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> ### THIS IS A GAP PACKAGE FOR COMPUTING NON-COMMUTATIVE GROBNER BASES
<#GAPDoc Label="Example18">
<Section Label="Example18"><Heading>The dihedral group of order 8</Heading>
In this example (Example 1 from Linton <Cite
Key="MR94k:20022"/>) the two-sided relations give the group algebra
of the group with presentation
<M>\langle a,b \mid a^4=b^2=(ab)^2=1\rangle</M>, the dihedral group of order 8.
It is possible to construct a permutation module of degree 4, over a field
<C>k</C>. In this example <C>k</C> will be the field of rational numbers.
<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 1 (for more information about the info
level, see Chapter <Ref Chap="Info"/>).
Now enter the relations as GAP polynomials. It is possible to enter them with
and without module generators. First it is shown how to enter the relations
without using a module. It is possible to enter them with a free associative
algebra with one over the field (the rational numbers) (see also <Ref
BookName="Reference" Label="FreeAssociativeAlgebraWithOne"/>). For
convenience we use the variables <C>a</C> and <C>b</C> for the generators of
the algebra and <C>e</C> for the one of the algebra.
<Listing><![CDATA[
gap> A:=FreeAssociativeAlgebraWithOne(Rationals, "a", "b");
<algebra-with-one over Rationals, with 2 generators>
gap> a:=A.a;;b:=A.b;;e:=One(A);;
]]></Listing>
First the relations are converted into NP format, see Section <Ref
Sect="NP"/>, after which the function <Ref Func="SGrobnerModule"
Style="Text"/> is called to calculate a Gröbner basis record.
<Listing><![CDATA[
gap> GBR:=SGrobnerModule(GP2NPList(prefixrels),GP2NPList(twosidrels));;
#I number of entered polynomials is 3
#I number of polynomials after reduction is 3
#I End of phase I
#I End of phase II
#I End of phase III
#I End of phase IV
#I The computation took 0 msecs.
#I number of entered polynomials is 7
#I number of polynomials after reduction is 7
#I End of phase I
#I End of phase II
#I End of phase III
#I End of phase IV
#I The computation took 4 msecs.
]]></Listing>
The record GBR has two members: the two-sided relations <C>GBR.ts</C> and the
prefix relations <C>GBR.p</C>. It is possible to print these using the
function <Ref Func="PrintNPList" Style="Text"/>:
<Listing><![CDATA[
gap> PrintNPList(GBR.ts);
b^2 - 1
aba - b
ba^2 - a^2b
bab - a^3
a^4 - 1
a^3b - ba
gap> PrintNPList(GBR.p);
[ b - 1 ]
[ a^3 - ab ]
[ a^2b - a^2 ]
]]></Listing>
It is now possible to calculate the standard basis of the quotient module
with the function <Ref Func="BaseQM" Style="Text"/>. This function has as
arguments the Gröbner basis record <C>GBR</C>, the number of generators of
the algebra (2), the number of generators of the module (1),
and a variable <C>maxno</C> for returning partial bases
(0 means full basis).
<Listing><![CDATA[
gap> B:=BaseQM(GBR,2,1,0);;
gap> PrintNPList(B);
[ 1 ]
[ a ]
[ a^2 ]
[ ab ]
]]></Listing>
It is also possible to use a module with one generator to enter these
relations:
It is possible to use the two-sided Gröbner basis which was already
calculated.
<Listing><![CDATA[
gap> GBR:=SGrobnerModule(GP2NPList(prefixrelsdom),GBR.ts);;
#I number of entered polynomials is 6
#I number of polynomials after reduction is 6
#I End of phase I
#I End of phase II
#I End of phase III
#I End of phase IV
#I The computation took 4 msecs.
#I number of entered polynomials is 7
#I number of polynomials after reduction is 7
#I End of phase I
#I End of phase II
#I End of phase III
#I End of phase IV
#I The computation took 0 msecs.
gap> PrintNPList(GBR.p);;
[ b - 1 ]
[ a^3 - ab ]
[ a^2b - a^2 ]
gap> B:=BaseQM(GBR,2,1,0);;
gap> PrintNPList(B);
[ 1 ]
[ a ]
[ a^2 ]
[ ab ]
]]></Listing>
To compute the image of right multiplication of the basis element
<C>B[Length(B)]</C> of the module with the quotient algebra element
corresponding to
<M>ab</M> we use the function <Ref Func="MulQM" Style="Text"/> with
arguments <C>B[Length(B)]</C>, <C>GB2NP(a*b)</C>, and <C>GBR</C>
We subsequently use <Ref Func="PrintNP" Style="Text"/> to display the result
as a 1-dimensional vector with an entry from <M>A</M>.
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