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gap> ######################### BEGIN COPYRIGHT MESSAGE #########################
GBNP - computing Gröbner bases of noncommutative polynomials
Copyright 2001-2010 by Arjeh M. Cohen, Dié A.H. Gijsbers, Jan Willem
Knopper, Chris Krook. Address: Discrete Algebra and Geometry (DAM) group
at the Department of Mathematics and Computer Science of Eindhoven
University of Technology.

For acknowledgements see the manual. The manual can be found in several
formats in the doc subdirectory of the GBNP distribution. The
acknowledgements formatted as text can be found in the file chap0.txt.

GBNP is free software; you can redistribute it and/or modify it under
the terms of the Lesser GNU General Public License as published by the
Free Software Foundation (FSF); either version 2.1 of the License, or
(at your option) any later version. For details, see the file 'LGPL' in
the doc subdirectory of the GBNP distribution or see the FSF's own site:
https://www.gnu.org/licenses/lgpl.html
gap> ########################## END COPYRIGHT MESSAGE ##########################

gap> ### filename = "example03.g"
gap> ### authors Cohen & Gijsbers

gap> ### THIS IS A GAP PACKAGE FOR COMPUTING NON-COMMUTATIVE GROBNER BASES
gap>
gap> ### Last change: August 22 2001.
gap> ### amc

<#GAPDoc Label="Example03">
<Section Label="Example03"><Heading>The order of the Weyl group of type E<M>_6</M></Heading>
In order to show how the order of a finite group of manageable size with a manageable
presentation can be computed, we determine the order of the Weyl group of type E<M>_6</M>.
This number is well known to be 51840.
<P/>
First load the package and set the standard infolevel <Ref
InfoClass="InfoGBNP" Style="Text"/> to 1 and the time infolevel <Ref
Func="InfoGBNPTime" Style="Text"/> to 2 (for more information about the info
level, see Chapter <Ref Chap="Info"/>).

<Listing><![CDATA[
gap> LoadPackage("gbnp", false);
true
gap> SetInfoLevel(InfoGBNP,1);
gap> SetInfoLevel(InfoGBNPTime,2);
]]></Listing>

Then input the relations in NP format (see <Ref Sect="NP"/>).
They come from the presentation of the Weyl group as a Coxeter group.
This means that there are six variables, one for each generator.
We let the corresponding variables be printed as <M>r_1</M>, ..., <M>r_6</M> by means of
<Ref Func="GBNP.ConfigPrint" Style="Text"/>
<Listing><![CDATA[
gap> GBNP.ConfigPrint(6,"r");
]]></Listing>

The relations are binomial and represent the group relations,
which express that the generators are involutions (that is,
have order 2) and that the orders of the products of any two generators
is specified by the Coxeter diagram (see any book on Coxeter groups for details).
The relations will be
assigned to <C>KI</C>.

<Listing><![CDATA[
gap> k1 := [[[1,3,1],[3,1,3]],[1,-1]];;
gap> k2 := [[[4,3,4],[3,4,3]],[1,-1]];;
gap> k3 := [[[4,2,4],[2,4,2]],[1,-1]];;
gap> k4 := [[[4,5,4],[5,4,5]],[1,-1]];;
gap> k5 := [[[6,5,6],[5,6,5]],[1,-1]];;
gap> k6 := [[[1,2],[2,1]],[1,-1]];;
gap> k7 := [[[1,4],[4,1]],[1,-1]];;
gap> k8 := [[[1,5],[5,1]],[1,-1]];;
gap> k9 := [[[1,6],[6,1]],[1,-1]];;
gap> k10 := [[[2,3],[3,2]],[1,-1]];;
gap> k11 := [[[2,5],[5,2]],[1,-1]];;
gap> k12 := [[[2,6],[6,2]],[1,-1]];;
gap> k13 := [[[3,5],[5,3]],[1,-1]];;
gap> k14 := [[[3,6],[6,3]],[1,-1]];;
gap> k15 := [[[4,6],[6,4]],[1,-1]];;
gap> k16 := [[[1,1],[]],[1,-1]];;
gap> k17 := [[[2,2],[]],[1,-1]];;
gap> k18 := [[[3,3],[]],[1,-1]];;
gap> k19 := [[[4,4],[]],[1,-1]];;
gap> k20 := [[[5,5],[]],[1,-1]];;
gap> k21 := [[[6,6],[]],[1,-1]];;
gap> KI := [k1,k2,k3,k4,k5,k6,k7,k8,k9,k10,
>        k11,k12,k13,k14,k15,k16,k17,k18,k19,k20,k21
>       ];;
]]></Listing>

The relations can be shown with <Ref Func="PrintNPList" Style="Text"/>:

<Listing><![CDATA[
gap> PrintNPList(KI);
r.1r.3r.1 - r.3r.1r.3
r.4r.3r.4 - r.3r.4r.3
r.4r.2r.4 - r.2r.4r.2
r.4r.5r.4 - r.5r.4r.5
r.6r.5r.6 - r.5r.6r.5
r.1r.2 - r.2r.1
r.1r.4 - r.4r.1
r.1r.5 - r.5r.1
r.1r.6 - r.6r.1
r.2r.3 - r.3r.2
r.2r.5 - r.5r.2
r.2r.6 - r.6r.2
r.3r.5 - r.5r.3
r.3r.6 - r.6r.3
r.4r.6 - r.6r.4
r.1^2 - 1
r.2^2 - 1
r.3^2 - 1
r.4^2 - 1
r.5^2 - 1
r.6^2 - 1
]]></Listing>

The Gröbner basis can now be calculated with
<Ref Func="SGrobner" Style="Text"/>:

<Listing><![CDATA[
gap> GB := SGrobner(KI);;
#I  number of entered polynomials is 21
#I  number of polynomials after reduction is 21
#I  End of phase I
#I  End of phase II
#I  End of phase III
#I  Time needed to clean G :0
#I  End of phase IV
#I  The computation took 132 msecs.
gap> PrintNPList(GB);
r.1^2 - 1
r.2r.1 - r.1r.2
r.2^2 - 1
r.3r.2 - r.2r.3
r.3^2 - 1
r.4r.1 - r.1r.4
r.4^2 - 1
r.5r.1 - r.1r.5
r.5r.2 - r.2r.5
r.5r.3 - r.3r.5
r.5^2 - 1
r.6r.1 - r.1r.6
r.6r.2 - r.2r.6
r.6r.3 - r.3r.6
r.6r.4 - r.4r.6
r.6^2 - 1
r.3r.1r.2 - r.2r.3r.1
r.3r.1r.3 - r.1r.3r.1
r.4r.2r.4 - r.2r.4r.2
r.4r.3r.4 - r.3r.4r.3
r.5r.4r.5 - r.4r.5r.4
r.6r.5r.6 - r.5r.6r.5
r.4r.3r.1r.4 - r.3r.4r.3r.1
r.5r.4r.2r.5 - r.4r.5r.4r.2
r.5r.4r.3r.5 - r.4r.5r.4r.3
r.6r.5r.4r.6 - r.5r.6r.5r.4
r.4r.2r.3r.4r.2 - r.3r.4r.2r.3r.4
r.4r.2r.3r.4r.3 - r.2r.4r.2r.3r.4
r.5r.4r.2r.3r.5 - r.4r.5r.4r.2r.3
r.5r.4r.3r.1r.5 - r.4r.5r.4r.3r.1
r.6r.5r.4r.2r.6 - r.5r.6r.5r.4r.2
r.6r.5r.4r.3r.6 - r.5r.6r.5r.4r.3
r.4r.2r.3r.1r.4r.2 - r.3r.4r.2r.3r.1r.4
r.5r.4r.2r.3r.1r.5 - r.4r.5r.4r.2r.3r.1
r.6r.5r.4r.2r.3r.6 - r.5r.6r.5r.4r.2r.3
r.6r.5r.4r.3r.1r.6 - r.5r.6r.5r.4r.3r.1
r.4r.2r.3r.1r.4r.3r.1 - r.2r.4r.2r.3r.1r.4r.3
r.5r.4r.2r.3r.4r.5r.4 - r.4r.5r.4r.2r.3r.4r.5
r.6r.5r.4r.2r.3r.1r.6 - r.5r.6r.5r.4r.2r.3r.1
r.6r.5r.4r.2r.3r.4r.6 - r.5r.6r.5r.4r.2r.3r.4
r.5r.4r.2r.3r.1r.4r.5r.4 - r.4r.5r.4r.2r.3r.1r.4r.5
r.6r.5r.4r.2r.3r.1r.4r.6 - r.5r.6r.5r.4r.2r.3r.1r.4
r.6r.5r.4r.2r.3r.1r.4r.3r.6 - r.5r.6r.5r.4r.2r.3r.1r.4r.3
r.6r.5r.4r.2r.3r.4r.5r.6r.5 - r.5r.6r.5r.4r.2r.3r.4r.5r.6
r.5r.4r.2r.3r.1r.4r.3r.5r.4r.3 - r.4r.5r.4r.2r.3r.1r.4r.3r.5r.4
r.6r.5r.4r.2r.3r.1r.4r.5r.6r.5 - r.5r.6r.5r.4r.2r.3r.1r.4r.5r.6
r.5r.4r.2r.3r.1r.4r.3r.5r.4r.2r.3 - r.4r.5r.4r.2r.3r.1r.4r.3r.5r.4r.2
r.6r.5r.4r.2r.3r.1r.4r.3r.5r.6r.5 - r.5r.6r.5r.4r.2r.3r.1r.4r.3r.5r.6
r.6r.5r.4r.2r.3r.1r.4r.3r.5r.4r.6r.5r.4 - r.5r.6r.5r.4r.2r.3r.1r.4r.3r.5r.4r.\
6r.5
r.6r.5r.4r.2r.3r.1r.4r.3r.5r.4r.2r.6r.5r.4r.2 - r.5r.6r.5r.4r.2r.3r.1r.4r.3r.\
5r.4r.2r.6r.5r.4
]]></Listing>

The base of the quotient algebra can be calculated with <Ref Func="BaseQA"
Style="Text"/>, which has as arguments a Gröbner basis <C>GB</C>, a number
of symbols <C>6</C> and a maximum terms to be found (here 0 is entered, for
a full base) . Since it is very long we will not print it here.

<Listing><![CDATA[
gap> B:=BaseQA(GB,6,0);;
]]></Listing>

The dimension of the quotient algebra can be calculated with
<Ref Func="DimQA" Style="Text"/>, the arguments are the Gröbner basis
<C>GB</C> and the number of symbols <C>6</C>. Since <Ref
InfoClass="InfoGBNPTime" Style="Text"/> is set to 2, we get
timing information from <Ref Func="DimQA" Style="Text"/>:

<Listing><![CDATA[
gap> DimQA(GB,6);
#I  The computation took 172 msecs.
51840
]]></Listing>

Note that the calculation of the dimension takes almost as long as
calculating the base. Since we have already calculated a base <C>B</C> it is
much more efficient to calculate the dimension with <C>Length</C>:

<Listing><![CDATA[
gap> Length(B);
51840
]]></Listing>

</Section>
<#/GAPDoc>

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