scholarly journals Leonard pairs having specified end-entries

2015 ◽  
Vol 465 ◽  
pp. 43-64
Author(s):  
Kazumasa Nomura
Keyword(s):  
Leonard Pairs

scholarly journals Self-dual Leonard pairs

2019 ◽  
Vol 7 (1) ◽  
pp. 1-19
Author(s):  
Kazumasa Nomura ◽  
Paul Terwilliger
Keyword(s):  
Vector Space ◽  
The Self ◽  
The Other ◽  
Endomorphism Algebra ◽  
Comprehensive Description ◽  
Leonard Pairs ◽  
Positive Dimension ◽  
Linear Maps ◽  
Leonard Pair ◽  
Dual Case

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.

scholarly journals Leonard pairs having LB–TD form

2014 ◽  
Vol 455 ◽  
pp. 1-21 ◽  
Author(s):  
Kazumasa Nomura
Keyword(s):  
Leonard Pairs

scholarly journals Balanced Leonard pairs

2007 ◽  
Vol 420 (1) ◽  
pp. 51-69 ◽  
Author(s):  
Kazumasa Nomura ◽  
Paul Terwilliger
Keyword(s):  
Leonard Pairs

scholarly journals LEONARD PAIRS AND THE ASKEY–WILSON RELATIONS

2004 ◽  
Vol 03 (04) ◽  
pp. 411-426 ◽  
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.

scholarly journals The universal DAHA of type (C1∨,C1) and Leonard pairs of q-Racah type

2017 ◽  
Vol 533 ◽  
pp. 14-83 ◽  
Author(s):  
Kazumasa Nomura ◽  
Paul Terwilliger
Keyword(s):  
Leonard Pairs

scholarly journals Three mutually adjacent Leonard pairs

2005 ◽  
Vol 408 ◽  
pp. 19-39 ◽  
Author(s):  
Brian Hartwig
Keyword(s):  
Leonard Pairs
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