Affine transformations of a Leonard pair

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Kazumasa Nomura
Paul Terwilliger

Abstract

Let K denote a field and let V denote a vector space over K withfinite positive
dimension. An ordered pair is considered of linear transformations A : V → V and A : V → V that
satisfy (i) and (ii) below:


(i) There exists a basis for V withrespect to whichthe matrix representing A is irreducible
tridiagonal and the matrix representing A is diagonal.


(ii) There exists a basis for V withrespect to whichthe matrix representing A is irreducible
tridiagonal and the matrix representing A is diagonal.


Sucha pair is called a Leonard pair on V . Let ξ, ζ, ξ, ζ denote scalars in K with ξ, ξ nonzero, and
note that ξA + ζI, ξ∗A∗ + ζ∗I is a Leonard pair on V . Necessary and sufficient conditions are given
for this Leonard pair to be isomorphic to A, A. Also given are necessary and sufficient conditions
for this Leonard pair to be isomorphic to the Leonard pair A, A.

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