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The subconstituent algebra of an association scheme. I. (English) Zbl 0785.05089
From the author’s abstract: We introduce a method for studying commutative association schemes with “many” vanishing intersection numbers and/or Krein parameters, and apply the method to the $$P$$- and $$Q$$-polynomial schemes. Let $$Y$$ denote any commutative association scheme, and fix any vertex $$x$$ of $$Y$$. We introduce a non-commutative, associative, semi-simple $$\mathbb{C}$$-algebra $$T=T(x)$$ whose structure reflects the combinatorial structure of $$Y$$. We call $$T$$ the subconstituent algebra of $$Y$$ with respect to $$x$$. Roughly speaking, $$T$$ is a combinatorial analog of the centralizer algebra of the stabilizer of $$x$$ in the automorphism group of $$Y$$.
In general, the struture of $$T$$ is not determined by the intersection numbers of $$Y$$, but these parameters do give some information. Indeed, we find a relation among the generators of $$T$$ for each vanishing intersection number or Krein parameter.
We identify a class of irreducible $$T$$-modules whose structure is especially simple, and say the members of this class are thin. Expanding on this, we say $$Y$$ is thin if every irreducible $$T(y)$$-module is thin for every vertex $$y$$ of $$Y$$. We compute the possible thin, irreducible $$Y$$-modules when $$Y$$ is $$P$$- and $$Q$$-polynomial. The ones with sufficiently large dimension are indexed by four bounded integer parameters. If $$Y$$ is assumed to be thin, then “sufficiently large dimension” means “dimension at least four”.
We give a combinatorial characterization of the thin $$P$$- and $$Q$$- polynomial schemes, and supply a number of examples of these objects. For each example, we show which irreducible $$T$$-modules actually occur.
We close with some conjectures and open problems.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05C12 Distance in graphs
Zbl 0785.05090
Full Text:
##### References:
 [1] Askey, R.; Wilson, J., A set of orthogonal polynomials that generalize the Racah coefficients on $$6-j$$ symbols, SIAM J. Math. Anal., 10, 1008-1016, (1979) · Zbl 0437.33014 [2] R. Askey, J. Wilson, “Some basic hypergeometric orthogonal polynomials that generalize the Jacobi polynomials,” Mem. Amer. Math. Soc.319, 1985. · Zbl 0572.33012 [3] E. Bannai, T. Ito, “Algebraic Combinatorics I: Association Schemes,” Benjamin-Cummings Lecture Note 58. Menlo Park, 1984. [4] Bannai, E.; Ito, T., Current researches on algebraic combinatorics, Graphs Combin., 2, 287-308, (1986) · Zbl 0685.05030 [5] E. Bannai, “On extremal finite sets in the sphere and other metric spaces, algebraic, extremal and metric combinatorics,” 1986 (Montreal, PQ, 1986), 13-38, London Math. Soc. Lecture Note Ser., 131, Cambridge Univ. Press, Cambridge-New York, 1988. [6] Bannai, E.; Nevai, Paul (ed.), Orthogonal polynomials in coding theory and algebraic combinatorics, 25-53, (1990), Boston [7] Bier, T., Lattices associated to the Rectangular Association Scheme, Ars Combin., 23A, 41-50, (1987) · Zbl 0624.05021 [8] Bier, T., Totient-numbers of the rectangular association scheme, Graphs Combin., 6, 5-15, (1990) · Zbl 0696.05013 [9] Biggs, N., Some Odd graph theory, Ann. New York Acad. Sci., 319, 71-81, (1979) [10] J. Van Bon, “Affine distance-transitive groups,” thesis, C.W.I., The Netherlands, 1990. [11] A. Brouwer, A. Cohen, A. Neumaier, Distance-Regular Graphs, Springer Verlag, New York, 1989. [12] Brouwer, A.; Hemmeter, J., A new family of distance-regular graphs and the (0,1,2)-cliques in dual polar graphs, European J. Combin., 13, 71-79, (1992) · Zbl 0762.05041 [13] Calderbank, A.; Delsarte, P.; Sloane, N., A strengthening of the Assmus-Mattson theorem, IEEE Trans. Inform. Theory, 37, 1261-1268, (1991) · Zbl 0734.94018 [14] A. Calderbank, P. Delsarte, “The concept of a $$(t, r)$$-regular design as an extension of the classical concept of a t-design,” preprint. · Zbl 0783.05028 [15] A. Calderbank, P. Delsarte, “On error-correcting codes and invariant linear forms,” preprint. · Zbl 0768.94020 [16] Cameron, P., Dual polar spaces, Geom. Dedicata, 12, 75-85, (1982) · Zbl 0473.51002 [17] Cameron, P.; Goethals, J.; Seidel, J., The Krein condition, spherical designs, Norton algebras, and permutation groups, Indag. Math., 40, 196-206, (1978) · Zbl 0399.20001 [18] S. Y. Choi, “An upper bound on the diameter of certain distance-regular graphs.” Proceedings of the Twentieth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1989). Congr. Numer. 70 (1990), 195-198. [19] Clayton, R., Perfect multiple coverings in metric schemes, No. 20, 51-64, (1990), New York-Berlin [20] C. Curtis, I. Reiner, “Representation Theory of Finite Groups and Associative Algebras”, Interscience, New York, 1962. · Zbl 0131.25601 [21] C. Delorme, “Distance bi-regular bipartite graphs,” European J. Combin., submitted. · Zbl 0804.05031 [22] P. Delsarte, “An algebraic approach to the association schemes of coding theory,” Philips Res. Reports Suppl.10 (1973). [23] Delsarte, P., Association schemes and $$t$$-designs in regular semi-lattices, J. Combin. Theory Ser. A, 20, 230-243, (1976) · Zbl 0342.05020 [24] Doob, M., On graph products and association schemes, Utilitas Math., 1, 291-302, (1972) · Zbl 0315.05130 [25] Dunkl, C., An addition theorem for some $$q$$-Hahn polynomials, Monatsh. Math., 85, 5-37, (1977) · Zbl 0345.33010 [26] Egawa, Y., Association schemes of quadratic forms, J. Combin. Theory Ser. A, 38, 1-14, (1985) · Zbl 0564.05014 [27] Egawa, Y., Characterization of $$H(n, q)$$ by the parameters, J. Combin. Theory Ser. A, 31, 108-125, (1981) · Zbl 0472.05056 [28] Faradzhev, I.; Ivanov, A. A., Distance-transitive representations of groups $$G$$ with PSL$$\_{}\{2\}(q) {ie386-1} G$${ie386-2} PΓL$$\_{}\{2\}(q),$$ European J. Combin., 11, 347-356, (1990) [29] Faradzhev, I.; Ivanov, A. A.; Klin, M., Galois correspondence between permutation groups and cellular rings (association schemes), Graphs Combin., 6, 303-332, (1990) · Zbl 0764.05099 [30] Gasper, G.; Rahman, M., Basic hypergeometric series, (1990), Cambridge · Zbl 0695.33001 [31] Godsil, C.; Shawe-Taylor, J., Distance-regularized graphs are distance-regular or distance-biregular, J. Combin. Theory Ser. B, 43, 14-24, (1987) · Zbl 0616.05041 [32] Hemmeter, J., The large cliques in the graph of quadratic forms, European J. Combin., 9, 395-410, (1988) · Zbl 0672.05044 [33] Hemmeter, J.; Woldar, A., Classification of the maximal cliques of size ≥$$q + 4$$ in the quadratic forms graph in odd characteristic, European J. Combin., 11, 433-450, (1990) · Zbl 0762.05089 [34] Hemmeter, J.; Woldar, A., On the maximal cliques of the quadratic forms graph in even characteristic, European J. Combin., 11, 119-126, (1990) · Zbl 0717.05040 [35] Huang, T., A characterization of the association schemes of bilinear forms, European J. Combin., 8, 159-173, (1987) · Zbl 0675.05064 [36] Huang, T., Further results on the E-K-R theorem for the distance regular graphs $$H\_{}\{q\}(k,n),$$ Bull. Inst. Math. Acad. Sinica, 16, 347-356, (1988) · Zbl 0711.05044 [37] T. Huang and M. Laurant, “$$(s,r$$:μ)-nets and alternating forms graphs,” submitted. [38] Ito, T.; Munemasa, A.; Yamada, M., Amorphous association schemes over the Galois rings of characteristic 4, European J Combin., 12, 513-526, (1991) · Zbl 0762.05098 [39] Ivanov, A. A., Distance-transitive graphs that admit elations, Izv. Akad. Nauk SSSR Ser. Mat., 53, 971-1000, (1989) [40] Ivanov, A.; Muzichuk, M.; Ustimenko, V., On a new family of $$(P$$ and $$Q)$$-polynomial schemes, European J. Combin., 10, 337-345, (1989) · Zbl 0709.05015 [41] Ivanov, A.; Shpektorov, S., The association schemes of the dual polar spaces of type $$2A\_{}\{\textit{2d}−1\}(p f)$$ are characterized by their parameters if $$d$$≥ 3, Linear Algebra Appl., 114, 133-139, (1989) [42] Ivanov, A.; Shpektorov, S., Characterization of the association schemes of Hermitian forms over GF(22), Geom. Dedicata, 30, 23-33, (1989) [43] Kawanaka, N.; Matsuyama, H., A twisted version of the Frobenius-Schur indicator and multiplicity-free permutation representations, Hokkaido Math. J., 19, 495-508, (1990) · Zbl 0791.20006 [44] E. Lambeck, “Contributions to the theory of distance-regular graphs,” Thesis, Eindhoven, 1990. · Zbl 0758.05047 [45] Leonard, D., Directed distance-regular graphs with the $$Q$$-polynomial property, J. Combin. Theory Ser. B, 48, 191-196, (1990) · Zbl 0723.05065 [46] Leonard, D., Non-symmetric, metric, cometric association schemes are self-dual, J. Combin. Theory Ser. B, 51, 244-247, (1991) · Zbl 0754.05076 [47] Leonard, D., Orthogonal polynomials, duality, and association schemes, SIAM J. Math. Anal., 13, 656-663, (1982) · Zbl 0495.33006 [48] Ma, S. L., On association schemes, Schur rings, strongly regular graphs and partial difference sets, Ars. Combin., 27, 211-220, (1989) · Zbl 0666.05015 [49] N. Manickam, “Distribution invariants of association schemes,” Seventeenth Manitoba Conference on Numerical Mathematics and Computing (Winnipeg, MB, 1987), Congr. Numer. 61 (1988), 121-131. [50] Moon, A., Characterization of the Odd graphs $$O\_{}\{k\}$$ by the parameters, Discrete Math., 42, 91-97, (1982) · Zbl 0494.05035 [51] Munemasa, A., On non-symmetric $$P$$-and $$Q$$-polynomial association schemes, J. Combin. Theory Ser. B, 51, 314-328, (1991) · Zbl 0754.05078 [52] Neumaier, A., Characterization of a class of distance-regular graphs, J. Reine Angew. Math., 357, 182-192, (1985) · Zbl 0552.05042 [53] Neumaier, A., Krein conditions and near polygons, J. Combin. Theory Ser. A, 54, 201-209, (1990) · Zbl 0738.05025 [54] Nomura, K., Distance-regular graphs of Hamming type, J. Combin. Theory Ser. B, 50, 160-167, (1990) · Zbl 0719.05070 [55] Nomura, K., Intersection diagrams of distance-biregular graphs, J. Combin. Theory Ser. B, 50, 214-221, (1990) · Zbl 0719.05071 [56] Nomura, K., On local structure of a distance-regular graph of Hamming type, J. Combin. Theory Ser. B, 47, 120-123, (1989) · Zbl 0628.05054 [57] Powers, D., Semi-regular graphs and their algebra, Linear and Multilinear Algebra, 24, 27-37, (1988) · Zbl 0697.05041 [58] Praeger, C. E.; Saxl, J.; Yokoyama, K., Distance-transitive graphs and finite simple groups, Proc. London Math. Soc., 55, 1-21, (1987) · Zbl 0619.05026 [59] Rifa, J.; Huguet, L., Classification of a class of distance-regular graphs via completely regular codes, Discrete Appl. Math., 26, 289-300, (1990) · Zbl 0687.05015 [60] Rifa, J., On the construction of completely regular linear codes from distance-regular graphs, No. 356, 376-393, (1989), Berlin-New York [61] Sloane, N.; Askey, R. (ed.), An introduction to association schemes and coding theory, 225-260, (1975), New York [62] Sole, P., A Lloyd theorem in weakly metric association schemes, European J. Combin., 10, 189-196, (1989) · Zbl 0722.05061 [63] Stanton, D., Harmonics on posets, J. Combin. Theory Ser. A, 40, 136-149, (1985) · Zbl 0573.06001 [64] Stanton, D., Some $$q$$-Krawtchouk polynomials on Chevalley groups, Amer. J. Math., 102, 625-662, (1980) · Zbl 0448.33019 [65] Suzuki “, H.$$, t$$-designs in $$H(d,$$q), Hokkaido Math. J., 19, 403-415, (1990) · Zbl 0726.05013 [66] Terwilliger, P., A characterization of $$P$$-and $$Q$$-polynomial association schemes, J. Combin. Theory Ser. A, 45, 8-26, (1987) · Zbl 0663.05016 [67] Terwilliger, P., A class of distance-regular graphs that are $$Q$$-polynomial, J. Combin. Theory Ser. B, 40, 213-223, (1986) · Zbl 0579.05029 [68] P. Terwilliger, “A generalization of Jackson”s _{8φ7} sum,” in preparation. [69] Terwilliger, P., Counting 4-vertex configurations in $$P$$-and $$Q$$-polynomial association schemes, Algebras Groups Geom., 2, 541-554, (1985) · Zbl 0592.05011 [70] P. Terwilliger, “Leonard pairs and the $$q$$-Racah polynomials,” in preparation. · Zbl 1075.05090 [71] Terwilliger, P., Root systems and the Johnson and Hamming graphs, European J. Combin., 8, 73-102, (1987) · Zbl 0614.05048 [72] Terwilliger, P., The classification of distance-regular graphs of type IIB, Combinatorica, 8, 125-132, (1988) · Zbl 0727.05050 [73] Terwilliger, P., The classification of finite connected hypermetric spaces, Graphs Combin., 3, 293-298, (1987) · Zbl 0618.05043 [74] Terwilliger, P., The incidence algebra of a uniform poset, No. 20, 193-212, (1990), New York · Zbl 0737.05032 [75] Terwilliger, P., The Johnson graph $$J(d,r)$$ is unique if $$(d,r)$$≠ (8,2), Discrete Math., 58, 175-189, (1986) · Zbl 0587.05038 [76] Terwilliger, P.$$, P$$-and $$Q$$-polynomial schemes with $$q$$= −1, J. Combin. Theory Ser. B, 42, 64-67, (1987) · Zbl 0675.05016 [77] Weichsel, P., Distance-regular graphs in block form, Linear Algebra Appl., 126, 135-148, (1989) · Zbl 0715.05051 [78] Yamada, M., Distance-regular digraphs of girth 4 over an extension ring of ℤ/4ℤ, Graphs Combin., 6, 381-394, (1990) · Zbl 0721.05028
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