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Abstract

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 Qpolynomial schemes. Let Y denote any commutative association scheme, and fix any vertex x of Y. We introduce a non-commutative, associative, semi-simple 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 Pand 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.

Quantum hypothesis testing in many-body systems

“…It is also interesting that this problem seems related to coding theory and combinatorics. In Appendix C, we show that the Gram matrix is an element of the Terwilliger algebra [62] of the Hamming cube H = {0, 1} n (see [63,64]). This is done by identifying the labels ξ( E) as subsets of H, given by the supports of the bit strings E. In this way we obtain the explicit expansion…”

“…The Hamming cube H n = {0, 1} n is the set of binary strings of length n with Hamming distance as the metric. The Terwilliger algebra of the Hamming cube [62,64] is an algebraic structure which is useful in combinatorics and coding theory (see [63] and references therein). We proceed as in [63], and identify the binary strings a 1 a 2 • • • a n with their support, the subset X of labels i for which the bit a i in the string takes value 1.…”

On the Terwilliger algebra of certain family of bipartite distance-regular graphs with Δ_2 = 0

“…Let T = T (x) denote the subalgebra of Mat X (C) generated by M , M * . We call T the Terwilliger algebra of Γ with respect to x [16,Definition 3.3]. Recall M is generated by A, so T is generated by A and the dual idempotents.…”

Scaling Limits for the Gibbs States on Distance-Regular Graphs with Classical Parameters

“…The subalgebra Ã(Γ) of M X (C) generated by the quantum components A + , A − , and A • of A is non-commutative unless |X| = 1, and is contained in the Terwilliger algebra of Γ with respect to o; cf. [34,35,36]. See [41] for discussions on when the two algebras are equal.…”