How to recognize a Leonard pair

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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Matrix units associated with the split basis of a Leonard pair

Linear Algebra and its Applications ◽

10.1016/j.laa.2006.03.009 ◽

2006 ◽

Vol 418 (2-3) ◽

pp. 775-787 ◽

Cited By ~ 16

Author(s):

Kazumasa Nomura ◽

Paul Terwilliger

Keyword(s):

Leonard Pair ◽

Matrix Units

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The end-parameters of a Leonard pair

Linear Algebra and its Applications ◽

10.1016/j.laa.2014.08.004 ◽

2014 ◽

Vol 462 ◽

pp. 88-109 ◽

Cited By ~ 1

Author(s):

Kazumasa Nomura

Keyword(s):

Leonard Pair

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Some trace formulae involving the split sequences of a Leonard pair

Linear Algebra and its Applications ◽

10.1016/j.laa.2005.08.019 ◽

2006 ◽

Vol 413 (1) ◽

pp. 189-201 ◽

Cited By ~ 19

Author(s):

Kazumasa Nomura ◽

Paul Terwilliger

Keyword(s):

Trace Formulae ◽

Leonard Pair

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Transition maps between the 24 bases for a Leonard pair

Linear Algebra and its Applications ◽

10.1016/j.laa.2009.03.007 ◽

2009 ◽

Vol 431 (5-7) ◽

pp. 571-593 ◽

Cited By ~ 1

Author(s):

Kazumasa Nomura ◽

Paul Terwilliger

Keyword(s):

Leonard Pair

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A Linear Map Acts as a Leonard Pair with Each of the Generators of Usl2

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2020/3593296 ◽

2020 ◽

Vol 2020 ◽

pp. 1-9

Author(s):

Hasan Alnajjar

Keyword(s):

Linear Combination ◽

Closed Field ◽

Algebraically Closed Field ◽

Linear Map ◽

Leonard Pair

Let ℱ denote an algebraically closed field with a characteristic not two. Fix an integer d≥3; let x, y, and z be the equitable basis of sl2 over ℱ. Let V denote an irreducible sl2-module with dimension d+1; let A∈EndV. In this paper, we show that if each of the pairs A,x, A,y, and A,z acts on V as a Leonard pair, then these pairs are of Krawtchouk type. Moreover, A is a linear combination of 1, x, y, and z.

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