Totally bipartite tridiagonal pairs

There is a concept in linear algebra called a tridiagonal pair. The concept was motivated by the theory of $Q$-polynomial distance-regular graphs. We give a tutorial introduction to tridiagonal pairs, working with a special case as a concrete example. The special case is called totally bipartite, or totally bipartite (TB). Starting from first principles, we give an elementary but comprehensive account of TB tridiagonal pairs. The following topics are discussed: (i) the notion of a TB tridiagonal system; (ii) the eigenvalue array; (iii) the standard basis and matrix representations; (iv) the intersection numbers; (v) the Askey--Wilson relations; (vi) a recurrence involving the eigenvalue array; (vii) the classification of TB tridiagonal systems; (viii) self-dual TB tridiagonal pairs and systems; (ix) the $\mathbb{Z}_3$-symmetric Askey--Wilson relations; (x) some automorphisms and antiautomorphisms associated with a TB tridiagonal pair; and (xi) an action of the modular group ${\rm PSL}_2(\mathbb{Z})$ associated with a TB tridiagonal pair.

Tridiagonal pairs of type III with height one

Let K denote an algebraically closed field with characteristic 0. Let V denote a vector space over K with finite positive dimension, and let A, A∗ denote a tridiagonal pair on V of diameter d. Let V0, . . . , Vd denote a standard ordering of the eigenspaces of A on V , and let θ0, . . . , θd denote the corresponding eigenvalues of A. It is assumed that d ≥ 3. Let ρi denote the dimension of Vi. The sequence ρ0, ρ1, . . . , ρd is called the shape of the tridiagonal pair. It is known that ρ0 = 1 and there exists a unique integer h (0 ≤ h ≤ d/2) such that ρi−1 < ρi for 1 ≤ i ≤ h, ρi−1 = ρi for h < i ≤ d − h, and ρi−1 > ρi for d − h < i ≤ d. The integer h is known as the height of the tridiagonal pair. In this paper, it is showed that the shape of a tridiagonal pair of type III with height one is either 1, 2, 2, . . ., 2, 1 or 1, 3, 3, 1. In each case, an interesting basis is found for V and the actions of A, A∗ on this basis are described.

HOW TO SHARPEN A TRIDIAGONAL PAIR

Let 𝔽 denote a field and let V denote a vector space over 𝔽 with finite positive dimension. We consider a pair of linear transformations A : V → V and A* : V → V that satisfy the following conditions: (i) each of A, A* is diagonalizable; (ii) there exists an ordering [Formula: see text] of the eigenspaces of A such that A* Vi ⊆ Vi-1 + Vi + Vi+1 for 0 ≤ i ≤ d, where V-1 = 0 and Vd+1 = 0; (iii) there exists an ordering [Formula: see text] of the eigenspaces of A* such that [Formula: see text] for 0 ≤ i ≤ δ, where [Formula: see text] and [Formula: see text]; (iv) there is no subspace W of V such that AW ⊆ W, A* W ⊆ W, W ≠ 0, W ≠ V. We call such a pair a tridiagonal pair on V. It is known that d = δ, and for 0 ≤ i ≤ d the dimensions of Vi, [Formula: see text], Vd-i, [Formula: see text] coincide. Denote this common dimension by ρi and call A, A*sharp whenever ρ0 = 1. Let T denote the 𝔽-subalgebra of End 𝔽(V) generated by A, A*. We show: (i) the center Z(T) is a field whose dimension over 𝔽 is ρ0; (ii) the field Z(T) is isomorphic to each of E0TE0, EdTEd, [Formula: see text], [Formula: see text], where Ei (resp. [Formula: see text]) is the primitive idempotent of A (resp. A*) associated with Vi (resp. [Formula: see text]); (iii) with respect to the Z(T)-vector space V the pair A, A* is a sharp tridiagonal pair.