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 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.

Compatibility and companions for Leonard pairs

“…42 we get (ii). (ii) ⇒ (i) By the construction, B i,i−1 = A i,i−1 = 1 for 1 ≤ i ≤ d. By Corollary 6.42 and (ii), we haveB i−1,i = A i−1,i for 1 ≤ i ≤ d. Now useLemma 11.5.…”

“…Leonard pairs have applications to many other areas of mathematics and physics, such as Lie theory [4,21,22,25,29,39], quantum groups [1, 2, 12-14, 26, 28, 30], spin models [16][17][18]41], double affine Hecke algebras [23,24,32,33,40], partially ordered sets [35,36,45,55], and exactly solvable models in statistical mechanics [5][6][7][8][9][10][11]. For more information about Leonard pairs and related topics, see [39,42,44,46,48].…”

Distance-regular graphs, the subconstituent algebra, and the $Q$-polynomial property

“…Comprehensive treatments can be found in [3,32,36,37,[67][68][69]78]. In addition, there are papers about the thin condition [10,21,24,64,69,82], irreducible T -modules with endpoint one [30], Γ being bipartite [6,13,14,41], Γ being almost-bipartite [9,42], Γ being dual bipartite [22], Γ being almost dual bipartite [23], Γ being 2-homogeneous [15,17,18,53], Γ being tight [56], Γ being a hypercube [27], Γ being a Doob graph [63], Γ being a Johnson graph [49,62], Γ being a Grassmann graph [48], Γ being a dual polar graph [84], Γ having a spin model in the Bose-Mesner algebra [16,52]. Some miscellaneous topics about irreducible T -modules can be found in [26,34,35,40,43,44,55,59,60,…”

“…Let h be given, and pick z ∈ X such that ∂(x, z) = h. We compute the (x, z)-entry of each term in (53). We do this using Lemma 19.1 (with ℓ = 1 and y = x) along with (52).…”

Compatibility and companions for Leonard pairs

“…Leonard pairs have applications to many other areas of mathematics and physics, such as Lie theory [25,39,4,21,22,29], quantum groups [1,12,28,30,13,2,14,26], spin models [17,41,18,16], double affine Hecke algebras [40,23,24,32,33], partially ordered sets [35,45,55,36], and exactly solvable models in statistical mechanics [5,6,7,8,9,10,11]. For more information about Leonard pairs and related topics, see [48,46,39,42,44].…”