Combinatorics

Discrete mathematics

Physics

Vector space

Pure mathematics

Dual (grammatical number)

Dimension (graph theory)

Mathematics

Symmetric matrix

Endomorphism

Literature

Automorphism

Eigenvalues and eigenvectors

Duality (order theory)

Quantum mechanics

Invertible matrix

Field (mathematics)

Diagonalizable matrix

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 −1 is 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.

Self-dual Leonard pairs