Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy (i), (ii) below:
(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 {\em Leonard pair} on $V$. It is known that there exists a basis for $V$ with respect to which the matrix representing $A$ is lower bidiagonal and the matrix representing $A^*$ is upper bidiagonal. In this paper we give some formulae involving the matrix units associated with this basis.
Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy (i), (ii) below: (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 {\em Leonard pair} on $V$. In the literature on Leonard pai…
We introduce and discuss an Erdős-Ko-Rado basis for the underlying vector space of a Leonard system $Φ= (A; A^*; \{E_i\}_{i=0}^d ; \{E_i^* \}_{i=0}^d)$ that satisfies a mild condition on the eigenvalues of $A$ and $A^*$. We describe the transition matrices to/from other known bases, as well as the matrices representing $A$ and $A^*$ with respect to the new basis. We also discuss how these results can be viewed as a generalization of the linear programming method used previously in the proofs of the "Erdős-Ko-Rado theorems" for several classical families of $Q$-polynomial distance-regular graph…
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 $A:V\to V$ and $A^*:V\to V$ 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…
Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\to V$ and $A^*: V\to V$ that satisfy (i) and (ii) below: (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 $V$. In this paper, we characterize the Leonard pairs using the noti…
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 $A:V\to V$ and $B:V\to V$ which satisfy both (i), (ii) below. (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is irreducible tridiagonal and the matrix representing $B$ is diagonal; (ii) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $B$ is irreducible tridiagonal. We call such a pair a Leonard pair on $V$. We introduce two canonical for…
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 $A : V \to V$ and $A^* : V \to V$ that satisfy (i) and (ii) below: (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 $V$. Let $x$, $c$, $x^*$, $c…
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 $A:V\to V$ and $A^*:V\to V$ that satisfy conditions (i), (ii) below. (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 diagonal and the matrix representing $A^*$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. Let $A,A^*$ den…
Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A : V \to V$ and $A^* : V \to V$ that satisfy (i) and (ii) below: (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 {\it Leonard pair} on $V$. Let $\cal X$ denote the se…
Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\rightarrow V$ and $A^{*}: V\rightarrow V$ that satisfy (i) and (ii) below. (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 $V$. In the literature, there are some paramet…
Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\to V$ and $A^*: V\to V$ that satisfy (i) and (ii) below. (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 $V$. Arlene Pascasio recently obtained a characterization of the $Q$…
Let $V$ denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations $A: V\rightarrow V$ and $A^*: V\rightarrow V$ that satisfy (i) and (ii) below. (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 $V$. Very roughly speaking, a Leonard pair is a line…
Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. By a Leonard pair on $V$ we mean an ordered pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following two conditions: (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. Let $v^*_0, ..., v^*_d$ (resp. $v_0, ..…
Let $V$ denote a vector space with finite positive dimension. We consider a pair of linear transformations $A : V \to V$ and $A^* : V \to V$ that satisfy (i) and (ii) below: (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 $V$. In an earlier paper we described 24 special bases for $V$. One featu…
Let $V$ denote a vector space with finite positive dimension. We consider a pair of linear transformations $A : V \to V$ and $A^* : V \to V$ that satisfy (i) and (ii) below: (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 {\em Leonard pair} on $V$. Let $v_0,v_1,...,v_d$ (resp. $w_0,w_1,...,w_d$) denote a basis…
Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy both conditions below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal and the matrix representing $A^*$ is irreducible tridiagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is diagonal and the matrix representing $A$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. Referring to the above Leonar…
Let $K$ denote a field, and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*: V \to V$ that satisfy (i), (ii) below: (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 {\em Leonard pair} on $V$. In this paper we investigate th…
Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of linear transformations $A:V\to V$ and $A^*:V\to V$ that satisfy both conditions below: (i) There exists a basis for $V$ with respect to which the matrix representing $A$ is diagonal, and the matrix representing $A^*$ is irreducible tridiagonal. (ii) There exists a basis for $V$ with respect to which the matrix representing $A^*$ is diagonal, and the matrix representing $A$ is irreducible tridiagonal. We call such a pair a Leonard pair on $V$. Refining this notion a bit, we in…
Let $V$ denote a vector space over C with finite positive dimension. By a {\em Leonard triple} on $V$ we mean an ordered triple of linear operators on $V$ such that for each of these operators there exists a basis of $V$ with respect to which the matrix representing that operator is diagonal and the matrices representing the other two operators are irreducible tridiagonal. Let $D$ denote a positive integer and let $Q_D$ denote the graph of the $D$-dimensional hypercube. Let $X$ denote the vertex set of $Q_D$ and let $A$ denote the adjacency matrix of $Q_D$. Fix $x \in X$ and let $A^*$ denote t…
Fix an algebraically closed field $\mathbb{F}$ and an integer $d \geq 3$. Let $V$ be a vector space over $\mathbb{F}$ with dimension $d+1$. A Leonard pair on $V$ is an ordered pair of diagonalizable linear transformations $A: V \to V$ and $A^* : V \to V$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Let $\{v_i\}_{i=0}^d$ (resp.\ $\{v^*_i\}_{i=0}^d$) be such an eigenbasis for $A$ (resp.\ $A^*$). For $0 \leq i \leq d$ define a linear transformation $E_i : V \to V$ such that $E_i v_i=v_i$ and $E_i v_j =0$ if $j \neq i$ $(0 \leq j \leq d)$. Define $E^*_i : …
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 $A:V\to V$ and $A^*:V\to V$ that satisfy conditions (i), (ii) below. (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 diagonal and the matrix representing $A^*$ is irreducible tridiagonal. We call such a pair a Leonard pair on $V$. We give an overview o…
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 $A:V\to V$ and $A^*:V\to V$ that satisfy conditions (i), (ii) below. (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 diagonal and the matrix representing $A^*$ is irreducible tridiagonal. We call such a pair a {\it Leonard pair} on $V$. The structure o…
This article discusses the representation theory of noncommutative algebras reality-based algebras with positive degree map over their field of definition. When the standard basis contains exactly two nonreal elements, the main result expresses the noncommutative simple component as a generalized quaternion algebra over its field of definition. The field of real numbers will always be a splitting field for this algebra, but there are noncommutative table algebras of dimension $6$ with rational field of definition for which it is a division algebra. The approach has other applications, one of w…
