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Kazumasa Nomura

Publications and source records attributed to Kazumasa Nomura.

At least 37 records · Page 2Linked to original sources
arXiv2014

Leonard pairs having specified end-entries

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 $E^*_i : V \to V$ in a similar way. The sequence $Φ=(A, \{E_i\}_{i=0}^d, A^*, \{E^*_i\}_{i=0}^d)$ is called a Leonard system on $V$ with diameter $d$. With respect to the basis $\{v_i\}_{i=0}^d$, let $\{þ_i\}_{i=0}^d$ (resp.\ $\{a^*_i\}_{i=0}^d$) be the diagonal entries of the matrix representing $A$ (resp.\ $A^*$). With respect to the basis $\{v^*_i\}_{i=0}^d$, let $\{θ^*_i\}_{i=0}^d$ (resp.\ $\{a_i\}_{i=0}^d$) be the diagonal entries of the matrix representing $A^*$ (resp.\ $A$). It is known that $\{θ_i\}_{i=0}^d$ (resp. $\{þ^*_i\}_{i=0}^d$) are mutually distinct, and the expressions $(θ_{i-1}-θ_{i+2})/(θ_i-θ_{i+1})$, $(θ^*_{i-1}-θ^*_{i+2})/(θ^*_i - θ^*_{i+1})$ are equal and independent of $i$ for $1 \leq i \leq d-2$. Write this common value as $β+ 1$. In the present paper we consider the "end-entries" $θ_0$, $θ_d$, $θ^*_0$, $θ^*_d$, $a_0$, $a_d$, $a^*_0$, $a^*_d$. We prove that a Leonard system with diameter $d$ is determined up to isomorphism by its end-entries and $β$ if and only if either (i) $β\neq \pm 2$ and $q^{d-1} \neq -1$, where $β=q+q^{-1}$, or (ii) $β= \pm 2$ and $\text{Char}(\mathbb{F}) \neq 2$.

math.RA↗
arXiv2014

The end-parameters of a Leonard pair

Fix an algebraically closed field $\F$ and an integer $d \geq 3$. Let $V$ be a vector space over $\F$ with dimension $d+1$. A Leonard pair on $V$ is a 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. There is an object related to a Leonard pair called a Leonard system. It is known that a Leonard system is determined up to isomorphism by a sequence of scalars $(\{þ_i\}_{i=0}^d, \{þ^*_i\}_{i=0}^d, \{\vphi_i\}_{i=1}^d, \{ϕ_i\}_{i=1}^d)$, called its parameter array. The scalars $\{þ_i\}_{i=0}^d$ (resp.\ $\{þ^*_i\}_{i=0}^d$) are mutually distinct, and the expressions $(þ_{i-2} - þ_{i+1})/(þ_{i-1}-þ_{i})$, $(þ^*_{i-2} - þ^*_{i+1})/(þ^*_{i-1}-þ^*_{i})$ are equal and independent of $i$ for $2 \leq i \leq d-1$. Write this common value as $β+1$. In the present paper, we consider the "end-parameters" $þ_0$, $þ_d$, $þ^*_0$, $þ^*_d$, $\vphi_1$, $\vphi_d$, $ϕ_1$, $ϕ_d$ of the parameter array. We show that a Leonard system is determined up to isomorphism by the end-parameters and $β$. We display a relation between the end-parameters and $β$. Using this relation, we show that there are up to inverse at most $\lfloor (d-1)/2 \rfloor$ Leonard systems that have specified end-parameters. The upper bound $\lfloor (d-1)/2 \rfloor$ is best possible.

math.RA↗
arXiv2014

Leonard pairs having LB-TD form

Fix an algebraically closed field $\mathbb{F}$ and an integer $d \geq 3$. Let $\text{Mat}_{d+1}(\mathbb{F})$ denote the $\mathbb{F}$-algebra consisting of the $(d+1) \times (d+1)$ matrices that have all entries in $\mathbb{F}$. We consider a pair of diagonalizable matrices $A,A^*$ in $\text{Mat}_{d+1}(\mathbb{F})$, each acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair is called a Leonard pair in $\text{Mat}_{d+1}(\mathbb{F})$. For a Leonard pair $A,A^*$ there is a nonzero scalar $q$ that is used to describe the eigenvalues of $A$ and $A^*$. In the present paper we find all Leonard pairs $A,A^*$ in $\text{Mat}_{d+1}(\mathbb{F})$ such that $A$ is lower bidiagonal with subdiagonal entries all $1$ and $A^*$ is irreducible tridiagonal, under the assumption that $q$ is not a root of unity. This gives a partial solution of a problem given by Paul Terwilliger.

math.RA↗
arXiv2012

Krawtchouk polynomials, the Lie algebra $\mathfrak{sl}_2$, and Leonard pairs

A Leonard pair is a pair of diagonalizable linear transformations of a finite-dimensional vector space, each of which acts in an irreducible tridiagonal fashion on an eigenbasis for the other one. In the present paper we give an elementary but comprehensive account of how the following are related: (i) Krawtchouk polynomials; (ii) finite-dimensional irreducible modules for the Lie algebra ${\mathfrak{sl}_2}$; (iii) a class of Leonard pairs said to have Krawtchouk type. Along the way we obtain elementary proofs of some well-known facts about Krawtchouk polynomials, such as the three-term recurrence, the orthogonality, the difference equation, and the generating function. The paper is a tutorial meant for a graduate student or a researcher unfamiliar with the above topics.

math.RT↗
arXiv2010

Tridiagonal matrices with nonnegative entries

In this paper we characterize the nonnegative irreducible tridiagonal matrices and their permutations, using certain entries in their primitive idempotents. Our main result is summarized as follows. Let $d$ denote a nonnegative integer. Let $A$ denote a matrix in $\matR$ and let $\{þ_i\}_{i=0}^d$ denote the roots of the characteristic polynomial of $A$. We say $A$ is multiplicity-free whenever these roots are mutually distinct and contained in $\R$. In this case $E_i$ will denote the primitive idempotent of $A$ associated with $þ_i$ $(0 \leq i \leq d)$. We say $A$ is symmetrizable whenever there exists an invertible diagonal matrix $Δ\in \matR$ such that $ΔA Δ^{-1}$ is symmetric. Let $Γ(A)$ denote the directed graph with vertex set $\{0,1,...,d\}$, where $i \rightarrow j$ whenever $i \neq j$ and $A_{ij} \neq 0$. Theorem: Assume that each entry of $A$ is nonnegative. Then the following are equivalent for $0 \leq s,t \leq d$: (i) The graph $Γ(A)$ is a bidirected path with endpoints $s$, $t$: (ii) The matrix $A$ is symmetrizable and multiplicity-free. Moreover the $(s,t)$-entry of $E_i$ times $(þ_i-þ_0)...(þ_i-þ_{i-1})(þ_i-þ_{i+1})...(þ_i-þ_d)$ is independent of $i$ for $0 \leq i \leq d$, and this common value is nonzero. Recently Kurihara and Nozaki obtained a theorem that characterizes the $Q$-polynomial property for symmetric association schemes. We view the above result as a linear algebraic generalization of their theorem.

math.CO↗
arXiv2010

A classification of sharp tridiagonal pairs

Let $F$ denote a field and let $V$ denote a vector space over $F$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i\rbrace_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering $\lbrace V^*_i\rbrace_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1}=0$ and $V^*_{δ+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a {\it tridiagonal pair} on $V$. It is known that $d=δ$ and for $ 0 \leq i \leq d$ the dimensions of $V_i,V_{d-i},V^*_i, V^*_{d-i}$ coincide. The pair $A,A^*$ is called {\it sharp} whenever ${\rm dim} V_0=1$. It is known that if $F$ is algebraically closed then $A,A^*$ is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the $μ$-conjecture.

math.RA↗
arXiv2009

On the shape of a tridiagonal pair

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 the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i\rbrace_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering $\lbrace V^*_i\rbrace_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1}=0$ and $V^*_{δ+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a {\it tridiagonal pair} on $V$. It is known that $d=δ$ and for $0 \leq i \leq d$ the dimensions of $V_i$, $V^*_i$, $V_{d-i}$, $V^*_{d-i}$ coincide; we denote this common dimension by $ρ_i$. In this paper we prove that $ρ_i \leq ρ_0 \binom{d}{i}$ for $0 \leq i \leq d$. It is already known that $ρ_0=1$ if $\K$ is algebraically closed.

math.RA↗
arXiv2009

Tridiagonal pairs of $q$-Racah type and the $μ$-conjecture

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 the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\lbrace V_i\rbrace_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering $\lbrace V^*_i\rbrace_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1}=0$ and $V^*_{δ+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a {\it tridiagonal pair} on $V$. It is known that $d=δ$ and for $0 \leq i \leq d$ the dimensions of $V_i$, $V_{d-i}$, $V^*_i$, $V^*_{d-i}$ coincide. We say the pair $A,A^*$ is {\it sharp} whenever $\dim V_0=1$. It is known that if $\K$ is algebraically closed then $A,A^*$ is sharp. A conjectured classification of the sharp tridiagonal pairs was recently introduced by T. Ito and the second author. Shortly afterwards we introduced a conjecture, called the {\em $μ$-conjecture}, which implies the classification conjecture. In this paper we show that the $μ$-conjecture holds in a special case called $q$-Racah.

math.RA↗
arXiv2009

Tridiagonal pairs and the $μ$-conjecture

Let $F$ denote a field and let $V$ denote a vector space over $F$ with finite positive dimension. We consider a pair of linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfy the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\{V_i\}_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering $\{V^*_i\}_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1}=0$ and $V^*_{δ+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a {\it tridiagonal pair} on $V$. It is known that $d=δ$ and for $0 \leq i \leq d$ the dimensions of $V_i$, $V_{d-i}$, $V^*_i$, $V^*_{d-i}$ coincide. We say the pair $A,A^*$ is {\it sharp} whenever $\dim V_0=1$. It is known that if $F$ is algebraically closed then $A,A^*$ is sharp. A conjectured classification of the sharp tridiagonal pairs was recently introduced by T. Ito and the second author. We present a result which supports the conjecture. Given scalars $\{þ_i\}_{i=0}^d$, $\{þ^*_i\}_{i=0}^d$ in $F$ that satisfy the known constraints on the eigenvalues of a tridiagonal pair, we define an $F$-algebra $T$ by generators and relations. We consider the algebra $e^*_0Te^*_0$ for a certain idempotent $e^*_0 \in T$. Let $R$ denote the polynomial algebra over $F$ involving $d$ variables.We display a surjective algebra homomorphism $μ: R \to e^*_0Te^*_0$. We conjecture that $μ$ is an isomorphism. We show that this $μ$-conjecture implies the classification conjecture, and that the $μ$-conjecture holds for $d\leq 5$.

math.RA↗
arXiv2008

The structure of a tridiagonal pair

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 the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering $\{V_i\}_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_i + V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering $\{V^*_i\}_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_i + V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1}=0$ and $V^*_{δ+1}=0$; (iv)there is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a tridiagonal pair on $V$. It is known that $d=δ$ and for $0 \leq i \leq d$ the dimensions of $V_i, V_{d-i}, V^*_i, V^*_{d-i}$ coincide. In this paper we show that the following (i)--(iv) hold provided that $K$ is algebraically closed: (i) Each of $V_0$, $V^*_0$, $V_d$, $V^*_d$ has dimension 1. (ii) There exists a nondegenerate symmetric bilinear form $(,)$ on $V$ such that $(Au,v)=(u,Av)$ and $(A^*u,v)=(u,A^*v)$ for all $u,v \in V$. (iii) There exists a unique anti-automorphism of $End(V)$ that fixes each of $A,A^*$. (iv) The pair $A,A^*$ is determined up to isomorphism by the data $(\{þ_i\}_{i=0}^d; \{þ^*_i\}_{i=0}^d; \{ζ_i\}_{i=0}^d)$, where $þ_i$ (resp. $þ^*_i$) is the eigenvalue of $A$ (resp. $A^*$) on $V_i$ (resp. $V^*_i$), and $\{ζ_i\}_{i=0}^d$ is the split sequence of $A,A^*$ corresponding to $\{þ_i\}_{i=0}^d$ and $\{þ^*_i\}_{i=0}^d$.

math.RA↗
arXiv2008

Towards a classification of the tridiagonal pairs

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. Let $End(V)$ denote the $K$-algebra consisting of all $K$-linear transformations from $V$ to $V$. We consider a pair $A,A^* \in End(V)$ that satisfy (i)--(iv) below: (i) Each of $A,A^*$ is diagonalizable. (ii) There exists an ordering $\{V_i\}_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$. (iii) There exists an ordering $\{V^*_i\}_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1}=0$ and $V^*_{δ+1}=0$. (iv) There is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a {\em tridiagonal pair} on $V$. Let $E^*_0$ denote the element of $End(V)$ such that $(E^*_0-I)V^*_0=0$ and $E^*_0V^*_i=0$ for $1 \leq i \leq d$. Let $D$ (resp. $D^*$) denote the $K$-subalgebra of $End(V)$ generated by $A$ (resp. $A^*$). In this paper we prove that the span of $E^*_0 D D^*DE^*_0$ equals the span of $E^*_0D E^*_0DE^*_0$, and that the elements of $E^*_0 D E^*_0$ mutually commute. We relate these results to some conjectures of Tatsuro Ito and the second author that are expected to play a role in the classification of tridiagonal pairs.

math.RA↗
arXiv2007

Sharp tridiagonal pairs

Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite positive dimension. We consider a pair of $K$-linear transformations $A:V \to V$ and $A^*:V \to V$ that satisfies the following conditions: (i) each of $A,A^*$ is diagonalizable; (ii) there exists an ordering ${V_i}_{i=0}^d$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$; (iii) there exists an ordering ${V^*_i}_{i=0}^δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1}=0$ and $V^*_{δ+1}=0$; (iv) there is no subspace $W$ of $V$ such that $AW \subseteq W$, $A^* W \subseteq W$, $W \neq 0$, $W \neq V$. We call such a pair a {\em tridiagonal pair} on $V$. It is known that $d=δ$ and for $0 \leq i \leq d$ the dimensions of $V_i$, $V_{d-i}$, $V^*_i$, $V^*_{d-i}$ coincide. We say the pair $A,A^*$ is {\em sharp} whenever $\dim V_0=1$. A conjecture of Tatsuro Ito and the second author states that if $K$ is algebraically closed then $A,A^*$ is sharp. In order to better understand and eventually prove the conjecture, in this paper we begin a systematic study of the sharp tridiagonal pairs.

math.RA↗
arXiv2007

Transition maps between the 24 bases for a Leonard pair

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 feature of these bases is that with respect to each of them the matrices that represent $A$ and $A^*$ are (i) diagonal and irreducible tridiagonal or (ii) irreducible tridiagonal and diagonal or (iii) lower bidiagonal and upper bidiagonal or (iv) upper bidiagonal and lower bidiagonal. For each ordered pair of bases among the 24, there exists a unique linear transformation from $V$ to $V$ that sends the first basis to the second basis; we call this the transition map. In this paper we find each transition map explicitly as a polynomial in $A,A^*$.

math.RA↗
arXiv2006

The split decomposition of a tridiagonal pair

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)--(iv) below: (i) Each of $A$, $A^*$ is diagonalizable. (ii) There exists an ordering $V_{0},V_{1},...,V_{d}$ of the eigenspaces of $A$ such that $A^* V_i \subseteq V_{i-1} + V_{i} + V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$, $V_{d+1}=0$. (iii) There exists an ordering $V^*_{0},V^*_{1},...,V^*_δ$ of the eigenspaces of $A^*$ such that $A V^*_i \subseteq V^*_{i-1} + V^*_{i} + V^*_{i+1}$ for $0 \leq i \leq δ$, where $V^*_{-1}=0$, $V^*_{δ+1}=0$. (iv) There is no subspace $W$ of $V$ such that both $AW \subseteq W$, $A^* W \subseteq W$, other than W=0 and $W=V$. We call such a pair a tridiagonal pair on $V$. In this note we obtain two results. First, we show that each of $A,A^*$ is determined up to affine transformation by the $V_i$ and $V^*_i$. Secondly, we characterize the case in which the $V_i$ and $V^*_i$ all have dimension one. We prove both results using a certain decomposition of $V$ called the split decomposition.

math.RA↗
arXiv2006

Affine transformations of a 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 $A : V \to V$ 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 $x$, $c$, $x^*$, $c^*$ denote scalars in $K$ with $x$, $x^*$ nonzero, and note that $xA+cI$, $x^*A^* + c^*I$ is a Leonard pair on $V$. We give necessary and sufficient conditions for this Leonard pair to be isomorphic to the Leonard pair $A$, $A^*$. We also give necessary and sufficient conditions for this Leonard pair to be isomorphic to the Leonard pair $A^*$, $A$.

math.RA↗
arXiv2006

The switching element for a Leonard pair

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 for $V$ referred to in (i) (resp. (ii)). We show that there exists a unique linear transformation $S: V \to V$ that sends $v_0$ to a scalar multiple of $v_d$, fixes $w_0$, and sends $w_i$ to a scalar multiple of $w_i$ for $1 \leq i \leq d$. We call $S$ the {\it switching element}. We describe $S$ from many points of view.

math.RA↗
arXiv2006

Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair

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 set of linear transformations $X:V \to V$ such that the matrix representing $X$ with respect to the basis (i) is tridiagonal and the matrix representing $X$ with respect to the basis (ii) is tridiagonal. We show that $\cal X$ is spanned by $I$, $A$, $A^*$, $AA^*$, $A^*A$, and these elements form a basis for $\cal X$ provided the dimension of $V$ is at least 3.

math.RA↗
arXiv2006

Matrix units associated with the split basis of a Leonard pair

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.

math.RA↗