A Mathematica Notebook for Computing the Homology of Iterated Products of Groups

Álvarez, Víctor; Armario, J. A.; Frau, M. D.; Real, Pedro

doi:10.1007/11832225_5

Cited by 4 publications

(7 citation statements)

References 13 publications

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“…Furthermore, the method works over other semidirect products of groups (as well as iterated products of groups), even though the fibre groups K may not be a finitely generated Abelian group (see Remarks 3 and 5). An extended version of the method for iterated products of central extensions and semidirect products of finitely generated Abelian groups has been implemented in Mathematica by the authors (see [1,3]). Some calculations with this package have led to the finding new cocyclic Hadamard matrices [2,4].…”

Section: On the Computation Of The Homology Of Groupsmentioning

confidence: 99%

“…The element of B 0 (A) corresponding to the identity element of (ground ring) is denoted by [ ] and the element sā 1 …”

Section: Example 1 the Dihedral Groupmentioning

confidence: 99%

“…This is the case of the iterated products of central extensions and semidirect products of finitely generated Abelian groups considered in [1,3].…”

Section: Remarkmentioning

confidence: 99%

“…All the executions and examples of this section have been worked out with aid of the Mathematica 4.0 notebook [3] described in [1]. We now include some calculations for dihedral groups and an iterated product of a central extension by a semidirect product of finite abelian groups by means of their homological models.…”

Section: Examplesmentioning

confidence: 99%

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Homological models for semidirect products of finitely generated Abelian groups

Álvarez

Armario

Frau

et al. 2012

AAECC

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Let G be a semidirect product of finitely generated Abelian groups. We provide a method for constructing an explicit contraction (special homotopy equivalence) from the reduced bar construction of the group ring of G, B(Z Z[G]), to a much smaller DGA-module hG. Such a contraction is called a homological model for G and is used as the input datum in the methods described in Álvarez et al. (J Symb Comput 44:558-570, 2009;2012) for calculating a generating set for representative 2-cocycles and n-cocycles over G, respectively. These computations have led to the finding of new cocyclic Hadamard matrices (Álvarez et al. in 2006).

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Section: On the Computation Of The Homology Of Groupsmentioning

confidence: 99%

“…The element of B 0 (A) corresponding to the identity element of (ground ring) is denoted by [ ] and the element sā 1 …”

Section: Example 1 the Dihedral Groupmentioning

confidence: 99%

“…This is the case of the iterated products of central extensions and semidirect products of finitely generated Abelian groups considered in [1,3].…”

Section: Remarkmentioning

confidence: 99%

Section: Examplesmentioning

confidence: 99%

See 2 more Smart Citations

Homological models for semidirect products of finitely generated Abelian groups

Álvarez

Armario

Frau

et al. 2012

AAECC

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“…The second one takes advantage of the inflation and transgression maps and was settled in [13] for those groups for which the word problem is solvable. The third one applies to groups for which a homological model is known [6][7][8], and has been implemented in Mathematica [3,4]. The theoretical background is explained in [5].…”

Section: Introductionmentioning

confidence: 99%

A system of equations for describing cocyclic Hadamard matrices

Álvarez¹,

Armario²,

Frau³

et al. 2008

J of Combinatorial Designs

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Abstract. Given a basis B = {f1, . . . , f k } for 2-cocycles f : G × G → {±1} over a group G of order |G| = 4t, we describe a non-linear system of 4t − 1 equations and k indeterminates xi over Z Z2, 1 ≤ i ≤ k, whose solutions determine the whole set of cocyclic Hadamard matrices over G, in the sense that (x 1 , . . . , x k ) is a solution of the system if and only if the 2-cocycle f = f , gj)). Furthermore, the study of any isolated equation of the system provides upper and lower bounds on the number of coboundary generators in B which have to be combined to form a cocyclic Hadamard matrix coming from a special class of cocycles. We include some results on the families of groups Z Z 2 2 × Z Z t and D4t. A deeper study of the system provides some more nice properties. For instance, in the case of dihedral groups D 4t , we have found that it suffices to check t instead of the 4t rows of M f , to decide the Hadamard character of the matrix (for a special class of cocycles f ).

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On higher dimensional cocyclic Hadamard matrices

Álvarez

Armario

Frau

et al. 2014

AAECC

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Provided that a cohomological model for G is known, we describe a method for constructing a basis for n-cocycles over G, from which the whole set of n-dimensional n-cocyclic matrices over G may be straightforwardly calculated. Focusing in the case n = 2 (which is of special interest, e.g. for looking for cocyclic Hadamard matrices), this method provides a basis for 2-cocycles in such a way that representative 2-cocycles are calculated all at once, so that there is no need to distinguish between inflation and transgression 2-cocycles (as it has traditionally been the case until now). When n > 2, this method provides an uniform way of looking for higher dimensional n-cocyclic Hadamard matrices for the first time. We illustrate the method with some examples, for n = 2, 3. In particular, we give some examples of improper 3-dimensional 3-cocyclic Hadamard matrices. Keywords: (co)homological model cocyclic matrix proper/improper higher dimensional Hadamard matrix.

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A Mathematica Notebook for Computing the Homology of Iterated Products of Groups

Cited by 4 publications

References 13 publications

Homological models for semidirect products of finitely generated Abelian groups

Homological models for semidirect products of finitely generated Abelian groups

A system of equations for describing cocyclic Hadamard matrices

On higher dimensional cocyclic Hadamard matrices

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