The shape of a tridiagonal pair.
Let \(V\) be a vector space over the field \(F\) with finite positive dimension. By a tridiagonal pair on \(V\), we mean a pair \((A,A^*)\) of linear transformations \(V\) that satisfy the following conditions: (i) each of \(A,A^*\) is diagonalizable, (ii) there exists an ordering \(V_0,\dots,V_d\) of the eigenspaces of \(A\) such that \(A^*V_i\subseteq V_{i-1}+V_i+V_{i+1}\) (\(0\leq i\leq d\)), where \(V_{-1}=V_{d+1}=0\), (iii) there exists an ordering \(V_0^*,...,V_\delta^*\) of the eigenspaces of \(A^*\) such that \(AV_i^*\subseteq V_{i-1}^*+V_i^*+V_{i+1}^*\) (\(0\leq i\leq \delta\)), where \(V_{-1}^*=V_{\delta+1}^*=0\), (iv) there does not exist a subspace \(W\) of \(V\), \(0\neq W\neq V\) such that \(AW\subseteq W\), \(A^*W\subseteq W\). It is known that \(d=\delta\) and the spaces \(V_i,V_i^*\) have the same dimension \(\rho_i\). The sequence (\(\rho_0,\dots,\rho_d\)) is symmetric and unimodal and it is called the shape vector of \(A,A^*\). Theorem 1.6. Let \(A,A^*\) be a tridiagonal pair on \(V\), \(F\) be an algebraically closed field with characteristic 0. If \(q\in F\) is a nonzero scalar that is not a root of unity, \(\beta=q^2+q^{-2}\) and \([A,A^2A^*-\beta AA^*A+A^*A^2]=0\), \([A^*,A^{*2}A-\beta A^*AA^*+AA^{*2}]=0\) (where \([r,s]=rs-sr\)), then \(\rho_i\leq {d\choose i}\) (\(0\leq i\leq d\)).
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