Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of $K$-linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfies the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering ${ V_i}_{ i=0}^d$ 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 $V_{ -1}=0$ and $V_{ d+1}=0$; (iii) there exists an ordering ${ V^*_i}_{ i=0}^\delta$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{ i-1} + V^*_{ i} + V^*_{ i+1}$ for $0 \leq i \leq \delta$, where $V^*_{ -1}=0$ and $V^*_{ \delta+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a { \em tridiagonal pair} on $V$. It is known that $d=\delta$ and for $0 \leq i \leq d$ the dimensions of $V_i$, $V_{ d-i}$, $V^*_i$, $V^*_{ d-i}$ coincide. We say the pair $A,A^*$ is { \em sharp} whenever $\dim V_0=1$. A conjecture of Tatsuro Ito and the second author states that if $K$ is algebraically closed then $A,A^*$ is sharp. In order to better understand and eventually prove the conjecture, in this paper we begin a systematic study of the sharp tridiagonal pairs.
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