Representations Of Cyclic Groups Research Articles

Classification of ℤ3-Equivariant Simple Function Germs

The present paper deals with the classification of multivariate holomorphic function germs that are equivariant simple under representations of cyclic groups. We obtain a complete classification of such function germs of two and three variables for all possible nontrivial ℤ3-actions. Our main classification methods generalize those used for the classification of simple germs in the nonequivariant case.

О классификации особенностей, эквивариантно простых относительно представлений циклических групп

О классификации особенностей, эквивариантно простых относительно представлений циклических групп

On Representations of Cyclic Groups Over a Ring of Integers

The purpose of this article to determine and classify the indecomposable RG-lattices, where R is a certain ring of algebraic integers, and G is a cyclic group of prime order.

Depth of modular invariant rings

It is well-known that the ring of invariants associated to a non-modular representation of a finite group is Cohen-Macaulay and hence has depth equal to the dimension of the representation. For modular representations the ring of invariants usually fails to be Cohen-Macaulay and computing the depth is often very difficult. In this paper1 we obtain a simple formula for the depth of the ring of invariants for a family of modular representations. This family includes all modular representations of cyclic groups. In particular, we obtain an elementary proof of the celebrated theorem of Ellingsrud and Skjelbred [6].

Z 3-Graded exterior differential calculus and gauge theories of higher order

We present a possible generalization of the exterior differential calculus, based on the operator d such that d^3=0, but d^2\not=0. The first and second order differentials generate an associative algebra; we shall suppose that there are no binary relations between first order differentials, while the ternary products will satisfy the cyclic relations based on the representation of cyclic group Z_3 by cubic roots of unity. We shall attribute grade 1 to the first order differentials and grade 2 to the second order differentials; under the associative multiplication law the grades add up modulo 3. We show how the notion of covariant derivation can be generalized with a 1-form A, and we give the expression in local coordinates of the curvature 3-form. Finally, the introduction of notions of a scalar product and integration of the Z_3-graded exterior forms enables us to define variational principle and to derive the differential equations satisfied by the curvature 3-form. The Lagrangian obtained in this way contains the invariants of the ordinary gauge field tensor F_{ik} and its covariant derivatives D_i F_{km}.

On Diederichsen's formula for extensions of lattices

and thus R, may be viewed as ZG-lattice. Indeed, every RR,-module becomes a ZG-module by virtue of the homomorphism ZG + R, . In Diederichsen’s fundamental article [2] on integral representations of cyclic groups, one of the vital steps was the computation of Ext&(R, , R,) when m and n are divisors of g. Diederichsen proved the following remarkable results: Ext&(R, , R,) = 0 unless m/n = pt for some prime p and some nonzero t E 2. If m/n = pt for some prime p and some nonzero integer t (possibly negative), then Ext&(R, , R,) . 1s isomorphic to the direct sum of v(Z) copies of z, where Z = Z/p2 and E = min(m, n). Diederichsen’s proof, though elementary in nature, depended on complicated manipulations with roots of unity. In this article we shall give an extremely

Integral Representations of Cyclic Groups of Order p 2

Integral Representations of Cyclic Groups of Order p 2

Representations of Cyclic Groups in Rings of Integers, II

Let Gk be a cyclic group of order k, and let ZGk denote its group ring over the ring Z of rational integers. We denote by n(ZGk) the number of non-isomorphic indecomposable left ZGk-modules having finite Z-bases. In 1938 Diederichsen [2] proved that n(ZG,) is finite for p a rational prime, and gave an incorrect proof that n(ZG4) is infinite. Recently Roiter [6] and Troy [8] independently showed that n(ZG4) = 9. The present authors [4] have shown more generally that n(ZGP2) is finite for all primes p. The aim of this paper is to establish the somewhat surprising result that n(ZGP3) is infinite for each prime p. Several interesting consequences of this result should be pointed out. To begin with, note that GP3 is a homomorphic image of Gp, for each r ? 3, and so each ZGP3-module may be viewed also as a ZGpr-module. Therefore n(ZGpr) _ n(ZGP3), which shows that n(ZGpr) is infinite for each r > 3. Next, let Hp denote (for the moment) a p-Sylow subgroup of an arbitrary finite group G. The authors have shown [4] that if n(ZHp) is infinite for some p, then also n(ZG) is infinite. They proved furthermore that n(ZHp) is necessarily infinite if Hp is non-cyclic. Combining this result with that of the present paper, it follows that n(ZHp) is finite if and only if Hp is cyclic of order p or p2. Thus, the possible groups G for which n(ZG) is finite are small in number; in fact, if n(ZG) is finite, then for each p dividing the order of G each p-Sylow subgroup of G must be cyclic of order p or p2. Whether the converse is true is as yet unknown.t Let Z* denote the ring of p-adic integers in the p-adic completion of the rational field. Define n(Z*Gpr) to be the number of non-isomorphic indecomposable left Z*Gpr-modules having finite Z*-bases. In their previous paper [4], the authors have shown that n(ZGpr) is finite if and only if n(Z*Gpr)is finite. It is therefore sufficienttoprovethatn(Z*GP3) is infinite. Section 1 contains some general remarks about extensions of one direct

Representations of Cyclic Groups in Rings of Integers, I

Representations of Cyclic Groups in Rings of Integers, I