On Schur Rings over Cyclic Groups, II

In the paper, we develop further the properties of Schur rings over infinite groups, with particular emphasis on the virtually cyclic group Z × Zp. We provide structure theorems for primitive sets in these Schur rings.In the case of Fermat and safe primes, a complete classification theorem is proven which states that all Schur rings over Z × Zp are traditional. We also draw analogs between Schur rings over Z × Zp and partitions of difference sets over Zp.

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“…As S contains the singleton {z}, S = S Z × S p . 4 As S Z and S p are automorphic, S is automorphic as well.…”

Section: The Structure Of Primitive Sets Of Schur Rings Over Z × Z Nmentioning

confidence: 99%

On Schur Rings over Infinite Groups

Bastian

Brewer

Humphries

et al. 2019

Algebr Represent Theor

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“…See [11] for further treatment of wedge products. Theorem 1.3 (Leung and Man [11,10]). Let G = Z n and let S be a Schur ring over G. Then S is trivial, an orbit ring, a dot product of Schur rings, or a wedge product of Schur rings.…”

Section: Schur Ringsmentioning

confidence: 99%

Primitive Idempotents of Schur Rings

Misseldine

2014

Algebr Represent Theor

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In this paper, we explore the nature of central idempotents of Schur rings over finite groups. We introduce the concept of a lattice Schur ring and explore properties of these kinds of Schur rings. In particular, the primitive, central idempotents of lattice Schur rings are completely determined. For a general Schur ring S, S contains a maximal lattice Schur ring, whose central, primitive idempotents form a system of pairwise orthogonal, central idempotents in S. We show that if S is a Schur ring with rational coefficients over a cyclic group, then these idempotents are always primitive and are spanned by the normal subgroups contained in S. Furthermore, a Wedderburn decomposition of Schur rings over cyclic groups is given. Some examples of Schur rings over non-cyclic groups will also be explored.Keywords: Schur Ring, cyclic group, primitive idempotent, group ring, Wedderburn decomposition AMS Classification: 20C05, 17C27, 16D70In 1950, Perlis and Walker [17] published the following result on the rational group algebra of a finite abelian group:Theorem. Let ζ n = e 2πi/n ∈ C. Let G be a finite abelian group of order n.where a d is the number of cyclic subgroups (or cyclic quotients) of G of order d. In particular, if G = Z n is a cyclic group of order n, thenOne consequence of the above decomposition is a solution to the isomorphism problem of group rings over finite abelian groups with integer coefficients. Since then, several other results about rational group algebras have been published

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“…A more general product operation, the wedge product of association schemes, was recently introduced and investigated in [88]. The term goes back to [76,77] who used it for recursive classification of S-rings over cyclic groups. In a similar situation Evdokimov and Ponomarenko [34] are speaking about wreath product of S-rings (unfortunately, their terminology does not coincide with traditional one).…”

Section: 4mentioning

confidence: 99%

Automorphism Groups of Rational Circulant Graphs

Klin¹,

Kovács²

2012

Electron. J. Combin.

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The paper concerns the automorphism groups of Cayley graphs over cyclic groups which have a rational spectrum (rational circulant graphs for short). With the aid of the techniques of Schur rings it is shown that the problem is equivalent to consider the automorphism groups of orthogonal group block structures of cyclic groups. Using this observation, the required groups are expressed in terms of generalized wreath products of symmetric groups.

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On Schur Rings over Cyclic Groups, II

Cited by 60 publications

References 5 publications

On Schur Rings over Infinite Groups

On Schur Rings over Infinite Groups

Primitive Idempotents of Schur Rings

Automorphism Groups of Rational Circulant Graphs

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