Leonard triples extended from a given totally almost bipartite Leonard pair of Bannai/Ito type

Abstract Let F denote a field and let V denote a vector space over F with finite positive dimension. Consider a pair A, A* of diagonalizable F-linear maps on V, each of which acts on an eigenbasis for the other one in an irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the self-dual case in which there exists an automorphism of the endomorphism algebra of V that swaps A and A*. Such an automorphism is unique, and called the duality A ↔ A*. In the present paper we give a comprehensive description of this duality. In particular,we display an invertible F-linearmap T on V such that the map X → TXT−1is the duality A ↔ A*. We express T as a polynomial in A and A*. We describe how T acts on 4 flags, 12 decompositions, and 24 bases for V.

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LEONARD PAIRS AND THE ASKEY–WILSON RELATIONS

Journal of Algebra and Its Applications ◽

10.1142/s0219498804000940 ◽

2004 ◽

Vol 03 (04) ◽

pp. 411-426 ◽

Cited By ~ 78

Author(s):

PAUL TERWILLIGER ◽

RAIMUNDAS VIDUNAS

Keyword(s):

Vector Space ◽

Linear Transformations ◽

Leonard Pairs ◽

Positive Dimension ◽

The Matrix ◽

Leonard Pair

Let K denote a field and let V denote a vector space over K with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A*:V→V which satisfy the following two properties: (i) There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A* is diagonal. (ii) There exists a basis for V with respect to which the matrix representing A* is irreducible tridiagonal and the matrix representing A is diagonal. We call such a pair a Leonard pair on V. Referring to the above Leonard pair, we show there exists a sequence of scalars β,γ,γ*,ϱ,ϱ*,ω,η,η* taken from K such that both [Formula: see text] The sequence is uniquely determined by the Leonard pair provided the dimension of V is at least 4. The equations above are called the Askey–Wilson relations.

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