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.
In particular this package provides efficient methods for finding conjugating elements, conjugacy classes, and centralisers. The implementations are based on [BNRW22] and references therein.
Here we include a list of operations that take advantage of the generic representation of wreath product elements.
We include python scripts in the dev/ 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
In the following let G = K ≀ H be a wreath product, where H ≤ Sym(m).
In GAP the wreath product G can be given in one of the following representations :
Generic Representation
Permutation Representation in Imprimitive Action
Permutation Representation in Product Action
Matrix Representation
Further let x, y ∈ P = K ≀ Sym(m) be elements of the parent wreath product P which is given in the same representation as G.
The following operations use implementations that exploit the generic representation and (sparse) wreath cycle decompositions :
RepresentativeAction(G, x, y)
Centraliser(G, x)
ConjugacyClasses(G)
Here we assume that G is given in some permutation representation.
The following operations use implementations that exploit the generic representation and (sparse) wreath cycle decompositions :
CycleIndex(G)
Here we assume that G is given in generic representation.
The following operations use implementations that exploit the generic representation and (sparse) wreath cycle decompositions :
Order(x)
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