Schur Rings Research Articles

Using Schur Rings to Produce GRRs for Dihedral Groups

If Γ is a finite group and G a graph such that Aut(G) ≡ Γ acts regularly on V(G), then we say that G is a graphical regular representation (GRR) of Γ. The question asking which finite groups have at least one GRR was an important question in algebraic graph theory and it was completely solved as a result of work done by several researchers. However, it remains a challenge to discern whether a group known to have GRRs has GRRs with specific properties, such as being trivalent. In this paper, we shall be deriving simple conditions on the parameters of a subset of a dihedral group for easily constructing trivalent graphical regular representations (GRR) of the group. Specifically, we shall prove the following: Let n be an odd integer greater than 5 and let r, s, and t be integers less than n such that the difference of any two of them is relatively prime to n. If 3r – 2s = t (mod n), then Cay(Dn, {abr, abs, abt}) is a GRR of Dn. We will also be looking at very convenient corollaries of this result. But another main aim of this paper is to show how a simple use of Schur rings can be used to derive such results. This paper therefore also serves as a review of some basic results about Schur rings which we feel should be among the standard armory of an algebraic graph theorist.

Schur rings over free Abelian group of rank two

Schur rings are subrings of the group ring of the group G afforded by a partition of G into finite sets. In this paper, Schur rings over a free abelian group of rank two are classified under the assumption that one of the direct factors is a union of some basic sets. There are eight different types, and all but one type of which are traditional.

Schur rings over Sp(n,2) and multiplicity one subgroups

We study commutative Schur rings over the symplectic groups Sp(n,2) that contain the class C of symplectic transvections. We find the possible partitions of C determined by the Schur ring. We show how this restricts the possibilities for multiplicity one subgroups of Sp(n,2).

Constructing linked systems of relative difference sets via Schur rings

Constructing linked systems of relative difference sets via Schur rings

Schur rings over infinite dihedral group

Schur rings over the infinite dihedral group Z ⋊ Z 2 are studied according to properties of Schur rings over infinite groups and the classification of Schur rings over infinite cyclic groups. Schur rings over Z ⋊ Z 2 are classified under the assumption that Z is an A -subgroup. Those Schur rings are proved to be traditional.

On Schur rings over infinite groups III

On Schur rings over infinite groups III

Construction of rank 4 self‐dual association schemes inducing three partial geometric designs

AbstractXu characterized rank 4 self‐dual association schemes inducing three partial geometric designs by their character tables. We construct such association schemes as Schur rings over Abelian 2‐groups.

On a Huge Family of Non-Schurian Schur Rings

In his famous monograph on permutation groups, H. Wielandt gives an example of a Schur ring over an elementary abelian group of order $p^2$ ($p>3$ is a prime), which is non-schurian, that is, it is the transitivity module of no permutation group. Generalizing this example, we construct a huge family of non-schurian Schur rings over elementary abelian groups of even rank.

Splitting fields of spectra of circulant graphs

Given a circulant graph Cay(Zn,S), we precisely determine its splitting field and algebraic degree, i.e. the least algebraic extension K|Q of the rationals, which contains all eigenvalues of Cay(Zn,S), and its respective degree [K:Q]. This generalizes a result of So who classified all integral circulant graphs. We prove that there is a deep connection between Schur rings and the splitting field of circulant graphs, which are subfields of cyclotomic fields, and solve the inverse Galois problem for circulant graphs showing that every finite abelian extension of the rationals is the splitting field of some circulant graph. Moreover, we deduce some new necessary criteria for isospectrality of circulant graphs and give a graph-theoretical interpretation of the algebraic degree of circulant graphs: we prove that the algebraic degree of a circulant graph is determined by its automorphism group.

Enumeration techniques on cyclic Schur rings

Any Schur ring is uniquely determined by a partition of the elements of the group. In this paper we present a general technique for enumerating Schur rings over cyclic groups using traditional Schur rings. We also survey recent efforts to enumerate Schur rings over cyclic groups of specific orders.

Schur rings over

For the direct product of infinite cyclic group and a cyclic group of order 3, the Schur rings over it are classified. In particular, all the Schur rings are proved to be traditional.

Separability of Schur Rings Over Abelian Groups of Odd Order

An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every algebraic isomorphism from the $S$-ring in question to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial isomorphism. A finite group $G$ is said to be separable with respect to $\mathcal{K}$ if every $S$-ring over $G$ is separable with respect to $\mathcal{K}$. We prove that every abelian group $G$ of order $9p$, where $p$ is a prime, is separable with respect to the class of all finite abelian groups. Modulo previously obtained results, this completes a classification of noncyclic abelian groups of odd order that are separable with respect to the class of all finite abelian groups. Also this implies that the Weisfeiler-Leman dimension of the class of Cayley graphs over $G$ is at most 2.

Schur rings over infinite groups II

We construct non-traditional Schur rings over various infinite groups, including surface groups, some free products with amalgamation, PSL(2,Z), and some groups related to crystallographic groups. Here non-traditional means that the Schur rings are not direct products, wedge products or orbit Schur rings. In certain cases we are able to investigate eigenvalues and eigenvectors for these Schur rings.

Eparability of Schur Rings Over an Abelian Group of Order 4p

An $S$-ring (a Schur ring) is said to be separable with respect to a class of groups $\mathcal{K}$ if every its algebraic isomorphism to an $S$-ring over a group from $\mathcal{K}$ is induced by a combinatorial isomorphism. We prove that every Schur ring over an abelian group $G$ of order $4p$, where $p$ is a prime, is separable with respect to the class of abelian groups. This implies that the Weisfeiler-Leman dimension of the class of Cayley graphs over $G$ is at most 2.

CI-Property for Decomposable Schur Rings over an Abelian Group

A Schur ring over a finite group is said to be decomposable if it is the generalized wreath product of Schur rings over smaller groups. In this paper we establish a sufficient condition for a decomposable Schur ring over the direct product of elementary abelian groups to be a CI-Schur ring. By using this condition we offer short proofs for some known results on the CI-property for decomposable Schur rings over an elementary abelian group of rank at most 5.

Difference Sets Disjoint from a Subgroup

We study finite groups G having a non-trivial, proper subgroup H and $$D \subset G {\setminus } H, D \cap D^{-1}=\emptyset ,$$ such that the multiset $$\{ xy^{-1}:x,y \in D\}$$ has every non-identity element occur the same number of times (such a D is called a difference set). We show that $$|G|=|H|^2$$ , and that $$|D \cap Hg|=|H|/2$$ for all $$g \notin H$$ . We show that H is contained in every normal subgroup of index 2, and other properties. We give a 2-parameter family of examples of such groups. We show that such groups have Schur rings with four principal sets, and that, further, these difference sets determine DRADs.

On Schur Rings over Infinite Groups

Schur rings are a type of subring of the group ring that is spanned by a partition of the group that meets certain conditions. Past literature has exclusively focused on the finite group case. This paper extends many classic results about Schur rings to the infinite groups, including Leung-Man's classification of Schur rings over finite cyclic groups. Additionally, this classification will be extended to all torsion-free locally cyclic groups.

Counting Schur rings over cyclic groups

Any Schur ring is uniquely determined by a partition of the elements of the group. An open question in the study of Schur rings is determining which partitions of the group induce a Schur ring. Although a structure theorem is available for Schur rings over cyclic groups, it is still a difficult problem to count all the partitions. For example, Kovacs, Liskovets, and Poschel determine formulas to count the number of wreath-indecomposable Schur rings. In this paper we solve the problem of counting the number of all Schur rings over cyclic groups of prime power order and draw some parallels with Higman's PORC conjecture.

Groups whose proper Schur rings are commutative

A Schur ring [Formula: see text] is called Dedekind if the formal sum of every [Formula: see text]-subgroup is in the center of [Formula: see text]. In this paper, we find all finite groups [Formula: see text] such that every proper Schur ring over [Formula: see text] is Dedekind. As a corollary of our main theorem, we find all finite groups [Formula: see text] such that every proper Schur ring over [Formula: see text] is commutative.

Tamaschke’s Results on Schur Rings and a Generalization of Association Schemes

The concepts of an association scheme and a coherent configuration are generalized by analyzing a relationship between S-rings (Schur rings) and association schemes. In this connection, Tamaschke’s results on S-rings and other generalizations of association schemes are discussed.