Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSchurity and separability of quasiregular coherent configurations
120
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
A permutation group is said to be quasiregular if every its transitive constituent 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 homogeneous components have distributive lattices of normal subgroups.
rate research
Read More
A Cartesian decomposition of a coherent configuration $cal X$ is defined as a special set of its parabolics that form a Cartesian decomposition of the underlying set. It turns out that every tensor decomposition of $cal X$ comes from a certain Cartesian decomposition. It is proved that if the coherent configuration $cal X$ is thick, then there is a unique maximal Cartesian decomposition of $cal X$, i.e., there is exactly one internal tensor decomposition of $cal X$ into indecomposable components. In particular, this implies an analog of the Krull--Schmidt theorem for the thick coherent configurations. A polynomial-time algorithm for finding the maximal Cartesian decomposition of a thick coherent configuration is constructed.
Let $G$ be a finite group. There is a natural Galois correspondence between the permutation groups containing $G$ as a regular subgroup, and the Schur rings (S-rings) over~$G$. The problem we deal with in the paper, is to characterize those S-rings that are closed under this correspondence, when the group $G$ is cyclic (the schurity problem for circulant S-rings). It is proved that up to a natural reduction, the characteristic property of such an S-ring is to be a certain algebraic fusion of its coset closure introduced and studied in the paper. Basing on this characterization we show that the schurity problem is equivalent to the consistency of a modular linear system associated with a circulant S-ring under consideration. As a byproduct we show that a circulant S-ring is Galois closed if and only if so is its dual.
The generalized wreath product of permutation groups is introduced. By means of it we study the schurity problem for S-rings over a cyclic group $G$ and the automorphism groups of them. Criteria for the schurity and non-schurity of the generalized wreath product of two such S-rings are obtained. As a byproduct of the developed theory we prove that $G$ is a Schur group whenever the total number $Omega(n)$ of prime factors of the integer $n=|G|$ is at most 3. Moreover, we describe the structure of a non-schurian S-ring over $G$ when $Omega(n)=4$. The latter result implies in particular that if $n=p^3q$ where $p$ and $q$ are primes, then $G$ is a Schur group.
A matrix is emph{simple} if it is a (0,1)-matrix and there are no repeated columns. Given a (0,1)-matrix $F$, we say a matrix $A$ has $F$ as a emph{configuration}, denoted $Fprec A$, if there is a submatrix of $A$ which is a row and column permutation of $F$. Let $|A|$ denote the number of columns of $A$. Let $mathcal{F}$ be a family of matrices. We define the extremal function $text{forb}(m, mathcal{F}) = max{|A|colon A text{ is an }m-text{rowed simple matrix and has no configuration } Finmathcal{F}}$. We consider pairs $mathcal{F}={F_1,F_2}$ such that $F_1$ and $F_2$ have no common extremal construction and derive that individually each $text{forb}(m, F_i)$ has greater asymptotic growth than $text{forb}(m, mathcal{F})$, extending research started by Anstee and Koch.
Let $G$ be a 4-chromatic maximal planar graph (MPG) with the minimum degree of at least 4 and let $C$ be an even-length cycle of $G$.If $|f(C)|=2$ for every $f$ in some Kempe equivalence class of $G$, then we call $C$ an unchanged bichromatic cycle (UBC) of $G$, and correspondingly $G$ an unchanged bichromatic cycle maximal planar graph (UBCMPG) with respect to $C$, where $f(C)={f(v)| vin V(C)}$. For an UBCMPG $G$ with respect to an UBC $C$, the subgraph of $G$ induced by the set of edges belonging to $C$ and its interior (or exterior), denoted by $G^C$, is called a base-module of $G$; in particular, when the length of $C$ is equal to four, we use $C_4$ instead of $C$ and call $G^{C_4}$ a 4-base-module. In this paper, we first study the properties of UBCMPGs and show that every 4-base-module $G^{C_4}$ contains a 4-coloring under which $C_4$ is bichromatic and there are at least two bichromatic paths with different colors between one pair of diagonal vertices of $C_4$ (these paths are called module-paths). We further prove that every 4-base-module $G^{C_4}$ contains a 4-coloring (called decycle coloring) for which the ends of a module-path are colored by distinct colors. Finally, based on the technique of the contracting and extending operations of MPGs, we prove that 55-configurations and 56-configurations are reducible by converting the reducibility problem of these two classes of configurations into the decycle coloring problem of 4-base-modules.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
