Two linear transformations each tridiagonal with respect to an eigenbasis of the other; the TD- D canonical form and the LB- UB canonical form

By Paul M. Terwilliger. 2003

135 citations recorded by the source corpus.

Abstract Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A : V → V and B : V → V which satisfy both (i), (ii) below. (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing B is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing B is irreducible tridiagonal. We call such a pair a Leonard pair on V. We introduce two canonical forms for Leonard pairs. We call these the TD–D canonical form and the LB–UB canonical form. In the TD–D canonical form the Leonard pair is represented by an irreducible tridiagonal matrix and a diagonal matrix, subject to a certain normalization. In the LB–UB canonical form the Leonard pair is represented by a lower bidiagonal matrix and an upper bidiagonal matrix, subject to a certain normalization. We describe the two canonical forms in detail. As an application we obtain the following results. Given square matrices A , B over K , with A tridiagonal and B diagonal, we display a necessary and sufficient condition for A , B to represent a Leonard pair. Given square matrices A , B over K , with A lower bidiagonal and B upper bidiagonal, we display a necessary and sufficient condition for A , B to represent a Leonard pair. We briefly discuss how Leonard pairs correspond to the q-Racah polynomials and some related polynomials in the Askey scheme. We present some open problems concerning Leonard pairs.

Subfields: Artificial Intelligence, Electrical and Electronic Engineering, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a89e34f61c417441df0e12df1e63217844a228f2

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Electrical and Electronic Engineering

In Geometry and Topology

Citations in the corpus

A LIMIT q = −1 FOR THE BIG Q-JACOBI POLYNOMIALS (2012, methodology): “The Askey-Wilson polynomials are described by the AW(3)-algebra [20], [17].”
The shape of a tridiagonal pair (2003, unknown): “There is a classification of Leonard pairs in [12].”
The Tetrahedron algebra, the Onsager algebra, and the sl2 loop algebra (2007, background): “Over the years O has been investigated many times in connection with solvable lattice models [1], [2], [3], [5], [6], [7], [8], [15], [16], [22], [26], [27], [28], [30], [39] representation theory [13], [14], [21] Kac-Moody algebras [12], [32], [33] tridiagonal pairs [23], [24], [36], [37], [38] and”
Tridiagonal pairs of q-Racah type (2008, background): “1], and these are classified up to isomorphism [53,56].”
Dual polar graphs, the quantum algebra U_q(sl_2), and Leonard systems of dual q-Krawtchouk type (2012, background): “[26] Let d denote a nonnegative integer and let V denote a vector space over C with dimension d+ 1.”
Dunkl shift operators and Bannai-Ito polynomials (2011, background): “Further developments of the theory of BI polynomials and their applications can be found in papers by Terwilliger [29], [30], Curtin [8], [9] and Vidūnas [31].”