The Terwilliger algebra of a distance-regular graph that supports a spin model

Caughman, John S. ; Wolff, Nadine

Journal of Algebraic Combinatorics, Tome 21 (2005) no. 3, pp. 289-310.

Résumé

Summary: Let $Gamma$ denote a distance-regular graph with vertex set $X$, diameter $Dge$ 3, valency $kge$ 3, and assume $Gamma$ supports a spin model $W$. Write $W = sum_{ i = 0} ^{ D} t _{ i} A _{ i}$ where $A _{ i}$ is the $i$th distance-matrix of $Gamma$. To avoid degenerate situations we assume $Gamma$ is not a Hamming graph and $t _{ i}notin{ t _{0}, - t _{0} }$ for 1 $leileD$. In an earlier paper Curtin and Nomura determined the intersection numbers of $Gamma$ in terms of $D$ and two complex parameters $eegr$ and $q$. We extend their results as follows. Fix any vertex $xisinX$ and let $T = T( x)$ denote the corresponding Terwilliger algebra. Let $U$ denote an irreducible $T$-module with endpoint $r$ and diameter $d$. We obtain the intersection numbers $c _{ i}( U), b _{ i}( U), a _{ i}( U)$ as rational expressions involving $r, d, D, eegr$ and $q$. We show that the isomorphism class of $U$ as a $T$-module is determined by $r$ and $d$. We present a recurrence that gives the multiplicities with which the irreducible $T$-modules appear in the standard module. We compute these multiplicites explicitly for the irreducible $T$-modules with endpoint at most 3. We prove that the parameter $q$ is real and we show that if $Gamma$ is not bipartite, then $q > 0$ and $eegr$ is real.

Détail

Classification : Primary, 05E30
Keywords: key words distance-regular graph, spin model, Terwilliger algebra