Near-bipartite Leonard pairs

Abstract

Let $\F$ denote a field, and let $V$ denote a vector space over $\F$ with finite positive dimension. A Leonard pair on $V$ is an ordered pair of diagonalizable $\F$-linear maps $A:V\to V$ and $A^{\ast }:V\to V$ that each act on an eigenbasis for the other in an irreducible tridiagonal fashion. Let $A,A^{\ast }$ denote a Leonard pair on $V$. Let $\{v_i{\}}_{i=0}^d$ denote an eigenbasis for $A^{\ast }$ on which $A$ acts in an irreducible tridiagonal fashion. For $0\le i\le d$ define an $\F$-linear map $E_i^{\ast }:V\to V$ such that $E_i^{\ast }{v}_i=v_i$ and $E_i^{\ast }{v}_j=0$ if $j\ne i$ $(0\le j\le d)$. The map $F={\sum }_{i=0}^d{E}_i^{\ast }A{E}_i^{\ast }$ is called the flat part of $A$. The Leonard pair $A,A^{\ast }$ is bipartite whenever $F=0$. The Leonard pair $A,A^{\ast }$ is said to be near-bipartite whenever the pair $A-F,A^{\ast }$ is a Leonard pair on $V$. In this case, the Leonard pair $A-F,A^{\ast }$ is bipartite, and called the bipartite contraction of $A,A^{\ast }$. Let $B,B^{\ast }$ denote a bipartite Leonard pair on $V$. By a near-bipartite expansion of $B,B^{\ast }$ we mean a near-bipartite Leonard pair on $V$ with bipartite contraction $B,B^{\ast }$. In the present paper we have three goals. Assuming $\F$ is algebraically closed, (i) we classify up to isomorphism the near-bipartite Leonard pairs over $\F$; (ii) for each near-bipartite Leonard pair over $\F$ we describe its bipartite contraction; (iii) for each bipartite Leonard pair over $\F$ we describe its near-bipartite expansions.