Tridiagonal pairs of Krawtchouk type

Abstract

Let $K$ denote an algebraically closed field with characteristic 0 and let $V$ denote a vector space over $K$ with finite positive dimension. Let $A,A^{\ast }$ denote a tridiagonal pair on $V$ with diameter $d$. We say that $A,A^{\ast }$ has Krawtchouk type whenever the sequence $\{d-2i{\}}_{i=0}^d$ is a standard ordering of the eigenvalues of $A$ and a standard ordering of the eigenvalues of $A^{\ast }$. Assume $A,A^{\ast }$ has Krawtchouk type. We show that there exists a nondegenerate symmetric bilinear form $<,>$ on $V$ such that $<Au,v>=<u,Av>$ and $<A^{\ast }u,v>=<u,A^{\ast }v>$ for $u,v\in V$. We show that the following tridiagonal pairs are isomorphic: (i) $A,A^{\ast }$; (ii) $-A,-A^{\ast }$; (iii) $A^{\ast },A$; (iv) $-A^{\ast },-A$. We give a number of related results and conjectures.

20 pages

Cited by in corpus (3)

• Towards a classification of the tridiagonal pairs
• Sharp tridiagonal pairs
• The structure of a tridiagonal pair