×

zbMATH — the first resource for mathematics

Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra. (English) Zbl 1035.05104
A distance-regular graph \(\Gamma\) with diameter \(D\) and Bose-Mesner algebra \({\mathbb M}\) is considered. Let \(A_i\) be the \(i\)-adjacency matrix of \(\Gamma\). For \(\theta\in {\mathbb C}\) let \({\mathbb M}(\theta)\) consist of those \(Y\in {\mathbb M}\) such that \((A-\theta I)Y\in {\mathbb C}A_D\) and \({\mathbb M}(\infty)={\mathbb C}A_D\). By a pseudo primitive idempotent for \(\theta\in {\mathbb C}\cup \{\infty\}\) we mean a nonzero element of \({\mathbb M}(\theta)\). Let \(X\) denote the vertex set of \(\Gamma\) and \(x\in X\). Let \(T=T(x)\) denote the subalgebra of \(\text{Mat}_X({\mathbb C})\) generated by \(A,E_0^*,\dots,E_D^*\), where \(A\) is the adjacency matrix of \(\Gamma\) and \(E_i^*\) is the projection onto the \(i\)th subconstituent of \(\Gamma\) with respect to \(x\). \(T\) is called the Terwilliger algebra of \(\Gamma\) with respect to \(x\). An irreducible \(T\)-module \(W\) is thin whenever dim \(E_i^*W\leq 1\) for \(0\leq i\leq D\). The endpoint of \(W\) is min\(\{i\;| \;E_i^*W\neq 0\}\). Let \(V={\mathbb C}^X\) denote the standard \(T\)-module. Fix \(0\neq v\in E_1^*V\cap {\mathbf 1}^\bot\) and define \(({\mathbb M},v)=\{P\in {\mathbb M}\;| \;Pv\in E_D^*V\}\). The following are equivalent: (i) dim(\({\mathbb M},v)\geq 2\); (ii) \(v\) is contained in a thin irreducible \(T\)-module with endpoint 1 (Theorem 1.1). Let \(W\) be the \(T\)-module which satisfies (ii). Then \(E_1^*W\) is a one-dimensional eigenspace for \(E_1^*AE_1^*\). Let \(\eta\) denote the corresponding eigenvalue and \(\tilde \eta=-1-b_1(1+\eta)^{-1}\) if \(\eta \neq -1\) and \(\tilde \eta=\infty\) if \(\eta=-1\). Then \(({\mathbb M},v)\) has a basis \((J,E)\), where \(J\) has all entries 1 and \(E\) is a pseudo primitive idempotent for \(\tilde \eta\) (Theorem 1.2).

MSC:
05E30 Association schemes, strongly regular graphs
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Balmaceda, J.M.; Oura, M., The Terwilliger algebras of the group association schemes of S5 and A5, Kyushu J. math, 48, 2, 221-231, (1994) · Zbl 0821.05059
[2] Bannai, E.; Ito, T., Algebraic combinatorics I: association schemes, (1984), Benjamin/Cummings London · Zbl 0555.05019
[3] Bannai, E.; Munemasa, A., The Terwilliger algebras of group association schemes, Kyushu J. math, 49, 1, 93-102, (1995) · Zbl 0839.05095
[4] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance-regular graphs, (1989), Springer-Verlag Berlin · Zbl 0747.05073
[5] Caughman, J., The Terwilliger algebras of bipartite P- and Q-polynomial association schemes, Discrete math, 196, 65-95, (1999) · Zbl 0924.05067
[6] Curtin, B., Bipartite distance-regular graphs I, Graphs combin, 15, 2, 143-158, (1999) · Zbl 0927.05083
[7] Curtin, B., Bipartite distance-regular graphs II, Graphs combin, 15, 4, 377-391, (1999) · Zbl 0939.05088
[8] B. Curtin, Distance-regular graphs which support a spin model are thin in: 16th British Combinatorial Conference, London, 1997, Discrete Math. 197-198 (1999) 205-216
[9] Curtin, B.; Nomura, K., Spin models and strongly hyper-self-dual bose – mesner algebras, J. algebraic combin, 13, 2, 173-186, (2001) · Zbl 0979.05111
[10] Egge, E., A generalization of the Terwilliger algebra, J. algebra, 233, 213-252, (2000) · Zbl 0960.05108
[11] Egge, E., The generalized Terwilliger algebra and its finite dimensional modules when d=2, J. algebra, 250, 178-216, (2002) · Zbl 1003.05107
[12] Go, J.T.; Terwilliger, P., Tight distance-regular graphs and the subconstituent algebra, European J. combin, 23, 793-816, (2002) · Zbl 1014.05070
[13] Hobart, S.A.; Ito, T., The structure of nonthin irreducible T-modules: ladder bases and classical parameters, J. algebraic combin, 7, 53-75, (1998) · Zbl 0911.05059
[14] Ishibashi, H.; Ito, T.; Yamada, M., Terwilliger algebras of cyclotomic schemes and Jacobi sums, European J. combin, 20, 5, 397-410, (1999) · Zbl 0944.05101
[15] Ishibashi, H., The Terwilliger algebras of certain association schemes over the Galois rings of characteristic 4, Graphs combin, 12, 1, 39-54, (1996) · Zbl 0852.05081
[16] Sagan, B.; Caughman, J.S., The multiplicities of a dual-thin Q-polynomial association scheme, Electron. J. combin, 8, 1, (2001), (electronic) · Zbl 0973.05081
[17] Tanabe, K., The irreducible modules of the Terwilliger algebras of Doob schemes, J. algebraic combin, 6, 173-195, (1997) · Zbl 0868.05056
[18] Terwilliger, P., A new feasibility condition for distance-regular graphs, Discrete math, 61, 311-315, (1986) · Zbl 0606.05045
[19] Terwilliger, P., The subconstituent algebra of an association scheme I, J. algebraic combin, 1, 4, 363-388, (1992) · Zbl 0785.05089
[20] Terwilliger, P., The subconstituent algebra of an association scheme II, J. algebraic combin, 2, 1, 73-103, (1993) · Zbl 0785.05090
[21] Terwilliger, P., The subconstituent algebra of an association scheme III, J. algebraic combin, 2, 2, 177-210, (1993) · Zbl 0785.05091
[22] Terwilliger, P., The subconstituent algebra of a distance-regular graph; thin modules with endpoint one, Linear algebra appl, 356, 157-187, (2002) · Zbl 1011.05066
[23] P. Terwilliger, An inequality involving the local eigenvalues of a distance-regular graph J. Algebraic Combin. (2003) · Zbl 1047.05045
[24] Tomiyama, M.; Yamazaki, N., The subconstituent algebra of a strongly regular graph, Kyushu J. math, 48, 2, 323-334, (1994) · Zbl 0842.05098
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.