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Hirotake Kurihara

Publications and source records attributed to Hirotake Kurihara.

14 recordsLinked to original sources
arXiv2026

Twisted primitive group association schemes

We give results on the question of whether the intersection numbers of a primitive group association scheme determine it up to combinatorial isomorphism. For $G=\operatorname{PSL}(2,q)$, where $q$ is an odd prime power with $q=11$ or $q\ge 17$, or $q=2^f$ with $f\ge3$, we construct a Schur partition that is algebraically isomorphic to the partition of $G$ into conjugacy classes but not combinatorially isomorphic to it. Consequently, the corresponding primitive group association schemes are not determined up to combinatorial isomorphism by their intersection numbers; in particular, they are non-separable. For $\mathfrak A_6$ and $\mathfrak A_8$, we also explicitly construct Schur partitions that are algebraically isomorphic to the corresponding partitions into conjugacy classes but not combinatorially isomorphic to them.

math.CO↗
arXiv2026

Imprimitive association schemes and elimination theory

We prove that a commutative association scheme is imprimitive if and only if it admits a multivariate $P$- or $Q$-polynomial structure with respect to an elimination-type monomial order. This provides a direct bridge between the classical theory of block and quotient schemes for imprimitive association schemes and elimination theory in computational commutative algebra. For an imprimitive multivariate $P$- or $Q$-polynomial association scheme, we determine the induced multivariate polynomial structures on the quotient and block schemes and describe their associated polynomials via explicit specializations, variable deletions, and rescalings of the original associated polynomials. At the level of zero-dimensional ideals, we show that the ideal of the block scheme is exactly an elimination ideal, whereas the ideal of the quotient scheme is obtained by adjoining the valency relations for the eliminated variables and then eliminating. As applications, we study direct products and crested products from the viewpoint of multivariate polynomiality, and we characterize the schemes that are multivariate $P$- or $Q$-polynomial with respect to every monomial order as precisely the direct products of univariate $P$- or $Q$-polynomial schemes. We also discuss formal duality, composition series, and several related open problems.

math.CO↗
arXiv2025

On the existence and non-existence of spherical $m$-stiff configurations

This paper investigates the existence of $m$-stiff configurations in the unit sphere $S^{d-1}$, which are spherical $(2m-1)$-designs that lie on $m$ parallel hyperplanes. We establish two non-existence results: (1) for each fixed integer $m > 5$, there exists no $m$-stiff configuration in $S^{d-1}$ for sufficiently large $d$; (2) for each fixed integer $d > 10$, there exists no $m$-stiff configuration in $S^{d-1}$ for sufficiently large $m$. Furthermore, we provide a complete classification of the dimensions where $m$-stiff configurations exist for $m=2,3,4,5$. We also determine the non-existence (and the existence) of $m$-stiff configurations in $S^{d-1}$ for small $d$ ($3 \leq d \leq 120$) with arbitrary $m$, and also for small $m$ ($6 \leq m \leq 10$) with arbitrary $d$. Finally, we conjecture that there is no $m$-stiff configuration in $S^{d-1}$ for $(d,m)$ with $d\geq 3$ and $m\geq 6$.

math.CO↗
arXiv2024

Classification of generalized Alexander quandles

The aim of this paper is to provide a new characterization of isomorphism classes of generalized Alexander quandles in terms of the underlying groups and their automorphisms. This extends the previous result [4, Theorem 1.4]. Additionally, we compute the number of generalized Alexander quandles up to quandle isomorphism arising from groups up to order 127 and their group automorphisms.

math.GT↗
arXiv2024

Bivariate $Q$-polynomial structures for the nonbinary Johnson scheme and the association scheme obtained from attenuated spaces

The study of $P$-polynomial association schemes (distance-regular graphs) and $Q$-polynomial association schemes, and in particular $P$- and $Q$-polynomial association schemes, has been a central theme not only in the theory of association schemes but also in the whole study of algebraic combinatorics in general. Leonard's theorem (1982) says that the spherical functions (or the character tables) of $P$- and $Q$-polynomial association schemes are described by Askey-Wilson orthogonal polynomials or their relatives. These polynomials are one-variable orthogonal polynomials. It seems that the new attempt to define and study higher rank $P$- and $Q$-polynomial association schemes had been hoped for, but had gotten only limited success. The first very successful attempt was initiated recently by Bernard-Crampé-d'Andecy-Vinet-Zaimi [arXiv:2212.10824], and then followed by Bannai-Kurihara-Zhao-Zhu [arXiv:2305.00707]. The general theory and some explicit examples of families of higher rank (multivariate) $P$- and/or $Q$-polynomial association schemes have been obtained there. The main purpose of the present paper is to prove that some important families of association schemes are shown to be bivariate $Q$-polynomial. Namely, we show that all the nonbinary Johnson association schemes and all the attenuated space association schemes are bivariate $Q$-polynomial. It should be noted that the parameter restrictions needed in the previous papers are completely lifted in this paper. Our proofs are done by explicitly calculating the Krein parameters of these association schemes. At the end, we mention some speculations and indications of what we can expect in the future study.

math.CO↗
arXiv2023

Multivariate P- and/or Q-polynomial association schemes

The classification problem of $P$- and $Q$-polynomial association schemes has been one of the central problems in algebraic combinatorics. Generalizing the concept of $P$- and $Q$-polynomial association schemes to multivariate cases, namely to consider higher rank $P$- and $Q$-polynomial association schemes, has been tried by some authors, but it seems that so far there were neither very well-established definition nor results. Very recently, Bernard, Crampé, d'Andecy, Vinet, and Zaimi [arXiv:2212.10824], defined bivariate $P$-polynomial association schemes, as well as bivariate $Q$-polynomial association schemes. In this paper, we study these concepts and propose a new modified definition concerning a general monomial order, which is more general and more natural and also easy to handle. We prove that there are many interesting families of examples of multivariate $P$- and/or $Q$-polynomial association schemes.

math.CO↗
arXiv2022

Generalized Alexander quandles of finite groups and their characterizations

The goal of this paper is to characterization generalized Alexander quandles of finite groups in the language of the underlying groups. Firstly, we prove that if finite groups $G$ are simple, then the quandle isomorphic classes of generalized Alexander quandles of $G$ one-to-one correspond to the conjugacy classes of the automorphism groups of $G$. This correspondence can be also claimed for the case of symmetric groups. Secondly, we give a characterization of generalized Alexander quandles of finite groups $G$ under some assumptions in terms of $G$. As corollaries of this characterization, we obtain several characterizations in some particular groups, e.g., abelian groups and dihedral groups. Finally, we perform a characterization of generalized Alexander quandles arising from groups with their order up to $15$.

math.GR↗
arXiv2022

The least Euclidean distortion constant of a distance-regular graph

In 2008, Vallentin made a conjecture involving the least distortion of an embedding of a distance-regular graph into Euclidean space. Vallentin's conjecture implies that for a least distortion Euclidean embedding of a distance-regular graph of diameter $d$, the most contracted pairs of vertices are those at distance $d$. In this paper, we confirm Vallentin's conjecture for several families of distance-regular graphs. We also provide counterexamples to this conjecture, where the largest contraction occurs between pairs of vertices at distance $d{-}1$. We suggest three alternative conjectures and prove them for several families of distance-regular graphs.

math.CO↗
arXiv2020

Homogeneous quandles arising from automorphisms of symmetric groups

Quandle is an algebraic system with one binary operation, but it is quite different from a group. Quandle has its origin in the knot theory and good relationships with the theory of symmetric spaces, so it is well-studied from points of view of both areas. In the present paper, we investigate a special kind of quandles, called generalized Alexander quandles $Q(G,ψ)$, which is defined by a group $G$ together with its group automorphism $ψ$. We develop the quandle invariants for generalized Alexander quandles. As a result, we prove that there is a one-to-one correspondence between generalized Alexander quandles arising from symmetric groups $\Sf_n$ and the conjugacy classes of $\Sf_n$ for $3 \leq n \leq 30$ with $n \neq 6,15$, and the case $n=6$ is also discussed.

math.GR↗
arXiv2012

An excess theorem for spherical 2-designs

We give an excess theorem for spherical 2-designs. This theorem is a dual version of the spectral excess theorem for graphs, which gives a characterization of distance-regular graphs, among regular graphs in terms of the eigenvalues and the excess. Here we give a characterization of Q-polynomial association schemes among spherical 2-designs.

math.CO↗
arXiv2011

Character tables of association schemes based on attenuated spaces

The set of subspaces of a given dimension in an attenuated space has a structure of a symmetric association scheme and this association scheme is called an association scheme based on an attenuated space. Association schemes based on attenuated spaces are generalizations of Grassmann schemes and bilinear forms schemes, and also $q$-analogues of non-binary Johnson schemes. Wang, Guo and Li computed the intersection numbers of association schemes based on attenuated spaces. The aim of this paper is to compute character tables of association schemes based on attenuated spaces using the method of Tarnanen, Aaltonen and Goethals. Moreover, we also prove that association schemes based on attenuated spaces include as a special case the $m$-flat association scheme, which is defined on the set of cosets of subspaces of a constant dimension in a vector space over a finite field.

math.CO↗