# Leonard pairs and Askey-Wilson relations

*Statistical Mechanics/Combinatorics Seminar*

*by Raimundas Vidunas*

*Institution:*Kyushu University, Japan

*Date: Thu 7th September 2006*

*Time: 3:15 PM*

*Location: Hercus Theatre, Deptartment of Physics, The University of Melbourne*

*Abstract*: A Leonard pair (A,B) is a pair of linear transformations on a

finite dimensional vector space, each having a tri-diagonal form with respect

to an eigenbasis of the other. They occur in representations of sl 2 and related algebras, or with some orthogonal polynomials and association schemes. If (A,B) is a Leonard pair, then A,B satisfy so called Askey-Wilson relations.

These are a pair of cubic non-commutative relations. One of the questions is, how many Leonard pairs can satisfy the same Askey-Wilson relations?

*For More Information:* Dr. Iwan Jensen Phone: +61 3 8344 5214 I.Jensen@ms.unimelb.edu.au