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Chapter 1.
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Chapter 2.
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Chapter 3.
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[]\T1/ptm/m/n/10.95 Frank Celler and Max Ne-un-ho-ef-fer. Xgap 4.23, a gap4 pa
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ccalgs/doc/chap0.txt 0000644 0216424 0000144 00000005223 14171241126 014337 0 ustar tobmoede users
[1X[5Xccalgs[105X[101X
[1XDescendants of nilpotent associative algebras and computation of coclass
graphs for such algebras[101X
Version 1.1
17 January 2022
Bettina Eick
Tobias Moede
Bettina Eick
Email: [7Xmailto:beick@tu-bs.de[107X
Homepage: [7Xhttp://www.iaa.tu-bs.de/beick/[107X
Tobias Moede
Email: [7Xmailto:t.moede@tu-bs.de[107X
Homepage: [7Xhttps://www.tu-braunschweig.de/iaa/personal/moede[107X
-------------------------------------------------------
[1XCopyright[101X
[33X[0;0Y© 2022 by Bettina Eick and Tobias Moede[133X
[33X[0;0Y[5Xccalgs[105X package is free software; you can redistribute it and/or modify it
under the terms of the GNU General Public License
([7Xhttp://www.fsf.org/licenses/gpl.html[107X) as published by the Free Software
Foundation; either version 2 of the License, or (at your option) any later
version.[133X
-------------------------------------------------------
[1XContents (ccalgs)[101X
1 [33X[0;0YIntroduction[133X
1.1 [33X[0;0YNilpotent associative algebras and canonical forms[133X
1.1-1 NAAlgByCanoForm
1.1-2 CanoFormByNAAlg
2 [33X[0;0YDescendants of nilpotent associative algebras[133X
2.1 [33X[0;0YImmediate descendants[133X
2.1-1 ImmediateDescendantsOfCanoForm
2.1-2 ImmediateDescendantsOfCanoForm
2.1-3 ImmediateDescendantsOfCanoForm
2.1-4 IsCapableCanoForm
2.2 [33X[0;0YClassifying nilpotent associative algebras by dimension[133X
2.2-1 CanoFormsByDimAndGens
2.2-2 CanoFormsByDim
2.3 [33X[0;0YEnumerating nilpotent associative algebras[133X
2.3-1 NumberOfImmediateDescendantsOfCanoForm
2.3-2 NumberOfImmediateDescendantsOfCanoForm
2.3-3 NumberOfImmediateDescendantsOfCanoForm
2.3-4 NumberOfNAAlgsByDimAndGens
2.3-5 NumberOfNAAlgsByDim
3 [33X[0;0YCoclass theory and coclass graphs[133X
3.1 [33X[0;0YCalculating coclass and determine coclass settledness[133X
3.1-1 CoclassOfCanoForm
3.1-2 IsCoclassSettledCanoForm
3.2 [33X[0;0YCalculating coclass graphs[133X
3.2-1 RootsOfCoclassGraph
3.2-2 RootsOfCoclassGraphByRank
3.2-3 DescendantTreeOfCanoForm
3.3 [33X[0;0YPlotting coclass graphs[133X
3.3-1 PlotDescendantTreeOfCanoForm
3.3-2 PlotCoclassGraph
3.3-3 GetSelectedCanoForms
[32X
ccalgs/doc/chap1.txt 0000644 0216424 0000144 00000003014 14171241126 014334 0 ustar tobmoede users
[1X1 [33X[0;0YIntroduction[133X[101X
[33X[0;0YThis package contains implementations of the algorithms described in [EM17].
Namely it contains a descendants algorithm for nilpotent associative
algebras and a method to calculate the roots of maximal descendant trees in
the coclass graph associated to the nilpotent associative [22XF[122X-algebras of
coclass [22Xr[122X. The implementation relies on the ModIsom package, especially the
representation of nilpotent associative [22XF[122X-algebras by canonical forms, see
[Eic11] for details.[133X
[1X1.1 [33X[0;0YNilpotent associative algebras and canonical forms[133X[101X
[33X[0;0YFor convenience a lot of the functions in this package are implemented for
canonical forms and for nilpotent associative algebras. Only the functions
taking canonical forms as input are documented. The corresponding functions
for nilpotent associative algebras are named similar to the ones for
canonical forms, replacing CanoForm by NAAlg. One can convert between the
two types using the following functions.[133X
[1X1.1-1 NAAlgByCanoForm[101X
[29X[2XNAAlgByCanoForm[102X( [3XC[103X ) [32X operation
[33X[0;0YConverts a canonical form [3XC[103X to a nilpotent associative algebra.[133X
[1X1.1-2 CanoFormByNAAlg[101X
[29X[2XCanoFormByNAAlg[102X( [3XA[103X ) [32X operation
[33X[0;0YConverts a nilpotent associative algebra [3XA[103X to a canonical form.[133X
ccalgs/doc/ccgraphs.xml 0000644 0216424 0000144 00000006406 14171237117 015130 0 ustar tobmoede users
Coclass theory and coclass graphs
Coclass theory and coclass graphs
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.
Calculating coclass and determine coclass settledness
<#Include Label="CoclassOfCanoForm">
<#Include Label="IsCoclassSettledCanoForm">
Calculating coclass graphs
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 , the second step is an iterated computation
of stepsize 1 descendants. The following functions implement these two steps.
<#Include Label="RootsOfCoclassGraph">
<#Include Label="RootsOfCoclassGraphByRank">
<#Include Label="DescendantTreeOfCanoForm">
Plotting coclass graphs
This section contains functions for plotting the constructed coclass graphs
and trees. Note that plotting requires the XGAP package, see .
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.
<#Include Label="PlotDescendantTreeOfCanoForm">
<#Include Label="PlotCoclassGraph">
<#Include Label="GetSelectedCanoForms">
ccalgs/doc/main.xml 0000644 0216424 0000144 00000003007 14171236611 014252 0 ustar tobmoede users
ccalgs">
<#Include Label="PKGVERSIONDATA">
]>
&ccalgs;
Descendants of nilpotent associative algebras and computation of coclass graphs for such algebras
Version &VERSION;
&RELEASEDATE;
Bettina Eick
beick@tu-bs.de
http://www.iaa.tu-bs.de/beick/
Tobias Moede
t.moede@tu-bs.de
https://www.tu-braunschweig.de/iaa/personal/moede
License
©right; 2022 by Bettina Eick and Tobias Moede
&ccalgs; package is free software;
you can redistribute it and/or modify it under the terms of the
http://www.fsf.org/licenses/gpl.html
as published by the Free Software Foundation; either version 2 of the License,
or (at your option) any later version.
<#Include SYSTEM "introduction.xml">
<#Include SYSTEM "generation.xml">
<#Include SYSTEM "ccgraphs.xml">
ccalgs/doc/generation.xml 0000644 0216424 0000144 00000005250 14171237132 015462 0 ustar tobmoede users
Descendants of nilpotent associative algebras
Descendants of nilpotent associative algebras
Immediate descendants
Let F be a finite field and let A and B be two nilpotent
associative F-algebras. Then B is called an immediate descendant
of A if A is isomorphic to B/B^{cl(B)}. The difference
dim(B)-dim(A) is called the stepsize of the descendant. A nilpotent associative
F-algebra is called capable, if it has descendants. In an
algorithm is described to calculate these immediate descendants. This algorithm and
the capability check is implemented in the following functions.
<#Include Label="ImmediateDescendantsOfCanoForm">
<#Include Label="IsCapableCanoForm">
Classifying nilpotent associative algebras by dimension
The descendant algorithm can be used for a classification of nilpotent
associative F-algebras by dimension. This is used in the following
functions.
<#Include Label="CanoFormsByDimAndGens">
<#Include Label="CanoFormsByDim">
Enumerating nilpotent associative algebras
Sometimes it will be enough to enumerate the number of immediate descendants
or the number of nilpotent associative algebras of a given dimension. This is
implemented in the following functions using orbit counting methods. However,
note that the enumeration of n-dimensional algebras still
requires the computation of the (n-1)-dimensional algebras.
<#Include Label="NumberOfImmediateDescendantsOfCanoForm">
<#Include Label="NumberOfNAAlgsByDimAndGens">
<#Include Label="NumberOfNAAlgsByDim">
ccalgs/doc/introduction.xml 0000644 0216424 0000144 00000003463 14171237140 016053 0 ustar tobmoede users
Introduction
Introduction
This package contains implementations of the algorithms described in
. Namely it contains a descendants algorithm for
nilpotent associative algebras and a method to calculate the roots
of maximal descendant trees in the coclass graph associated to the
nilpotent associative F-algebras of coclass r. The implementation
relies on the ModIsom package, especially the representation of
nilpotent associative F-algebras by canonical forms, see
for details.
Nilpotent associative algebras and canonical forms
For convenience a lot of the functions in this package are implemented for
canonical forms and for nilpotent associative algebras. Only the functions
taking canonical forms as input are documented. The corresponding functions
for nilpotent associative algebras are named similar to the ones for canonical
forms, replacing CanoForm by NAAlg. One can convert between the two types using
the following functions.
<#Include Label="NAAlgByCanoForm">
<#Include Label="CanoFormByNAAlg">
ccalgs/doc/chap2.txt 0000644 0216424 0000144 00000011743 14171241126 014345 0 ustar tobmoede users
[1X2 [33X[0;0YDescendants of nilpotent associative algebras[133X[101X
[1X2.1 [33X[0;0YImmediate descendants[133X[101X
[33X[0;0YLet [22XF[122X be a finite field and let [22XA[122X and [22XB[122X be two nilpotent associative
[22XF[122X-algebras. Then [22XB[122X is called an immediate descendant of [22XA[122X if [22XA[122X is isomorphic
to [22XB/B^cl(B)[122X. The difference [22Xdim(B)-dim(A)[122X is called the stepsize of the
descendant. A nilpotent associative [22XF[122X-algebra is called capable, if it has
descendants. In [EM17] an algorithm is described to calculate these
immediate descendants. This algorithm and the capability check is
implemented in the following functions.[133X
[1X2.1-1 ImmediateDescendantsOfCanoForm[101X
[29X[2XImmediateDescendantsOfCanoForm[102X( [3XC[103X, [3Xs[103X ) [32X operation
[33X[0;0YGiven a canonical form [3XC[103X this function returns the immediate descendants of
[3XC[103X with stepsize [3Xs[103X.[133X
[1X2.1-2 ImmediateDescendantsOfCanoForm[101X
[29X[2XImmediateDescendantsOfCanoForm[102X( [3XC[103X, [3Xslist[103X ) [32X operation
[33X[0;0YGiven a canonical form [3XC[103X this function returns the immediate descendants of
[3XC[103X with stepsizes in [3Xslist[103X.[133X
[1X2.1-3 ImmediateDescendantsOfCanoForm[101X
[29X[2XImmediateDescendantsOfCanoForm[102X( [3XC[103X ) [32X operation
[33X[0;0YGiven a canonical form [3XC[103X this function returns the immediate descendants of
[3XC[103X of all stepsizes.[133X
[1X2.1-4 IsCapableCanoForm[101X
[29X[2XIsCapableCanoForm[102X( [3XC[103X ) [32X property
[33X[0;0YThis returns whether the given canonical form [3XC[103X is capable, i.e. whether it
has descendants.[133X
[1X2.2 [33X[0;0YClassifying nilpotent associative algebras by dimension[133X[101X
[33X[0;0YThe descendant algorithm can be used for a classification of nilpotent
associative [22XF[122X-algebras by dimension. This is used in the following
functions.[133X
[1X2.2-1 CanoFormsByDimAndGens[101X
[29X[2XCanoFormsByDimAndGens[102X( [3XF[103X, [3Xd[103X, [3Xn[103X ) [32X operation
[33X[0;0YGiven a finite field [3XF[103X, a generator number [3Xd[103X and a dimension [3Xn[103X, this
function calculates canonical forms for the [3Xd[103X-generator nilpotent
associative [3XF[103X-algebras of dimension [3Xn[103X.[133X
[1X2.2-2 CanoFormsByDim[101X
[29X[2XCanoFormsByDim[102X( [3XF[103X, [3Xn[103X ) [32X operation
[33X[0;0YGiven a finite field [3XF[103X and a dimension [3Xn[103X, this function calculates canonical
forms for the nilpotent associative [3XF[103X-algebras of dimension [3Xn[103X.[133X
[1X2.3 [33X[0;0YEnumerating nilpotent associative algebras[133X[101X
[33X[0;0YSometimes it will be enough to enumerate the number of immediate descendants
or the number of nilpotent associative algebras of a given dimension. This
is implemented in the following functions using orbit counting methods.
However, note that the enumeration of [22Xn[122X-dimensional algebras still requires
the computation of the [22X(n-1)[122X-dimensional algebras.[133X
[1X2.3-1 NumberOfImmediateDescendantsOfCanoForm[101X
[29X[2XNumberOfImmediateDescendantsOfCanoForm[102X( [3XC[103X, [3Xs[103X ) [32X operation
[33X[0;0YGiven a canonical form [3XC[103X this returns the number of immediate descendants of
[3XC[103X with stepsize [3Xs[103X.[133X
[1X2.3-2 NumberOfImmediateDescendantsOfCanoForm[101X
[29X[2XNumberOfImmediateDescendantsOfCanoForm[102X( [3XC[103X, [3Xslist[103X ) [32X operation
[33X[0;0YGiven a canonical form [3XC[103X this returns the number of immediate descendants of
[3XC[103X with stepsizes in [3Xslist[103X.[133X
[1X2.3-3 NumberOfImmediateDescendantsOfCanoForm[101X
[29X[2XNumberOfImmediateDescendantsOfCanoForm[102X( [3XC[103X ) [32X operation
[33X[0;0YGiven a canonical form [3XC[103X this returns the number of immediate descendants of
[3XC[103X of all stepsizes.[133X
[1X2.3-4 NumberOfNAAlgsByDimAndGens[101X
[29X[2XNumberOfNAAlgsByDimAndGens[102X( [3XF[103X, [3Xd[103X, [3Xn[103X ) [32X operation
[33X[0;0YGiven a finite field [3XF[103X, a generator number [3Xd[103X and a dimension [3Xn[103X, this
function returns the number of [3Xd[103X-generator nilpotent associative [3XF[103X-algebras
of dimension [3Xn[103X.[133X
[1X2.3-5 NumberOfNAAlgsByDim[101X
[29X[2XNumberOfNAAlgsByDim[102X( [3XF[103X, [3Xn[103X ) [32X operation
[33X[0;0YGiven a finite field [3XF[103X and a dimension [3Xn[103X, this function returns the number
of nilpotent associative [3XF[103X-algebras of dimension [3Xn[103X.[133X
ccalgs/doc/chapBib.txt 0000644 0216424 0000144 00000001336 14171241126 014675 0 ustar tobmoede users
[1XReferences[101X
[[20XCN12[120X] [16XCeller, F. and Neunhoeffer, M.[116X, [17XXGAP 4.23, a GAP4 package, available from http://www.gap-system.org/Packages/xgap.html[117X (2012).
[[20XEic11[120X] [16XEick, B.[116X, [17XModIsom - Isomorphism testing and automorphism group computation for modular nilpotent associative algebras[117X (2011), ((A {{\sf GAP} 4} package, available from http://www.icm.tu-bs.de/~beick/so.html)).
[[20XEM17[120X] [16XEick, B. and Moede, T.[116X, [17XClassifying nilpotent associative algebras: small coclass and finite fields[117X, in Algorithmic and experimental methods in algebra, geometry, and number theory, Springer, Cham (2017), 213--229.
[32X
ccalgs/doc/chapInd.txt 0000644 0216424 0000144 00000002125 14171241126 014710 0 ustar tobmoede users
[1XIndex[101X
[2XCanoFormByNAAlg[102X 1.1-2
[2XCanoFormsByDim[102X 2.2-2
[2XCanoFormsByDimAndGens[102X 2.2-1
Coclass theory and coclass graphs 3.
[2XCoclassOfCanoForm[102X 3.1-1
Descendants of nilpotent associative algebras 2.
[2XDescendantTreeOfCanoForm[102X 3.2-3
[2XGetSelectedCanoForms[102X 3.3-3
[2XImmediateDescendantsOfCanoForm[102X, for a range of stepsizes 2.1-2
for a single stepsize 2.1-1
for all stepsizes 2.1-3
Introduction 1.
[2XIsCapableCanoForm[102X 2.1-4
[2XIsCoclassSettledCanoForm[102X 3.1-2
License .-1
[2XNAAlgByCanoForm[102X 1.1-1
[2XNumberOfImmediateDescendantsOfCanoForm[102X, for a range of stepsizes 2.3-2
for a single stepsize 2.3-1
for all stepsizes 2.3-3
[2XNumberOfNAAlgsByDim[102X 2.3-5
[2XNumberOfNAAlgsByDimAndGens[102X 2.3-4
[2XPlotCoclassGraph[102X 3.3-2
[2XPlotDescendantTreeOfCanoForm[102X 3.3-1
[2XRootsOfCoclassGraph[102X 3.2-1
[2XRootsOfCoclassGraphByRank[102X 3.2-2
-------------------------------------------------------
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graphs for such algebras\mbox{}}}\\
\vfill
{\Huge Version 1.1\mbox{}}\\[1cm]
{17 January 2022\mbox{}}\\[1cm]
\mbox{}\\[2cm]
{\Large \textbf{ Bettina Eick \mbox{}}}\\
{\Large \textbf{ Tobias Moede \mbox{}}}\\
\hypersetup{pdfauthor= Bettina Eick ; Tobias Moede }
\end{center}\vfill
\mbox{}\\
{\mbox{}\\
\small \noindent \textbf{ Bettina Eick } Email: \href{mailto://beick@tu-bs.de} {\texttt{beick@tu-bs.de}}\\
Homepage: \href{http://www.iaa.tu-bs.de/beick/} {\texttt{http://www.iaa.tu-bs.de/beick/}}}\\
{\mbox{}\\
\small \noindent \textbf{ Tobias Moede } Email: \href{mailto://t.moede@tu-bs.de} {\texttt{t.moede@tu-bs.de}}\\
Homepage: \href{https://www.tu-braunschweig.de/iaa/personal/moede} {\texttt{https://www.tu-braunschweig.de/iaa/personal/moede}}}\\
\end{titlepage}
\newpage\setcounter{page}{2}
{\small
\section*{Copyright}
\logpage{[ 0, 0, 1 ]}
\index{License} {\copyright} 2022 by Bettina Eick and Tobias Moede
\textsf{ccalgs} package is free software; you can redistribute it and/or modify it under the
terms of the \href{http://www.fsf.org/licenses/gpl.html} {GNU General Public License} as published by the Free Software Foundation; either version 2 of the License,
or (at your option) any later version. \mbox{}}\\[1cm]
\newpage
\def\contentsname{Contents\logpage{[ 0, 0, 2 ]}}
\tableofcontents
\newpage
\chapter{\textcolor{Chapter }{Introduction}}\label{Introduction}
\logpage{[ 1, 0, 0 ]}
\hyperdef{L}{X7DFB63A97E67C0A1}{}
{
\index{Introduction} This package contains implementations of the algorithms described in \cite{EMo15}. Namely it contains a descendants algorithm for nilpotent associative
algebras and a method to calculate the roots of maximal descendant trees in
the coclass graph associated to the nilpotent associative $F$-algebras of coclass $r$. The implementation relies on the ModIsom package, especially the
representation of nilpotent associative $F$-algebras by canonical forms, see \cite{modisom} for details.
\section{\textcolor{Chapter }{Nilpotent associative algebras and canonical forms}}\label{Nilpotent associative algebras and canonical forms}
\logpage{[ 1, 1, 0 ]}
\hyperdef{L}{X85FFFB1482865201}{}
{
For convenience a lot of the functions in this package are implemented for
canonical forms and for nilpotent associative algebras. Only the functions
taking canonical forms as input are documented. The corresponding functions
for nilpotent associative algebras are named similar to the ones for canonical
forms, replacing CanoForm by NAAlg. One can convert between the two types
using the following functions.
\subsection{\textcolor{Chapter }{NAAlgByCanoForm}}
\logpage{[ 1, 1, 1 ]}\nobreak
\hyperdef{L}{X7A8551DD78A8BEF3}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{NAAlgByCanoForm({\mdseries\slshape C})\index{NAAlgByCanoForm@\texttt{NAAlgByCanoForm}}
\label{NAAlgByCanoForm}
}\hfill{\scriptsize (operation)}}\\
Converts a canonical form \mbox{\texttt{\mdseries\slshape C}} to a nilpotent associative algebra. }
\subsection{\textcolor{Chapter }{CanoFormByNAAlg}}
\logpage{[ 1, 1, 2 ]}\nobreak
\hyperdef{L}{X8169C72F802898FA}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{CanoFormByNAAlg({\mdseries\slshape A})\index{CanoFormByNAAlg@\texttt{CanoFormByNAAlg}}
\label{CanoFormByNAAlg}
}\hfill{\scriptsize (operation)}}\\
Converts a nilpotent associative algebra \mbox{\texttt{\mdseries\slshape A}} to a canonical form. }
}
}
\chapter{\textcolor{Chapter }{Descendants of nilpotent associative algebras}}\label{Descendants of nilpotent associative algebras}
\logpage{[ 2, 0, 0 ]}
\hyperdef{L}{X858D322880E7FA8C}{}
{
\index{Descendants of nilpotent associative algebras}
\section{\textcolor{Chapter }{Immediate descendants}}\label{Immediate descendants}
\logpage{[ 2, 1, 0 ]}
\hyperdef{L}{X7D7C28297F6779FB}{}
{
Let $F$ be a finite field and let $A$ and $B$ be two nilpotent associative $F$-algebras. Then $B$ is called an immediate descendant of $A$ if $A$ is isomorphic to $B/B^{cl(B)}$. The difference $dim(B)-dim(A)$ is called the stepsize of the descendant. A nilpotent associative $F$-algebra is called capable, if it has descendants. In \cite{EMo15} an algorithm is described to calculate these immediate descendants. This
algorithm and the capability check is implemented in the following functions.
\subsection{\textcolor{Chapter }{ImmediateDescendantsOfCanoForm (for a single stepsize)}}
\logpage{[ 2, 1, 1 ]}\nobreak
\hyperdef{L}{X7AE4EC917E736A58}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{ImmediateDescendantsOfCanoForm({\mdseries\slshape C, s})\index{ImmediateDescendantsOfCanoForm@\texttt{ImmediateDescendantsOfCanoForm}!for a single stepsize}
\label{ImmediateDescendantsOfCanoForm:for a single stepsize}
}\hfill{\scriptsize (operation)}}\\
Given a canonical form \mbox{\texttt{\mdseries\slshape C}} this function returns the immediate descendants of \mbox{\texttt{\mdseries\slshape C}} with stepsize \mbox{\texttt{\mdseries\slshape s}}. }
\subsection{\textcolor{Chapter }{ImmediateDescendantsOfCanoForm (for a range of stepsizes)}}
\logpage{[ 2, 1, 2 ]}\nobreak
\hyperdef{L}{X808C6CB484406CD5}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{ImmediateDescendantsOfCanoForm({\mdseries\slshape C, slist})\index{ImmediateDescendantsOfCanoForm@\texttt{ImmediateDescendantsOfCanoForm}!for a range of stepsizes}
\label{ImmediateDescendantsOfCanoForm:for a range of stepsizes}
}\hfill{\scriptsize (operation)}}\\
Given a canonical form \mbox{\texttt{\mdseries\slshape C}} this function returns the immediate descendants of \mbox{\texttt{\mdseries\slshape C}} with stepsizes in \mbox{\texttt{\mdseries\slshape slist}}. }
\subsection{\textcolor{Chapter }{ImmediateDescendantsOfCanoForm (for all stepsizes)}}
\logpage{[ 2, 1, 3 ]}\nobreak
\hyperdef{L}{X84FDA1E48706AC93}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{ImmediateDescendantsOfCanoForm({\mdseries\slshape C})\index{ImmediateDescendantsOfCanoForm@\texttt{ImmediateDescendantsOfCanoForm}!for all stepsizes}
\label{ImmediateDescendantsOfCanoForm:for all stepsizes}
}\hfill{\scriptsize (operation)}}\\
Given a canonical form \mbox{\texttt{\mdseries\slshape C}} this function returns the immediate descendants of \mbox{\texttt{\mdseries\slshape C}} of all stepsizes. }
\subsection{\textcolor{Chapter }{IsCapableCanoForm}}
\logpage{[ 2, 1, 4 ]}\nobreak
\hyperdef{L}{X78B6696078EA17C5}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsCapableCanoForm({\mdseries\slshape C})\index{IsCapableCanoForm@\texttt{IsCapableCanoForm}}
\label{IsCapableCanoForm}
}\hfill{\scriptsize (property)}}\\
This returns whether the given canonical form \mbox{\texttt{\mdseries\slshape C}} is capable, i.e. whether it has descendants. }
}
\section{\textcolor{Chapter }{Classifying nilpotent associative algebras by dimension}}\label{Classifying nilpotent associative algebras by dimension}
\logpage{[ 2, 2, 0 ]}
\hyperdef{L}{X7831B9438023FC40}{}
{
The descendant algorithm can be used for a classification of nilpotent
associative $F$-algebras by dimension. This is used in the following functions.
\subsection{\textcolor{Chapter }{CanoFormsByDimAndGens}}
\logpage{[ 2, 2, 1 ]}\nobreak
\hyperdef{L}{X86D653DD829028ED}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{CanoFormsByDimAndGens({\mdseries\slshape F, d, n})\index{CanoFormsByDimAndGens@\texttt{CanoFormsByDimAndGens}}
\label{CanoFormsByDimAndGens}
}\hfill{\scriptsize (operation)}}\\
Given a finite field \mbox{\texttt{\mdseries\slshape F}}, a generator number \mbox{\texttt{\mdseries\slshape d}} and a dimension \mbox{\texttt{\mdseries\slshape n}}, this function calculates canonical forms for the \mbox{\texttt{\mdseries\slshape d}}-generator nilpotent associative \mbox{\texttt{\mdseries\slshape F}}-algebras of dimension \mbox{\texttt{\mdseries\slshape n}}. }
\subsection{\textcolor{Chapter }{CanoFormsByDim}}
\logpage{[ 2, 2, 2 ]}\nobreak
\hyperdef{L}{X85BDCDA47C7322A4}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{CanoFormsByDim({\mdseries\slshape F, n})\index{CanoFormsByDim@\texttt{CanoFormsByDim}}
\label{CanoFormsByDim}
}\hfill{\scriptsize (operation)}}\\
Given a finite field \mbox{\texttt{\mdseries\slshape F}} and a dimension \mbox{\texttt{\mdseries\slshape n}}, this function calculates canonical forms for the nilpotent associative \mbox{\texttt{\mdseries\slshape F}}-algebras of dimension \mbox{\texttt{\mdseries\slshape n}}. }
}
\section{\textcolor{Chapter }{Enumerating nilpotent associative algebras}}\label{Enumerating nilpotent associative algebras}
\logpage{[ 2, 3, 0 ]}
\hyperdef{L}{X8177D216867C5E1B}{}
{
Sometimes it will be enough to enumerate the number of immediate descendants
or the number of nilpotent associative algebras of a given dimension. This is
implemented in the following functions using orbit counting methods. However,
note that the enumeration of $n$-dimensional algebras still requires the computation of the $(n-1)$-dimensional algebras.
\subsection{\textcolor{Chapter }{NumberOfImmediateDescendantsOfCanoForm (for a single stepsize)}}
\logpage{[ 2, 3, 1 ]}\nobreak
\hyperdef{L}{X83F973F5788BAFA0}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{NumberOfImmediateDescendantsOfCanoForm({\mdseries\slshape C, s})\index{NumberOfImmediateDescendantsOfCanoForm@\texttt{Number}\-\texttt{Of}\-\texttt{Immediate}\-\texttt{Descendants}\-\texttt{Of}\-\texttt{Cano}\-\texttt{Form}!for a single stepsize}
\label{NumberOfImmediateDescendantsOfCanoForm:for a single stepsize}
}\hfill{\scriptsize (operation)}}\\
Given a canonical form \mbox{\texttt{\mdseries\slshape C}} this returns the number of immediate descendants of \mbox{\texttt{\mdseries\slshape C}} with stepsize \mbox{\texttt{\mdseries\slshape s}}. }
\subsection{\textcolor{Chapter }{NumberOfImmediateDescendantsOfCanoForm (for a range of stepsizes)}}
\logpage{[ 2, 3, 2 ]}\nobreak
\hyperdef{L}{X7CD6DF9078149628}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{NumberOfImmediateDescendantsOfCanoForm({\mdseries\slshape C, slist})\index{NumberOfImmediateDescendantsOfCanoForm@\texttt{Number}\-\texttt{Of}\-\texttt{Immediate}\-\texttt{Descendants}\-\texttt{Of}\-\texttt{Cano}\-\texttt{Form}!for a range of stepsizes}
\label{NumberOfImmediateDescendantsOfCanoForm:for a range of stepsizes}
}\hfill{\scriptsize (operation)}}\\
Given a canonical form \mbox{\texttt{\mdseries\slshape C}} this returns the number of immediate descendants of \mbox{\texttt{\mdseries\slshape C}} with stepsizes in \mbox{\texttt{\mdseries\slshape slist}}. }
\subsection{\textcolor{Chapter }{NumberOfImmediateDescendantsOfCanoForm (for all stepsizes)}}
\logpage{[ 2, 3, 3 ]}\nobreak
\hyperdef{L}{X833ABFBB7CC5F303}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{NumberOfImmediateDescendantsOfCanoForm({\mdseries\slshape C})\index{NumberOfImmediateDescendantsOfCanoForm@\texttt{Number}\-\texttt{Of}\-\texttt{Immediate}\-\texttt{Descendants}\-\texttt{Of}\-\texttt{Cano}\-\texttt{Form}!for all stepsizes}
\label{NumberOfImmediateDescendantsOfCanoForm:for all stepsizes}
}\hfill{\scriptsize (operation)}}\\
Given a canonical form \mbox{\texttt{\mdseries\slshape C}} this returns the number of immediate descendants of \mbox{\texttt{\mdseries\slshape C}} of all stepsizes. }
\subsection{\textcolor{Chapter }{NumberOfNAAlgsByDimAndGens}}
\logpage{[ 2, 3, 4 ]}\nobreak
\hyperdef{L}{X783487EA7CF561C0}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{NumberOfNAAlgsByDimAndGens({\mdseries\slshape F, d, n})\index{NumberOfNAAlgsByDimAndGens@\texttt{NumberOfNAAlgsByDimAndGens}}
\label{NumberOfNAAlgsByDimAndGens}
}\hfill{\scriptsize (operation)}}\\
Given a finite field \mbox{\texttt{\mdseries\slshape F}}, a generator number \mbox{\texttt{\mdseries\slshape d}} and a dimension \mbox{\texttt{\mdseries\slshape n}}, this function returns the number of \mbox{\texttt{\mdseries\slshape d}}-generator nilpotent associative \mbox{\texttt{\mdseries\slshape F}}-algebras of dimension \mbox{\texttt{\mdseries\slshape n}}. }
\subsection{\textcolor{Chapter }{NumberOfNAAlgsByDim}}
\logpage{[ 2, 3, 5 ]}\nobreak
\hyperdef{L}{X857964EA7CD4F287}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{NumberOfNAAlgsByDim({\mdseries\slshape F, n})\index{NumberOfNAAlgsByDim@\texttt{NumberOfNAAlgsByDim}}
\label{NumberOfNAAlgsByDim}
}\hfill{\scriptsize (operation)}}\\
Given a finite field \mbox{\texttt{\mdseries\slshape F}} and a dimension \mbox{\texttt{\mdseries\slshape n}}, this function returns the number of nilpotent associative \mbox{\texttt{\mdseries\slshape F}}-algebras of dimension \mbox{\texttt{\mdseries\slshape n}}. }
}
}
\chapter{\textcolor{Chapter }{Coclass theory and coclass graphs}}\label{Coclass theory and coclass graphs}
\logpage{[ 3, 0, 0 ]}
\hyperdef{L}{X8166B0DD7D2C5881}{}
{
\index{Coclass theory and coclass graphs} 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 \textsf{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.
\section{\textcolor{Chapter }{Calculating coclass and determine coclass settledness}}\label{Calculating coclass and determine coclass settledness}
\logpage{[ 3, 1, 0 ]}
\hyperdef{L}{X78A7D1CB82A1C9C3}{}
{
\subsection{\textcolor{Chapter }{CoclassOfCanoForm}}
\logpage{[ 3, 1, 1 ]}\nobreak
\hyperdef{L}{X87BC544A81F5CA13}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{CoclassOfCanoForm({\mdseries\slshape C})\index{CoclassOfCanoForm@\texttt{CoclassOfCanoForm}}
\label{CoclassOfCanoForm}
}\hfill{\scriptsize (attribute)}}\\
This returns the coclass of the given canonical form \mbox{\texttt{\mdseries\slshape C}}. }
\subsection{\textcolor{Chapter }{IsCoclassSettledCanoForm}}
\logpage{[ 3, 1, 2 ]}\nobreak
\hyperdef{L}{X8242D8F37EDEACA2}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{IsCoclassSettledCanoForm({\mdseries\slshape C})\index{IsCoclassSettledCanoForm@\texttt{IsCoclassSettledCanoForm}}
\label{IsCoclassSettledCanoForm}
}\hfill{\scriptsize (attribute)}}\\
Determines whether the given canonical form \mbox{\texttt{\mdseries\slshape C}} is coclass settled. }
}
\section{\textcolor{Chapter }{Calculating coclass graphs}}\label{Calculating coclass graphs}
\logpage{[ 3, 2, 0 ]}
\hyperdef{L}{X7901961F79F38397}{}
{
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 \cite{EMo15}, the second step is an iterated computation of stepsize $1$ descendants. The following functions implement these two steps.
\subsection{\textcolor{Chapter }{RootsOfCoclassGraph}}
\logpage{[ 3, 2, 1 ]}\nobreak
\hyperdef{L}{X79FB79337D5D69E3}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{RootsOfCoclassGraph({\mdseries\slshape F, r})\index{RootsOfCoclassGraph@\texttt{RootsOfCoclassGraph}}
\label{RootsOfCoclassGraph}
}\hfill{\scriptsize (operation)}}\\
Given a finite field \mbox{\texttt{\mdseries\slshape F}}, a coclass \mbox{\texttt{\mdseries\slshape r}} this function calculates the roots of the maximal descendant trees of the
associated coclass graph. }
\subsection{\textcolor{Chapter }{RootsOfCoclassGraphByRank}}
\logpage{[ 3, 2, 2 ]}\nobreak
\hyperdef{L}{X7FCF5C89781B51A8}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{RootsOfCoclassGraphByRank({\mdseries\slshape F, r, d})\index{RootsOfCoclassGraphByRank@\texttt{RootsOfCoclassGraphByRank}}
\label{RootsOfCoclassGraphByRank}
}\hfill{\scriptsize (operation)}}\\
Given a finite field \mbox{\texttt{\mdseries\slshape F}}, a coclass \mbox{\texttt{\mdseries\slshape r}} and a generator number \mbox{\texttt{\mdseries\slshape d}} this function calculates the roots with the given number of generators of the
maximal descendant trees of the associated coclass graph. }
\subsection{\textcolor{Chapter }{DescendantTreeOfCanoForm}}
\logpage{[ 3, 2, 3 ]}\nobreak
\hyperdef{L}{X841575E786F638A0}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{DescendantTreeOfCanoForm({\mdseries\slshape C, l})\index{DescendantTreeOfCanoForm@\texttt{DescendantTreeOfCanoForm}}
\label{DescendantTreeOfCanoForm}
}\hfill{\scriptsize (operation)}}\\
Given a canonical form \mbox{\texttt{\mdseries\slshape C}} for a nilpotent associative $F$-algebra, this iteratedly calculates \mbox{\texttt{\mdseries\slshape l}} levels of descendants of the same coclass as \mbox{\texttt{\mdseries\slshape C}}. }
}
\section{\textcolor{Chapter }{Plotting coclass graphs}}\label{Plotting coclass graphs}
\logpage{[ 3, 3, 0 ]}
\hyperdef{L}{X7EF2AC0886289E57}{}
{
This section contains functions for plotting the constructed coclass graphs
and trees. Note that plotting requires the XGAP package, see \cite{XGAP}. 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.
\subsection{\textcolor{Chapter }{PlotDescendantTreeOfCanoForm}}
\logpage{[ 3, 3, 1 ]}\nobreak
\hyperdef{L}{X856261CA7EBEDFC3}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{PlotDescendantTreeOfCanoForm({\mdseries\slshape C, d, P})\index{PlotDescendantTreeOfCanoForm@\texttt{PlotDescendantTreeOfCanoForm}}
\label{PlotDescendantTreeOfCanoForm}
}\hfill{\scriptsize (operation)}}\\
Given a canonical form \mbox{\texttt{\mdseries\slshape C}}, this plots the descendant tree of canonical forms of the same coclass as \mbox{\texttt{\mdseries\slshape C}} up to dimension \mbox{\texttt{\mdseries\slshape d}} into the GraphicPoset \mbox{\texttt{\mdseries\slshape P}}. }
\subsection{\textcolor{Chapter }{PlotCoclassGraph}}
\logpage{[ 3, 3, 2 ]}\nobreak
\hyperdef{L}{X781E94087A89E0C8}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{PlotCoclassGraph({\mdseries\slshape F, r, d, P})\index{PlotCoclassGraph@\texttt{PlotCoclassGraph}}
\label{PlotCoclassGraph}
}\hfill{\scriptsize (operation)}}\\
Given a finite field \mbox{\texttt{\mdseries\slshape F}} and a coclass \mbox{\texttt{\mdseries\slshape r}}, this plots the associated coclass graph up to dimension \mbox{\texttt{\mdseries\slshape d}} into the GraphicPoset \mbox{\texttt{\mdseries\slshape P}}. }
\subsection{\textcolor{Chapter }{GetSelectedCanoForms}}
\logpage{[ 3, 3, 3 ]}\nobreak
\hyperdef{L}{X805C948B7EA9D06E}{}
{\noindent\textcolor{FuncColor}{$\triangleright$\enspace\texttt{GetSelectedCanoForms({\mdseries\slshape P})\index{GetSelectedCanoForms@\texttt{GetSelectedCanoForms}}
\label{GetSelectedCanoForms}
}\hfill{\scriptsize (operation)}}\\
Returns a list of canonical forms corresponding to the selected vertices in
the GraphicPoset \mbox{\texttt{\mdseries\slshape P}}. }
}
}
\def\bibname{References\logpage{[ "Bib", 0, 0 ]}
\hyperdef{L}{X7A6F98FD85F02BFE}{}
}
\bibliographystyle{alpha}
\bibliography{manual}
\addcontentsline{toc}{chapter}{References}
\def\indexname{Index\logpage{[ "Ind", 0, 0 ]}
\hyperdef{L}{X83A0356F839C696F}{}
}
\cleardoublepage
\phantomsection
\addcontentsline{toc}{chapter}{Index}
\printindex
\newpage
\immediate\write\pagenrlog{["End"], \arabic{page}];}
\immediate\closeout\pagenrlog
\end{document}
ccalgs/doc/chap3.txt 0000644 0216424 0000144 00000011021 14171241126 014333 0 ustar tobmoede users
[1X3 [33X[0;0YCoclass theory and coclass graphs[133X[101X
[33X[0;0YRecall that for a nilpotent associative algebra [22XA[122X its coclass is defined as
[22Xcc(A) = dim(A) - cl(A)[122X. Fixing a finite field [22XF[122X and a non-negative integer [22Xr[122X
one can define the associated coclass graph. Its vertices correspond
one-to-one to the isomorphism types of nilpotent associative [22XF[122X-algebras of
coclass [22Xr[122X and there is a directed edge [22XA[122X to [22XB[122X if [22XA[122X is isomorphic to
[22XB/B^cl(B)[122X. Such an algebra is called coclass settled, if all its descendants
are of the same coclass as the algebra itself. The [5Xccalgs[105X 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.[133X
[1X3.1 [33X[0;0YCalculating coclass and determine coclass settledness[133X[101X
[1X3.1-1 CoclassOfCanoForm[101X
[29X[2XCoclassOfCanoForm[102X( [3XC[103X ) [32X attribute
[33X[0;0YThis returns the coclass of the given canonical form [3XC[103X.[133X
[1X3.1-2 IsCoclassSettledCanoForm[101X
[29X[2XIsCoclassSettledCanoForm[102X( [3XC[103X ) [32X attribute
[33X[0;0YDetermines whether the given canonical form [3XC[103X is coclass settled.[133X
[1X3.2 [33X[0;0YCalculating coclass graphs[133X[101X
[33X[0;0YWe 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
[22X1[122X descendants. The following functions implement these two steps.[133X
[1X3.2-1 RootsOfCoclassGraph[101X
[29X[2XRootsOfCoclassGraph[102X( [3XF[103X, [3Xr[103X ) [32X operation
[33X[0;0YGiven a finite field [3XF[103X, a coclass [3Xr[103X this function calculates the roots of
the maximal descendant trees of the associated coclass graph.[133X
[1X3.2-2 RootsOfCoclassGraphByRank[101X
[29X[2XRootsOfCoclassGraphByRank[102X( [3XF[103X, [3Xr[103X, [3Xd[103X ) [32X operation
[33X[0;0YGiven a finite field [3XF[103X, a coclass [3Xr[103X and a generator number [3Xd[103X this function
calculates the roots with the given number of generators of the maximal
descendant trees of the associated coclass graph.[133X
[1X3.2-3 DescendantTreeOfCanoForm[101X
[29X[2XDescendantTreeOfCanoForm[102X( [3XC[103X, [3Xl[103X ) [32X operation
[33X[0;0YGiven a canonical form [3XC[103X for a nilpotent associative [22XF[122X-algebra, this
iteratedly calculates [3Xl[103X levels of descendants of the same coclass as [3XC[103X.[133X
[1X3.3 [33X[0;0YPlotting coclass graphs[133X[101X
[33X[0;0YThis section contains functions for plotting the constructed coclass graphs
and trees. Note that plotting requires the XGAP package, see [CN12]. The
parameter [22XP[122X 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.[133X
[1X3.3-1 PlotDescendantTreeOfCanoForm[101X
[29X[2XPlotDescendantTreeOfCanoForm[102X( [3XC[103X, [3Xd[103X, [3XP[103X ) [32X operation
[33X[0;0YGiven a canonical form [3XC[103X, this plots the descendant tree of canonical forms
of the same coclass as [3XC[103X up to dimension [3Xd[103X into the GraphicPoset [3XP[103X.[133X
[1X3.3-2 PlotCoclassGraph[101X
[29X[2XPlotCoclassGraph[102X( [3XF[103X, [3Xr[103X, [3Xd[103X, [3XP[103X ) [32X operation
[33X[0;0YGiven a finite field [3XF[103X and a coclass [3Xr[103X, this plots the associated coclass
graph up to dimension [3Xd[103X into the GraphicPoset [3XP[103X.[133X
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