arXiv subjects

Kazumasa Nomura

Publications and source records attributed to Kazumasa Nomura.

At least 19 recordsLinked to original sources

arXiv2026

Variations on a circular Hessenberg pair

A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. A Hessenberg pair is an ordered pair of diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a Hessenberg fashion. A Hessenberg system $Φ$ an `oriented' version of a Hessenberg pair. It is known that $Φ$ is determined up to isomorphism by its parameter array; this consists of the eigenvalue sequence of $Φ$, the dual eigenvalue sequence of $Φ$, and a sequence of nonzero scalars $\{ϕ_i\}_{i=1}^d$ called the split sequence of $Φ$. We are interested in some types of Hessenberg matrices, said to be circular, quasi-circular, tridiagonal, and irreducible tridiagonal. We are interested in the families of Hessenberg systems for which the associated Hessenberg matrices have one of the above types. A Hessenberg system of irreducible tridiagonal type is often called a Leonard system. In this case the associated Hessenberg pair satisfies two relations, called the tridiagonal relations. We are interested in the family of Hessenberg systems for which the associated Hessenberg pair satisfies the tridiagonal relations. We are also interested in the family of Hessenberg systems for which the eigenvalue sequence and dual eigenvalue sequence satisfy a linear three-term recurrence. In the present paper we have two main goals. First, we show how the above families of Hessenberg systems are related to each other. Second, we describe each family in terms of the parameter array.

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arXiv2026

Circular Hessenberg pairs and the tridiagonal relations

A square matrix is said to be Hessenberg whenever each entry below the subdiagonal is zero, and each entry on the subdiagonal is nonzero. A Hessenberg matrix is called circular whenever the top-right corner entry is nonzero, and every other entry above the superdiagonal is zero. A circular Hessenberg pair consists of two diagonalizable linear maps on a nonzero finite-dimensional vector space, that each act on an eigenbasis of the other one in a circular Hessenberg fashion. In 2022, Jae-ho Lee conjectured that a circular Hessenberg pair satisfies two relations called the tridiagonal relations. In the present paper, we prove Lee's conjecture. Our proof is not elementary.

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arXiv2025

Spin Leonard pairs and the zero diagonal space

We consider a Leonard pair $A, A^*$ of linear maps on a vector space $V$ that has finite positive dimension. This Leonard pair $A,A^*$ is said to have spin whenever there exist invertible linear maps $W : V \to V$ and $W^* : V \to V$ such that $W A = A W$ and $W^* A^* = A^* W^*$ and $W A^* W^{-1} = (W^*)^{-1} A W^*$. Let $\{θ^*_i\}_{i=0}^d$ denote a standard ordering of the eigenvalues of $A^*$. There is a related sequence of scalars $\{a_i\}_{i=0}^d$ called intersection numbers. The Leonard pair $A,A^*$ is called self-dual whenever $\{θ^*_i\}_{i=0}^d$ is a standard ordering of the eigenvalues of $A$. We obtain the following results under the assumption that the ground field is algebraically closed and $d \geq 3$. We show that a Leonard pair $A,A^*$ on $V$ has spin if and only if both (i) $A,A^*$ is self-dual; (ii) there exist scalars $f_0,f_1,f_2, f_3$ (not all zero) such that $f_0 + f_1 θ^*_i + f_2 a_i + f_3 a_i θ^*_i = 0$ for $0 \leq i \leq d$. We also classify the Leonard pairs $A,A^*$ on $V$ that satisfy (ii) without assuming (i). To do this we bring in the following maps. For $0 \leq i \leq d$ let $E^*_i : V \to V$ denote the projection onto the $θ^*_i$-eigenspace of $A^*$. Let ${\mathcal Z}(A,A^*)$ denote the set of elements $X$ in $\text{Span}\{I, A^*, A, A A^*\}$ such that $E^*_i X E^*_i= 0$ for $0 \leq i \leq d$. We call ${\mathcal Z}(A,A^*)$ the zero diagonal space of $A,A^*$. As we will see, ${\mathcal Z}(A,A^*) \neq 0$ if and only if the above condition (ii) holds. As we investigate the case ${\mathcal Z}(A,A^*) \neq 0$ in detail, we break the problem into 13 cases called types; these are the $q$-Racah type and its relatives. For each type we give a necessary and sufficient condition for ${\mathcal Z}(A,A^*) \not=0$. For each type we give an explicit basis for ${\mathcal Z}(A,A^*)$.

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arXiv2024

The nucleus of the Johnson graph $J(N,D)$

In this paper, we describe the nucleus of the Johnson graph $Γ= J(N,D)$ with $N > 2D$. Let $X$ denote the vertex set of $Γ$. Let $A \in \text{Mat}_X({\mathbb C})$ denote the adjacency matrix of $Γ$. Let $\{E_i\}_{i=0}^D$ denote the $Q$-polynomial ordering of the primitive idempotents of $A$. Fix $x \in X$, and consider the corresponding dual adjacency matrix $A^*$ and dual primitive idempotents $\{E^*_i\}_{i=0}^D$. The subalgebra $T$ of $\text{Mat}_X({\mathbb C})$ generated by $A$, $A^*$ is called the subconstituent algebra of $Γ$ with respect to $x$. Let $V={\mathbb C}^X$ denote the standard module of $Γ$. For $0 \leq i \leq D$ define \[ {\mathcal N}_i = (E^*_0 V + E^*_1 V + \cdots + E^*_i V) \cap (E_0 V + E_1 V + \cdots + E_{D-i} V). \] It is known that the sum ${\mathcal N} = \sum_{i=0}^D {\mathcal N}_i$ is direct, and $\mathcal N$ is a $T$-module. The $T$-module $\mathcal N$ is called the nucleus of $Γ$ with respect to $x$. For $0 \leq i \leq D$ we construct a basis for ${\mathcal N}_i$ and a basis for $E^*_i {\mathcal N}$. From this we obtain two bases of $\mathcal N$. We give a combinatorial interpretation of these two bases. We give the transition matrices between these two bases. We also give the action of $A$, $A^*$ on these bases.

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arXiv2024

The Norton-balanced condition for $Q$-polynomial distance-regular graphs

Let $Γ$ denote a $Q$-polynomial distance-regular graph, with vertex set $X$ and diameter $D\geq 3$. The standard module $V$ has a basis $\lbrace {\hat x} \vert x \in X\rbrace$, where ${\hat x}$ denotes column $x$ of the identity matrix $I \in {\rm Mat}_X(\mathbb C)$. Let $E$ denote a $Q$-polynomial primitive idempotent of $Γ$. The eigenspace $EV$ is spanned by the vectors $\lbrace E {\hat x} \vert x \in X\rbrace$. It was previously known that these vectors satisfy a condition called the balanced set condition. In this paper, we introduce a variation on the balanced set condition called the Norton-balanced condition. The Norton-balanced condition involves the Norton algebra product on $EV$. We define $Γ$ to be Norton-balanced whenever $Γ$ has a $Q$-polynomial primitive idempotent $E$ such that the set $\lbrace E {\hat x} \vert x \in X\rbrace$ is Norton-balanced. We show that $Γ$ is Norton-balanced in the following cases: (i) $Γ$ is bipartite; (ii) $Γ$ is almost bipartite; (iii) $Γ$ is dual-bipartite; (iv) $Γ$ is almost dual-bipartite; (v) $Γ$ is tight; (vi) $Γ$ is a Hamming graph; (vii) $Γ$ is a Johnson graph; (viii) $Γ$ is the Grassmann graph $J_q(2D,D)$; (ix) $Γ$ is a halved bipartite dual-polar graph; (x) $Γ$ is a halved Hemmeter graph; (xi) $Γ$ is a halved hypercube; (xii) $Γ$ is a folded-half hypercube; (xiii) $Γ$ has $q$-Racah type and affords a spin model. Some theoretical results about the Norton-balanced condition are obtained, and some open problems are given.

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arXiv2024

Formal self-duality and numerical self-duality for symmetric association schemes

Let ${\mathcal X} = (X, \{R_i\}_{i=0}^d)$ denote a symmetric association scheme. Fix an ordering $\{E_i\}_{i=0}^d$ of the primitive idempotents of $\mathcal{X}$, and let $P$ (resp.\ $Q$) denote the corresponding first eigenmatrix (resp.\ second eigenmatrix) of $\mathcal X$. The scheme $\mathcal X$ is said to be formally self-dual (with respect to the ordering $\{E_i\}_{i=0}^d$) whenever $P=Q$. We define $\mathcal X$ to be numerically self-dual (with respect to the ordering $\{E_i\}_{i=0}^d$) whenever the intersection numbers and Krein parameters satisfy $p^h_{i,j} =q^h_{i,j}$ for $0 \leq h,i,j \leq d$. It is known that with respect to the ordering $\{E_i\}_{i=0}^d$, formal self-duality implies numerical self-duality. This raises the following question: is it possible that with respect to the ordering $\{E_i\}_{i=0}^d$, $\mathcal X$ is numerically self-dual but not formally self-dual? This is possible as we will show. We display an example of a symmetric association scheme and an ordering the primitive idempotents with respect to which the scheme is numerically self-dual but not formally self-dual. We have the following additional results about self-duality. Assume that $\mathcal X$ is $P$-polynomial. We show that the following are equivalent: (i) $\mathcal X$ is formally self-dual with respect to the ordering $\{E_i\}_{i=0}^d$; (ii) $\mathcal X$ is numerically self-dual with respect to the ordering $\{E_i\}_{i=0}^d$. Assume that the ordering $\{E_i\}_{i=0}^d$ is $Q$-polynomial. We show that the following are equivalent: (i) $\mathcal X$ is formally self-dual with respect to the ordering $\{E_i\}_{i=0}^d$; (ii) $\mathcal X$ is numerically self-dual with respect to the ordering $\{E_i\}_{i=0}^d$.

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arXiv2023

Spin models and distance-regular graphs of $q$-Racah type

Let $Γ$ denote a distance-regular graph, with vertex set $X$ and diameter $D\geq 3$. We assume that $Γ$ is formally self-dual and $q$-Racah type. We also assume that for each $x \in X$ the subconstituent algebra $T=T(x)$ contains a certain central element $Z=Z(x)$. We use $Z$ to construct a spin model $\sf W$ afforded by $Γ$. We investigate the combinatorial implications of $Z$. We reverse the logical direction and recover $Z$ from $\sf W$. We finish with some open problems.

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arXiv2023

Near-bipartite Leonard pairs

Let $\F$ denote a field, and let $V$ denote a vector space over $\F$ with finite positive dimension. A Leonard pair on $V$ is an ordered pair of diagonalizable $\F$-linear maps $A: V \to V$ and $A^* : V \to V$ that each act on an eigenbasis for the other in an irreducible tridiagonal fashion. Let $A,A^*$ denote a Leonard pair on $V$. Let $\{v_i\}_{i=0}^d$ denote an eigenbasis for $A^*$ on which $A$ acts in an irreducible tridiagonal fashion. For $0 \leq i \leq d$ define an $\F$-linear map $E^*_i : V \to V$ such that $E^*_i v_i = v_i$ and $E^*_i v_j = 0$ if $j \neq i$ $(0 \leq j \leq d)$. The map $F = \sum_{i=0}^d E^*_i A E^*_i$ is called the flat part of $A$. The Leonard pair $A,A^*$ is bipartite whenever $F=0$. The Leonard pair $A,A^*$ is said to be near-bipartite whenever the pair $A-F, A^*$ is a Leonard pair on $V$. In this case, the Leonard pair $A-F, A^*$ is bipartite, and called the bipartite contraction of $A,A^*$. Let $B,B^*$ denote a bipartite Leonard pair on $V$. By a near-bipartite expansion of $B,B^*$ we mean a near-bipartite Leonard pair on $V$ with bipartite contraction $B,B^*$. In the present paper we have three goals. Assuming $\F$ is algebraically closed, (i) we classify up to isomorphism the near-bipartite Leonard pairs over $\F$; (ii) for each near-bipartite Leonard pair over $\F$ we describe its bipartite contraction; (iii) for each bipartite Leonard pair over $\F$ we describe its near-bipartite expansions.

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arXiv2021

Compatibility and companions for Leonard pairs

In this paper, we introduce the concepts of compatibility and companion for Leonard pairs. These concepts are roughly described as follows. Let $\mathbb{F}$ denote a field, and let $V$ denote a vector space over $\mathbb{F}$ with finite positive dimension. A Leonard pair on $V$ is an ordered pair of diagonalizable $\mathbb{F}$-linear maps $A : V \to V$ and $A^* : V \to V$ that each act in an irreducible tridiagonal fashion on an eigenbasis for the other one. Leonard pairs $A,A^*$ and $B,B^*$ on $V$ are said to be compatible whenever $A^* = B^*$ and $[A,A^*] = [B,B^*]$, where $[r,s] = r s - s r$. For a Leonard pair $A,A^*$ on $V$, by a companion of $A,A^*$ we mean an $\mathbb{F}$-linear map $K: V \to V$ such that $K$ is a polynomial in $A^*$ and $A-K, A^*$ is a Leonard pair on $V$. The concepts of compatibility and companion are related as follows. For compatible Leonard pairs $A,A^*$ and $B,B^*$ on $V$, define $K = A-B$. Then $K$ is a companion of $A,A^*$. For a Leonard pair $A,A^*$ on $V$ and a companion $K$ of $A,A^*$, define $B = A-K$ and $B^* = A^*$. Then $B,B^*$ is a Leonard pair on $V$ that is compatible with $A,A^*$. Let $A,A^*$ denote a Leonard pair on $V$. We find all the Leonard pairs $B, B^*$ on $V$ that are compatible with $A,A^*$. For each solution $B, B^*$ we describe the corresponding companion $K = A-B$.

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arXiv2020

Idempotent systems and character algebras

We recently introduced the notion of an idempotent system. This linear algebraic object is motivated by the structure of an association scheme. There is a type of idempotent system, said to be symmetric. In the present paper we classify up to isomorphism the idempotent systems and the symmetric idempotent systems. We also describe how symmetric idempotent systems are related to character algebras.

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arXiv2020

Idempotent systems

In this paper we introduce the notion of an idempotent system. This linear algebraic object is motivated by the structure of an association scheme. We focus on a family of idempotent systems, said to be symmetric. A symmetric idempotent system is an abstraction of the primary module for the subconstituent algebra of a symmetric association scheme. We describe the symmetric idempotent systems in detail. We also consider a class of symmetric idempotent systems, said to be $P$-polynomial and $Q$-polynomial. In the topic of orthogonal polynomials there is an object called a Leonard system. We show that a Leonard system is essentially the same thing as a symmetric idempotent system that is $P$-polynomial and $Q$-polynomial.

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arXiv2019

Leonard pairs, spin models, and distance-regular graphs

A Leonard pair is an ordered pair of diagonalizable linear maps on a finite-dimensional vector space, that each act on an eigenbasis for the other one in an irreducible tridiagonal fashion. In the present paper we consider a type of Leonard pair, said to have spin. The notion of a spin model was introduced by V.F.R. Jones to construct link invariants. A spin model is a symmetric matrix over $\mathbb{C}$ that satisfies two conditions, called the type II and type III conditions. It is known that a spin model $\sf W$ is contained in a certain finite-dimensional algebra $N({\sf W})$, called the Nomura algebra. It often happens that a spin model $\sf W$ satisfies ${\sf W} \in {\sf M} \subseteq N({\sf W})$, where $\sf M$ is the Bose-Mesner algebra of a distance-regular graph $Γ$; in this case we say that $Γ$ affords $\sf W$. If $Γ$ affords a spin model, then each irreducible module for every Terwilliger algebra of $Γ$ takes a certain form, recently described by Caughman, Curtin, Nomura, and Wolff. In the present paper we show that the converse is true; if each irreducible module for every Terwilliger algebra of $Γ$ takes this form, then $Γ$ affords a spin model. We explicitly construct this spin model when $Γ$ has $q$-Racah type. The proof of our main result relies heavily on the theory of spin Leonard pairs.

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arXiv2018

Self-dual Leonard pairs

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 $A \leftrightarrow A^*$. In the present paper we give a comprehensive description of this duality. In particular, we display an invertible $\F$-linear map $T$ on $V$ such that the map $X \mapsto T X T^{-1}$ is the duality $A \leftrightarrow A^*$. 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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arXiv2017

Totally bipartite tridiagonal pairs

There is a concept in linear algebra called a tridiagonal pair. The concept was motivated by the theory of $Q$-polynomial distance-regular graphs. We give a tutorial introduction to tridiagonal pairs, working with a special case as a concrete example. The special case is called totally bipartite, or TB. Starting from first principles, we give an elementary but comprehensive account of TB tridiagonal pairs. The following topics are discussed: (i) the notion of a TB tridiagonal system; (ii) the eigenvalue array; (iii) the standard basis and matrix representations; (iv) the intersection numbers; (v) the Askey-Wilson relations; (vi) a recurrence involving the eigenvalue array; (vii) the classification of TB tridiagonal systems; (viii) self-dual TB tridiagonal pairs and systems; (ix) the $\mathbb{Z}_3$-symmetric Askey-Wilson relations; (x) some automorphisms and antiautomorphisms associated with a TB tridiagonal pair; (xi) an action of the modular group ${\rm PSL}_2(\mathbb{Z})$ associated with a TB tridiagonal pair.

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arXiv2017

The universal DAHA of type $(C_1^\vee,C_1)$ and Leonard pairs of $q$-Racah type

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. Let $\mathbb F$ denote an algebraically closed field, and fix a nonzero $q \in \mathbb F$ that is not a root of unity. The universal double affine Hecke algebra (DAHA) $\hat{H}_q$ of type $(C_1^\vee,C_1)$ is the associative $\mathbb F$-algebra defined by generators $\lbrace t_i^{\pm 1}\rbrace_{i=0}^3$ and relations (i) $t_it_i^{-1}=t_i^{-1}t_i=1$; (ii) $t_i+t_i^{-1}$ is central; (iii) $t_0t_1t_2t_3 = q^{-1}$. We consider the elements $X=t_3t_0$ and $Y=t_0t_1$ of $\hat{H}_q$. Let $\mathcal V$ denote a finite-dimensional irreducible $\hat{H}_q$-module on which each of $X$, $Y$ is diagonalizable and $t_0$ has two distinct eigenvalues. Then $\mathcal V$ is a direct sum of the two eigenspaces of $t_0$. We show that the pair $X+X^{-1}$, $Y+Y^{-1}$ acts on each eigenspace as a Leonard pair, and each of these Leonard pairs falls into a class said to have $q$-Racah type. Thus from $\mathcal V$ we obtain a pair of Leonard pairs of $q$-Racah type. It is known that a Leonard pair of $q$-Racah type is determined up to isomorphism by a parameter sequence $(a,b,c,d)$ called its Huang data. Given a pair of Leonard pairs of $q$-Racah type, we find necessary and sufficient conditions on their Huang data for that pair to come from the above construction.

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arXiv2015

An LR pair that can be extended to an LR triple

Fix an integer $d \geq 0$, a field $\mathbb{F}$, and a vector space $V$ over $\mathbb{F}$ with dimension $d+1$. By a decomposition of $V$ we mean a sequence $\{V_i\}_{i=0}^d$ of $1$-dimensional $\mathbb{F}$-subspaces of $V$ such that $V = \sum_{i=0}^d V_i$ (direct sum). Consider $\mathbb{F}$-linear transformations $A$, $B$ from $V$ to $V$. Then $A,B$ is called an LR pair whenever there exists a decomposition $\{V_i\}_{i=0}^d$ of $V$ such that $A V_i = V_{i-1}$ and $B V_i = V_{i+1}$ for $0 \leq i \leq d$, where $V_{-1}=0$ and $V_{d+1}=0$. By an LR triple we mean a $3$-tuple $A,B,C$ of $\mathbb{F}$-linear transformations from $V$ to $V$ such that any two of them form an LR pair. In the present paper, we consider how an LR pair $A,B$ can be extended to an LR triple $A,B,C$.

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arXiv2015

Linear transformations that are tridiagonal with respect to the three decompositions for an LR triple

Fix an integer $d \geq 0$, a field $\mathbb{F}$, and a vector space $V$ over $\mathbb{F}$ with dimension $d+1$. By a decomposition of $V$ we mean a sequence $\{V_i\}_{i=0}^d$ of $1$-dimensional subspaces of $V$ whose sum is $V$. For a linear transformation $A$ from $V$ to $V$, we say $A$ lowers $\{V_i\}_{i=0}^d$ whenever $A V_i = V_{i-1}$ for $0 \leq i \leq d$, where $V_{-1}=0$. We say $A$ raises $\{V_i\}_{i=0}^d$ whenever $A V_i = V_{i+1}$ for $0 \leq i \leq d$, where $V_{d+1}=0$. An ordered pair of linear transformations $A,B$ from $V$ to $V$ is called LR whenever there exists a decomposition $\{V_i\}_{i=0}^d$ of $V$ that is lowered by $A$ and raised by $B$. In this case the decomposition $\{V_i\}_{i=0}^d$ is uniquely determined by $A,B$; we call it the $(A,B)$-decomposition of $V$. Consider a $3$-tuple of linear transformations $A$, $B$, $C$ from $V$ to $V$ such that any two of $A$, $B$, $C$ form an LR pair on $V$. Such a $3$-tuple is called an LR triple on $V$. Let $α$, $β$, $γ$ be nonzero scalars in $\mathbb{F}$. The triple $αA, βB, γC$ is an LR triple on $V$, said to be associated to $A,B,C$. Let $\{V_i\}_{i=0}^d$ be a decomposition of $V$ and let $X$ be a linear transformation from $V$ to $V$. We say $X$ is tridiagonal with respect to $\{V_i\}_{i=0}^d$ whenever $X V_i \subseteq V_{i-1} + V_i + V_{i+1}$ for $0 \leq i \leq d$. Let $\cal X$ be the vector space over $\mathbb{F}$ consisting of the linear transformations from $V$ to $V$ that are tridiagonal with respect to the $(A,B)$ and $(B,C)$ and $(C,A)$ decompositions of $V$. There is a special class of LR triples, called $q$-Weyl type. In the present paper, we find a basis of $\cal X$ for each LR triple that is not associated to an LR triple of $q$-Weyl type.

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arXiv2015

Leonard pairs having zero-diagonal TD-TD form

Fix an algebraically closed field $\mathbb{F}$ and an integer $n \geq 1$. Let $\text{Mat}_n(\mathbb{F})$ denote the $\mathbb{F}$-algebra consisting of the $n \times n$ matrices that have all entries in $\mathbb{F}$. We consider a pair of diagonalizable matrices in $\text{Mat}_{n}(\mathbb{F})$, each acting in an irreducible tridiagonal fashion on an eigenbasis for the other one. Such a pair is called a Leonard pair in $\text{Mat}_{n}(\mathbb{F})$. In the present paper, we find all Leonard pairs $A,A^*$ in $\text{Mat}_{n}(\mathbb{F})$ such that each of $A$ and $A^*$ is irreducible tridiagonal with all diagonal entries $0$. This solves a problem given by Paul Terwilliger.

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