Quelle operations.xml Sprache: XML

<Chapter Label="Operations">
<Heading>Operations</Heading>

The generic representation of wreath product elements in wreath products of finite groups
and in particular their (sparse) wreath cycle decompositions
can be used to speed up certain computations in wreath products. <P/>

In particular this package provides efficient methods for finding conjugating elements, conjugacy classes, and centralisers.
The implementations are based on <Cite Key="WPE"/> and references therein.

<Section Label="Operations List">
<Heading>Operations List</Heading>

Here we include a list of operations that take advantage
of the generic representation of wreath product elements.<P/>

We include python scripts in the <C>dev/</C> directory that benchmark
the &WPE; and native &GAP; implementations of these operations separately.
The comparison of the runtimes supports the conclusion that the &WPE; implementations are
an order of magnitude faster than the native &GAP; implementations.
We can now solve these computational tasks for large wreath products
that were previously not feasible in &GAP;<P/>

<Subsection Label="Wreath Product Representations">
<Heading>Wreath Product Representations</Heading>

In the following let <M>G = K \wr H</M> be a wreath product, where <M>H \leq \textrm{Sym}(m)</M>.<P/>

In &GAP; the wreath product <M>G</M> can be given in one of the following representations :

<List>
<Item> Generic Representation</Item>
<Item> Permutation Representation in Imprimitive Action </Item>
<Item> Permutation Representation in Product Action </Item>
<Item> Matrix Representation </Item>
</List>

</Subsection>

<Subsection Label="Operations for all Representations">
<Heading>Operations for all Representations</Heading>

Further let <M>x, y \in P = K \wr \textrm{Sym}(m)</M>
be elements of the parent wreath product <M>P</M>
which is given in the same representation as <M>G</M>.<P/>



The following operations use implementations that exploit the generic representation and (sparse) wreath cycle decompositions :

<List>
<Item> <C>RepresentativeAction(G, x, y)</C> </Item>
<Item> <C>Centraliser(G, x)</C> </Item>
<Item> <C>ConjugacyClasses(G)</C> </Item>

</List>

</Subsection>

<Subsection Label="Operations for Permutation Representation">
<Heading>Operations for Permutation Representations</Heading>

Here we assume that <M>G</M> is given in some permutation representation.<P/>

The following operations use implementations that exploit the generic representation and (sparse) wreath cycle decompositions :

<List>
<Item> <C>CycleIndex(G)</C> </Item>
</List>

</Subsection>

<Subsection Label="Operations for Generic Representation">
<Heading>Operations for Generic Representation</Heading>

Here we assume that <M>G</M> is given in generic representation.<P/>

The following operations use implementations that exploit the generic representation and (sparse) wreath cycle decompositions :

<List>
<Item> <C>Order(x)</C> </Item>
</List>

</Subsection>

</Section>



</Chapter>

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