A Note on Thin P-Polynomial and Dual-Thin Q-PolynomialSymmetric Association Schemes
Journal of Algebraic Combinatorics: An International Journal
On the natural representation of S(Ω) into L2 (P(Ω)): discrete harmonics and Fourier transform
Journal of Combinatorial Theory Series A
Tight distance-regular graphs and the subconstituent algebra
European Journal of Combinatorics
The Terwilliger algebra of the hypercube
European Journal of Combinatorics
Journal of Computational and Applied Mathematics - Proceedings of the sixth international symposium on orthogonal polynomials, special functions and their applications
Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra
European Journal of Combinatorics - Special issue on algebraic combinatorics: in memory of J.J. Seidel
An Inequality Involving the Local Eigenvalues of a Distance-Regular Graph
Journal of Algebraic Combinatorics: An International Journal
Taut distance-regular graphs of even diameter
Journal of Combinatorial Theory Series B
Designs, Codes and Cryptography
Algebraic Characterizations of Graph Regularity Conditions
Designs, Codes and Cryptography
The Terwilliger Algebra of a Distance-Regular Graph that Supports a Spin Model
Journal of Algebraic Combinatorics: An International Journal
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the seventh international symposium on Orthogonal polynomials, special functions and applications
The Terwilliger Algebra Associated with a Set of Vertices in a Distance-Regular Graph
Journal of Algebraic Combinatorics: An International Journal
The Terwilliger algebra of a Hamming scheme H (d,q)
European Journal of Combinatorics
European Journal of Combinatorics
New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming
Journal of Combinatorial Theory Series A - Special issue in honor of Jacobus H. van Lint
Almost-bipartite distance-regular graphs with the Q-polynomial property
European Journal of Combinatorics
Imprimitive cometric association schemes: Constructions and analysis
Journal of Algebraic Combinatorics: An International Journal
Triangle-free distance-regular graphs
Journal of Algebraic Combinatorics: An International Journal
3-bounded property in a triangle-free distance-regular graph
European Journal of Combinatorics
Leonard triples and hypercubes
Journal of Algebraic Combinatorics: An International Journal
Some matrices associated with the split decomposition for a Q-polynomial distance-regular graph
European Journal of Combinatorics
Q-polynomial distance-regular graphs with a1=0 and a2≠0
European Journal of Combinatorics
Note: A note on triangle-free distance-regular graphs with a2≠0
Journal of Combinatorial Theory Series B
The subconstituent algebra of a Latin square
European Journal of Combinatorics
The spectra of the local graphs of the twisted Grassmann graphs
European Journal of Combinatorics
Distance-regular graphs and the q-tetrahedron algebra
European Journal of Combinatorics
New proofs of the Assmus-Mattson theorem based on the Terwilliger algebra
European Journal of Combinatorics
European Journal of Combinatorics
Commutative association schemes
European Journal of Combinatorics
Permutation group approach to association schemes
European Journal of Combinatorics
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the seventh international symposium on Orthogonal polynomials, special functions and applications
The Q-polynomial idempotents of a distance-regular graph
Journal of Combinatorial Theory Series B
Symmetric chains, Gelfand---Tsetlin chains, and the Terwilliger algebra of the binary Hamming scheme
Journal of Algebraic Combinatorics: An International Journal
New parameters of subsets in polynomial association schemes
Journal of Combinatorial Theory Series A
A spectral excess theorem for nonregular graphs
Journal of Combinatorial Theory Series A
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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 \Bbb {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 structure 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-moduIes 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 T-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.