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1 Resolutions in Hap
 1.1 The Standard Representation HapResolutionRep
 1.2 The HapLargeGroupResolutionRep Representation

1 Resolutions in Hap

This document is only concerned with the representation of resolutions in Hap. Note that it is not a part of Hap. The framework provided here is just an extension of Hap data types used in HAPcryst and HAPprime.

From now on, let G be a group and dots -> M_n-> M_n-1->dots-> M_1-> M_0-> Z be a resolution with free ZG modules M_i.

The elements of the modules M_i can be represented in different ways. This is what makes different representations for resolutions desirable. First, we will look at the standard representation (HapResolutionRep) as it is defined in Hap. After that, we will present another representation for infinite groups. Note that all non-standard representations must be sub-representations of the standard representation to ensure compatibility with Hap.

1.1 The Standard Representation HapResolutionRep

For every M_i we fix a basis and number its elements. Furthermore, it is assumed that we have a (partial) enumeration of the group of a resolution. In practice this is done by generating a lookup table on the fly.

In standard representation, the elements of the modules M_k are represented by lists -"words"- of pairs of integers. A letter [i,g] of such a word consists of the number of a basis element i or -i for its additive inverse and a number g representing a group element.

A HapResolution in HapResolutionRep representation is a component object with the components

Note that making HapResolutions immutable will make the .elts component immutable. As this lookup table might change during calculations, we do not recommend using immutable resolutions (in any representation).

1.2 The HapLargeGroupResolutionRep Representation

In this sub-representation of the standard representation, the module elements in this resolution are lists of groupring elements. So the lookup table .elts is not used as long as no conversion to standard representation takes place. In addition to the components of a HapResolution, a resolution in large group representation has the following components:

The effort of having two versions of boundary and dimension is necessary to keep the structure compatible with the usual Hap resolution.

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