Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A * : V → V that satisfy (i) and (ii) below:(i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A * is diagonal.(ii) There exists a basis for V with respect to which the matrix representing A * is irreducible tridiagonal and the matrix representing A is diagonal.We call such a pair a Leonard pair on V . Let X denote the set of linear transformations X : V → V such that the matrix representing X with respect to the basis (i) is tridiagonal and the matrix representing X with respect to the basis (ii) is tridiagonal. We show that X is spanned byand these elements form a basis for X provided the dimension of V is at least 3.(ii) There exists a basis for V with respect to which the matrix representing A * is irreducible tridiagonal and the matrix representing A is diagonal.
A spin model (for link invariants) is a square matrix W which satisfies certain axioms. For a spin model W , it is known that W T W −1 is a permutation matrix, and its order is called the index of W. F. Jaeger and K. Nomura found spin models of index 2, by modifying the construction of symmetric spin models from Hadamard matrices. The aim of this paper is to give a construction of spin models of an arbitrary even index from any Hadamard matrix. In particular, we show that our spin models of indices a power of 2 are new.
A Leonard pair is an ordered pair of diagonalizable linear maps on a finite-dimensional vector space, that each act on an eigenbasis for the other one in an irreducible tridiagonal fashion. In the present paper we consider a type of Leonard pair, said to have spin. The notion of a spin model was introduced by V.F.R. Jones to construct link invariants. A spin model is a symmetric matrix over C that satisfies two conditions, called the type II and type III conditions. It is known that a spin model W is contained in a certain finite-dimensional algebra N (W), called the Nomura algebra. It often happens that a spin model W satisfies W ∈ M ⊆ N (W), where M is the Bose-Mesner algebra of a distance-regular graph Γ; in this case we say that Γ affords W. If Γ affords a spin model, then each irreducible module for every Terwilliger algebra of Γ takes a certain form, recently described by Caughman, Curtin, Nomura, and Wolff. In the present paper we show that the converse is true; if each irreducible module for every Terwilliger algebra of Γ takes this form, then Γ affords a spin model. We explicitly construct this spin model when Γ has q-Racah type. The proof of our main result relies heavily on the theory of spin Leonard pairs. U as a spin Leonard pair, and W, W * act on U as a Boltzmann pair for this Leonard pair; see Proposition 11.10 and Lemma 11.11 below. Consequently the intersection numbers of U [28, Definition 11.1] are given by the formulas in [5, Theorem 8.5]. Another consequence for U is that its endpoint [25, Lemma 3.12] is equal to its dual endpoint [25, Lemma 3.9]. Now we describe our main result. Assume that our distance-regular graph Γ is formally self-dual and has q-Racah type [29, Definition 5.1]. Assume that for all x ∈ X each irreducible T(x)-module U is thin with matching endpoint and dual-endpoint, and the intersection numbers of U are given by the formulas in [5, Theorem 8.5]. Then there exists a spin model afforded by Γ; this spin model is explicitly constructed. Our main result is Theorem 14.7. We comment on the nature of this result. We have not discovered any new spin model to date. What we have shown, is that a new spin model would result from the discovery of a distance-regular graph with the right sort of irreducible T-modules.The proof of our main result relies heavily on the theory of spin Leonard pairs. In the first half of the paper we develop this theory, and obtain several new results that may be of independent interest; for instance Theorem 5.25 and Lemmas 6.11, 6.18. The paper is organized as follows. CONTENTS Leonard pairs and Leonard systemsWe now begin our formal argument. In this section we recall the notion of a Leonard pair. We use the following terms. A square matrix is said to be tridiagonal whenever each nonzero entry lies on either the diagonal, the subdiagonal, or the superdiagonal. A tridiagonal matrix is said to be irreducible whenever each entry on the subdiagonal is nonzero and each entry on the superdiagonal is nonzero. Let F denote a field.
Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and A * : V → V that satisfy the following two conditions:(i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A * is diagonal.(ii) There exists a basis for V with respect to which the matrix representing A * is irreducible tridiagonal and the matrix representing A is diagonal.
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