Totally bipartite tridiagonal pairs

Abstract

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 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 Z 3symmetric Askey-Wilson relations; (x) some automorphisms and antiautomorphisms associated with a TB tridiagonal pair; (xi) an action of the modular group PSL 2 (Z) associated with a TB tridiagonal pair.

Double Lowering Operators on Polynomial

“…(ii) ⇒ (i) Define ψ ∈ End(V ) such that ψτ i = ϑ i τ i−1 for 0 ≤ i ≤ N. We have ψ = 0 since N ≥ 1 and ϑ 1 = 1, τ 0 = 1. We show ψ ∈ L. To do this, it is convenient to first show that ψ satisfies (28). To obtain (28), for 0 ≤ j ≤ N we apply each side of (28) to τ j .…”

“…We show ψ ∈ L. To do this, it is convenient to first show that ψ satisfies (28). To obtain (28), for 0 ≤ j ≤ N we apply each side of (28) to τ j . Concerning the left-hand side of (28), we have ∆τ j = η j by (8).…”

“…We have shown that each side of (28) sends τ j → η j for 0 ≤ j ≤ N. Therefore (28) holds. By (28), ∆ is a polynomial in ψ. Consequently ψ∆ = ∆ψ. The map ψ satisfies Proposition 5.1(i), so ψ ∈ L by Proposition 5.1(i),(iii).…”

“…For 0 ≤ j ≤ N we compare the coefficient of ψ j on each side of (71). For the left-hand side these coefficients are obtained from (28). We require…”

“…Since that origin, the tridiagonal pair concept has found applications to all sorts of topics in special functions (orthogonal polynomials of the Askey-scheme [3,25,27,32,33], the Askey-Wilson algebra [14,35,40,42]), Lie theory (the sl 2 loop algebra [20], the tetrahedron algebra [15,21]), statistical mechanics (the Onsager algebra [12,16] and q-Onsager algebra [5,6,23,30,37,39]), and quantum groups (the equitable presentation [1,24,36], the quantum affine sl 2 algebra [2,18,19,22], L-operators [26,38]). For more information about the above topics, see [28,34] and the references therein.…”