The split decomposition of a tridiagonal pair
| dc.creator | Nomura, Kazumasa | |
| dc.creator | Terwilliger, Paul | |
| dc.date | 2006-12-16 | |
| dc.date.accessioned | 2026-07-07T07:35:40Z | |
| dc.date.available | 2026-07-07T07:35:40Z | |
| 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)--(iv) below: (i) Each of $A$, $A^*$ is diagonalizable. (ii) There exists an ordering $V_{0},V_{1},...,V_{d}$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$, $V_{d+1}=0$. (iii) There exists an ordering $V^*_{0},V^*_{1},...,V^*_δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1}=0$, $V^*_{δ+1}=0$. (iv) There is no subspace $W$ of $V$ such that both $AW \subseteq W$, $A^* W \subseteq W$, other than W=0 and $W=V$. We call such a pair a tridiagonal pair on $V$. In this note we obtain two results. First, we show that each of $A,A^*$ is determined up to affine transformation by the $V_i$ and $V^*_i$. Secondly, we characterize the case in which the $V_i$ and $V^*_i$ all have dimension one. We prove both results using a certain decomposition of $V$ called the split decomposition. | |
| dc.description | 7 pages | |
| dc.identifier | https://arxiv.org/abs/math/0612460 | |
| dc.identifier | http://arxiv.org/abs/math/0612460 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/120170 | |
| dc.subject | Rings and Algebras | |
| dc.subject | 05E35 | |
| dc.title | The split decomposition of a tridiagonal pair | |
| dc.type | text |