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BY 4.0 license Open Access Published by De Gruyter Open Access January 1, 2019

Self-dual Leonard pairs

  • Kazumasa Nomura EMAIL logo and Paul Terwilliger
From the journal Special Matrices


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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Received: 2018-05-08
Accepted: 2018-09-22
Published Online: 2019-01-01

© by Kazumasa Nomura, Paul Terwilliger, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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