Tridiagonal pairs of Krawtchouk type

TL;DR

This paper studies tridiagonal pairs of Krawtchouk type, establishing their symmetry properties, existence of invariant bilinear forms, and isomorphisms under certain transformations, contributing to the algebraic understanding of these structures.

Contribution

It characterizes Krawtchouk type tridiagonal pairs, proves the existence of a symmetric bilinear form invariant under both operators, and identifies isomorphisms among related pairs.

Findings

Existence of a nondegenerate symmetric bilinear form invariant under A and A*

Isomorphisms between pairs (A,A*), (-A,-A*), (A*,A), (-A*,-A)

Structural properties of Krawtchouk type tridiagonal pairs

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 $A,A^*$ denote a tridiagonal pair on $V$ with diameter $d$. We say that $A,A^*$ has Krawtchouk type whenever the sequence $\lbrace d-2i\rbrace_{i=0}^d$ is a standard ordering of the eigenvalues of $A$ and a standard ordering of the eigenvalues of $A^*$. Assume $A,A^*$ has Krawtchouk type. We show that there exists a nondegenerate symmetric bilinear form $< , >$ on $V$ such that $<Au,v>= < u,Av>$ and $<A^*u,v >= < u,A^*v>$ for $u,v\in V$. We show that the following tridiagonal pairs are isomorphic: (i) $A,A^*$; (ii) $-A,-A^*$; (iii) $A^*,A$; (iv) $-A^*,-A$. We give a number of related results and conjectures.