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Schurity and separability of quasiregular coherent configurations
Hirasaka, Mitsugu,Kim, Kijung,Ponomarenko, Ilia Elsevier 2018 Journal of algebra Vol.510 No.-
<P><B>Abstract</B></P> <P>A permutation group is said to be quasiregular if each of its transitive constituents is regular, and a quasiregular coherent configuration can be thought as a combinatorial analog of such a group: the transitive constituents are replaced by the homogeneous components. In this paper, we are interested in the question when the configuration is schurian, i.e., formed by the orbitals of a permutation group, or/and separable, i.e., uniquely determined by the intersection numbers. In these terms, an old result of Hanna Neumann is, in a sense, dual to the statement that the quasiregular coherent configurations with cyclic homogeneous components are schurian. In the present paper, we (a) establish the duality in a precise form and (b) generalize the latter result by proving that a quasiregular coherent configuration is schurian and separable if the groups associated with the homogeneous components have distributive lattices of normal subgroups.</P>
Isomorphism classes of association schemes induced by Hadamard matrices
Hirasaka, M.,Kim, K.,Yu, H. Academic Press 2016 European journal of combinatorics : Journal europ& Vol.51 No.-
<P>Every Hadamard matrix H of order n > 1 induces a graph with 4n vertices, called the Hadamard graph Gamma(H) of H. Since Gamma(H) is a distance-regular graph with diameter 4, it induces a 4-class association scheme (Omega, S) of order 4n. In this article we show a way to construct fission schemes of (Omega, S) under certain conditions, and for such a fission scheme we estimate the number of isomorphism classes with the same intersection numbers as the fission scheme. (C) 2015 Elsevier Ltd. All rights reserved.</P>
Characterization of balanced coherent configurations
Hirasaka, Mitsugu,Sharafdini, Reza Elsevier 2010 Journal of algebra Vol.324 No.8
<P><B>Abstract</B></P><P>Let <I>G</I> be a group acting on a finite set <I>Ω</I>. Then <I>G</I> acts on Ω×Ω by its entry-wise action and its orbits form the basis relations of a coherent configuration (or shortly scheme). Our concern is to consider what follows from the assumption that the number of orbits of <I>G</I> on <SUB>Ωi</SUB>×<SUB>Ωj</SUB> is constant whenever <SUB>Ωi</SUB> and <SUB>Ωj</SUB> are orbits of <I>G</I> on <I>Ω</I>. One can conclude from the assumption that the actions of <I>G</I> on <SUB>Ωi</SUB>'s have the same permutation character and are not necessarily equivalent. From this viewpoint one may ask how many inequivalent actions of a given group with the same permutation character there exist. In this article we will approach to this question by a purely combinatorial method in terms of schemes and investigate the following topics: (i) balanced schemes and their central primitive idempotents, (ii) characterization of reduced balanced schemes.</P>
CHARACTERIZATION OF FINITE COLORED SPACES WITH CERTAIN CONDITIONS
Hirasaka, Mitsugu,Shinohara, Masashi Korean Mathematical Society 2019 대한수학회지 Vol.56 No.3
A colored space is a pair (X, r) of a set X and a function r whose domain is $\(^X_2\)$. Let (X, r) be a finite colored space and $Y,\;Z{\subseteq}X$. We shall write $Y{\simeq}_rZ$ if there exists a bijection $f:Y{\rightarrow}Z$ such that r(U) = r(f(U)) for each $U{\in}\({^Y_2}\)$ where $f(U)=\{f(u){\mid}u{\in}U\}$. We denote the numbers of equivalence classes with respect to ${\simeq}_r$ contained in $\(^X_i\)$ by $a_i(r)$. In this paper we prove that $a_2(r){\leq}a_3(r)$ when $5{\leq}{\mid}X{\mid}$, and show what happens when equality holds.
Every Formula Not Shown -equivalenced association scheme is Frobenius
Hirasaka, M.,Kim, K. T.,Park, J. R. Springer Science + Business Media 2015 Journal of algebraic combinatorics Vol.41 No.1
<P>For a positive integer k we say that an association scheme with more than one point is k-equivalenced if each non-diagonal relation has valency k. In this paper we prove that every 3-equivalenced association scheme is Frobenius, that is, the set of relations coincides with the set of orbitals of a Frobenius group.</P>
Characterization of finite colored spaces with certain conditions
Mitsugu Hirasaka,Masashi Shinohara 대한수학회 2019 대한수학회지 Vol.56 No.3
A \textit{colored space} is {a pair} $(X,r)$ of a set $X$ and a function $r$ whose domain is $X\choose 2$. Let $(X,r)$ be a finite colored space and $Y,Z\subseteq X$. We shall write $Y\simeq_r Z$ if there exists a bijection $f:Y\to Z$ such that $r(U)=r(f(U))$ for each $U\in {Y\choose 2}$ {where $f(U)=\{f(u)\mid u\in U\}$}. We denote the numbers of equivalence classes with respect to $\simeq_r$ contained in {$X \choose i$ by $a_i(r)$}. In this paper we prove that $a_2(r)\leq a_3(r)$ when $5\leq |X|$, and show what happens {when equality} holds.
Elsevier 2009 European journal of combinatorics : Journal europ& Vol.30 No.1
<P><B>Abstract</B></P><P>Let PQ denote the set of n∈N such that n is a product of two primes with gcd(n,φ(n))=1 where φ is the Euler function. In this article we aim to find n∈PQ such that any imprimitive permutation group of degree n is multiplicity-free. Let R denote the set of such integers in PQ. Our main theorem shows that there are at most finitely many Fermat primes if and only if |PQ−R| is finite, whose proof is based on the classification of finite simple groups.</P>
Coherent configurations over copies of association schemes of prime order
Sharafdini, Reza,Hirasaka, Mitsugu University of Primorska Press 2017 Ars mathematica contemporanea Vol.12 No.1
<P>In this paper we consider a combinatorial analogue to this fact through the theory of coherent configurations, and give some arithmetic sufficient conditions for a coherent configuration with two homogeneous components of prime order to be uniquely determined by one of the homogeneous components.</P>
COMMUTATIVITY OF ASSOCIATION SCHEMES OF ORDER pq
Hanaki, Akihide,Hirasaka, Mitsugu The Youngnam Mathematical Society 2013 East Asian mathematical journal Vol.29 No.1
Let (X, S) be an association scheme where X is a finite set and S is a partition of $X{\times}X$. The size of X is called the order of (X, S). We define $\mathcal{C}$ to be the set of positive integers m such that each association scheme of order $m$ is commutative. It is known that each prime is belonged to $\mathcal{C}$ and it is conjectured that each prime square is belonged to $\mathcal{C}$. In this article we give a sufficient condition for a scheme of order pq to be commutative where $p$ and $q$ are primes, and obtain a partial answer for the conjecture in case where $p=q$.