The determinant of $AA^*-A^*A$ for a Leonard pair $A,A^*$
| dc.creator | Nomura, Kazumasa | |
| dc.creator | Terwilliger, Paul | |
| dc.date | 2005-11-26 | |
| dc.date.accessioned | 2026-07-07T06:51:45Z | |
| dc.date.available | 2026-07-07T06:51:45Z | |
| dc.description | Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*: V \to V$ that satisfy (i), (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 {\em Leonard pair} on $V$. In this paper we investigate the commutator $AA^*-A^*A$. Our results are as follows. First assume the dimension of $V$ is even. We show $AA^*-A^*A$ is invertible and display several attractive formulae for the determinant. Next assume the dimension of $V$ is odd. We show that the null space of $AA^*-A^*A$ has dimension 1. We display a nonzero vector in this null space. We express this vector as a sum of eigenvectors for $A$ and as a sum of eigenvectors for $A^*$. | |
| dc.description | 11 pages | |
| dc.identifier | https://arxiv.org/abs/math/0511641 | |
| dc.identifier | http://arxiv.org/abs/math/0511641 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/105092 | |
| dc.subject | Rings and Algebras | |
| dc.subject | Combinatorics | |
| dc.subject | 15A15 | |
| dc.title | The determinant of $AA^*-A^*A$ for a Leonard pair $A,A^*$ | |
| dc.type | text |