Recall that for a nilpotent associative algebra A its coclass is defined as cc(A) = dim(A) - cl(A). Fixing a finite field F and a non-negative integer r one can define the associated coclass graph. Its vertices correspond one-to-one to the isomorphism types of nilpotent associative F-algebras of coclass r and there is a directed edge A to B if A is isomorphic to B/B^cl(B). Such an algebra is called coclass settled, if all its descendants are of the same coclass as the algebra itself. The ccalgs package provides methods for calculating the coclass of a given nilpotent associative algebra and determining whether an algebra is coclass settled, i.e. whether all of its descendants have the same coclass as the algebra itself. Furthermore there are methods available for calculating finite parts of a coclass graph and plotting these finite parts using the XGAP package.
‣ CoclassOfCanoForm ( C ) | ( attribute ) |
This returns the coclass of the given canonical form C.
‣ IsCoclassSettledCanoForm ( C ) | ( attribute ) |
Determines whether the given canonical form C is coclass settled.
We usually divide the calculation of a coclass graph in two steps. The first step is the calculation of the roots of the maximal descendant trees as described in [EM17], the second step is an iterated computation of stepsize 1 descendants. The following functions implement these two steps.
‣ RootsOfCoclassGraph ( F, r ) | ( operation ) |
Given a finite field F, a coclass r this function calculates the roots of the maximal descendant trees of the associated coclass graph.
‣ RootsOfCoclassGraphByRank ( F, r, d ) | ( operation ) |
Given a finite field F, a coclass r and a generator number d this function calculates the roots with the given number of generators of the maximal descendant trees of the associated coclass graph.
‣ DescendantTreeOfCanoForm ( C, l ) | ( operation ) |
Given a canonical form C for a nilpotent associative F-algebra, this iteratedly calculates l levels of descendants of the same coclass as C.
This section contains functions for plotting the constructed coclass graphs and trees. Note that plotting requires the XGAP package, see [CN12]. The parameter P in all these functions must be a graphic poset created using the function GraphicPoset from the XGAP package. Once plotted it is also possible to recover the canonical forms or nilpotent associative algebras associated to a vertex. Currently the following functions are available.
‣ PlotDescendantTreeOfCanoForm ( C, d, P ) | ( operation ) |
Given a canonical form C, this plots the descendant tree of canonical forms of the same coclass as C up to dimension d into the GraphicPoset P.
‣ PlotCoclassGraph ( F, r, d, P ) | ( operation ) |
Given a finite field F and a coclass r, this plots the associated coclass graph up to dimension d into the GraphicPoset P.
‣ GetSelectedCanoForms ( P ) | ( operation ) |
Returns a list of canonical forms corresponding to the selected vertices in the GraphicPoset P.
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