The switching element for a Leonard pair
The authors continue their study of Leonard pairs and in this paper the attention is focused on the so-called switching element. Given a \(d+1\;(d>0)\) dimensional vector space \(V\), a pair of linear transformations \(A,A^*\) is called a Leonard pair if (i) there exists a basis \(v=\{v_i\}_{i=0}^d\) with mat\(_{v,v}\,A\) irreducible tri-diagonal and mat\(_{v,v}\,A^*\) diagonal, (ii) there exists a basis \(w=\{w_i\}_{i=0}^d\) with mat\(_{w,w}\,A\) irreducible tri-diagonal and mat\(_{w,w}\,A^*\) diagonal. (Note: the * in \(A^*\) does \textbf{not} mean the customary conjugate transpose \(\overline{A}^{\text{t}}\).) After a short introduction and some sections stating results from previous papers (Leonard system, \(D_4\)-action, parameter array, first and second split sequence), finally the switching element of the Leonard system \(\Phi\) connected with the Leonard pair \(\{A,A^*\}\) is introduced. The main property of the switching element \(S\) is: it is the unique linear transformation \(S:V\rightarrow V\) with \(Sv_0=\lambda v_d,\;Sw_0=w_0,\;Sw_i=\mu_i w_i\;(1\leq i\leq d)\). It is outside the scope of this review to give all the notions and notations explicitly her; for the initiated it will suffice to state that the body of the paper describes the switching element from various points of view: --- as a polynomial, - by its action on decompositions, --- by its action on the flags, --- its connection with the decomposition, --- as a matrix. The paper then gives for a Leonard pair a representation in matrix form and a characterization in terms of the switching element; finally, some open problems are stated.
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