subconstituent algebra or the Terwilliger algebra. An irreducible $T$-module $W$ is said to be $thin$ whenever dim $E* _{ i } Wle$ 1 for 0 $le$ i $leD$. By the $endpoint$ of $W$ we mean min${ i| E* _{ i } Wne$ 0. We show the following are equivalent: (i) Equality holds in the above inequality for 1 $leileD - 1$; (ii) Equality holds in the above inequality for $i = D - 1$; (iii) Every irreducible $T$-module with endpoint 1 is thin.

Détail

Classification : Primary, 05E30;, Secondary, 05E35,, 05C50
Keywords: distance-regular graph, association scheme, Terwilliger algebra, subconstituent algebra