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Kazumasa Nomura

Publications and source records attributed to Kazumasa Nomura.

39 records · Page 3Linked to original sources
arXiv2005

The determinant of $AA^*-A^*A$ for a Leonard pair $A,A^*$

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 the commutator $AA^*-A^*A$. Our results are as follows. First assume the dimension of $V$ is even. We show $AA^*-A^*A$ is invertible and display several attractive formulae for the determinant. Next assume the dimension of $V$ is odd. We show that the null space of $AA^*-A^*A$ has dimension 1. We display a nonzero vector in this null space. We express this vector as a sum of eigenvectors for $A$ and as a sum of eigenvectors for $A^*$.

math.RA↗
arXiv2005

Some trace formulae involving the split sequences 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 $V$. In the literature on Leonard pairs there exist two parameter sequences called the first split sequence and the second split sequence. We display some attractive formulae for the first and second split sequence that involve the trace function.

math.RA↗
arXiv2005

Balanced Leonard Pairs

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. $v_0, ..., v_d$) denote a basis for $V$ that satisfies (i) (resp. (ii)). For $0 \leq i \leq d$ let $a_i$ denote the coefficient of $v^*_i$, when we write $A v^*_i$ as a linear combination of $v^*_0, ..., v^*_d$, and let $a^*_i$ denote the coefficient of $v_i$, when we write $A^* v_i$ as a linear combination of $v_0..., v_d$. In this paper we show $a_0=a_d$ if and only if $a^*_0=a^*_d$. Moreover we show that for $d \geq 1$ the following are equivalent: (i) $a_0=a_d$ and $a_1=a_{d-1}$; (ii) $a^*_0=a^*_d$ and $a^*_1=a^*_{d-1}$; (iii) $a_i=a_{d-i}$ and $a^*_i=a^*_{d-i}$ for $0 \leq i \leq d$. We say $A$, $A^*$ is balanced whenever (i)--(iii) hold. We say $A$, $A^*$ is essentially bipartite (resp. essentially dual bipartite}) whenever $a_i$ (resp. $a^*_i$) is independent of $i$ for $0 \leq i \leq d$. Observe that if $A$, $A^*$ is essentially bipartite or dual bipartite, then $A$, $A^*$ is balanced. For $d \neq 2$ we show that if $A$, $A^*$ is balanced then $A$, $A^*$ is essentially bipartite or dual bipartite.

math.RA↗