A characterization of Leonard pairs using the notion of a tail

Edward Hanson (University of Wisconsin - Madison, USA)

Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\rightarrow V$ and $A^*: V\rightarrow V$ that satisfy (i) and (ii) below:
(i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $A^*$ is diagonal.
(ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is irreducible tridiagonal and the matrix representing $A$ is diagonal.
We call such a pair a {\it Leonard pair} on $V$. In this talk, we characterize the Leonard pairs using the notion of a tail. This notion is borrowed from algebraic graph theory.