2004/06/28 by Paul Terwilliger · 1 citation
Mathematics · Physics and Astronomy · #math.RA #math-ph #math.MP #msc:17B37 #msc:05E30
published as Rocky Mountain J. Math. 32 (2002), 827--888 · 50 pages, 1 figure
arxiv created 2004/06/28 · arxiv updated 2009/12/01
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→ V and A^*:V→ V that satisfy both conditions below: (i) There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^* is irreducible tridiagonal. (ii) There exists a basis for V with respect to which the matrix representing A^* is diagonal and the matrix representing A is irreducible tridiagonal. We call such a pair a \it Leonard pair on V. Referring to the above Leonard pair, we investigate 24 bases for V on which the action of A and A^* takes an attractive form. With respect to each of these bases, the matrices representing A and A^* are either diagonal, lower bidiagonal, upper bidiagonal, or tridiagonal.