Quelle example17.xml Sprache: XML

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 = "example17.g"
gap> ### authors Knopper

gap> ### THIS IS A GAP PACKAGE GBNP
gap> ### FOR COMPUTING WITH NON-COMMUTATIVE POLYNOMIALS

<#GAPDoc Label="Example17">
<Section Label="Example17"><Heading>An algebra over a finite field</Heading>
A small example over a field other than the rationals, using the conversion
functions from <Ref Sect="TransitionFunctions"/>.
The input relations define the symmetric group of degree 3,
denoted <M>S_3</M>.
<P/>
First load the package and set the standard infolevel <Ref
InfoClass="InfoGBNP" Style="Text"/> to 2 and the time infolevel <Ref
Func="InfoGBNPTime" Style="Text"/> to 1 (for more information about the info
level, see Chapter <Ref Chap="Info"/>).

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

Let <C>F</C> be the field GF(2). The relations can be
entered as elements of a free associative algebra with one <C>A</C> (see
<Ref BookName="Reference" Label="FreeAssociativeAlgebraWithOne"/>).

<Listing><![CDATA[
gap> F:=GF(2);;
gap> A:=FreeAssociativeAlgebraWithOne(F,"a","b");
<algebra-with-one over GF(2), with 2 generators>
gap> g:=GeneratorsOfAlgebraWithOne(A);
[ (Z(2)^0)*a, (Z(2)^0)*b ]
]]></Listing>

Enter the relations <M>\{a^2-1,b^2-1,(ab)^3-1\}</M>, convert them to NP-form,
see Section <Ref Sect="NP"/>, with <Ref Func="GP2NPList" Style="Text"/> and
print them with <Ref Func="PrintNPList" Style="Text"/>:

<Listing><![CDATA[
gap> KI_GP := [ g[1]^2-g[1]^0, g[2]^2-g[1]^0, (g[1]*g[2])^3-g[1]^0];
[ (Z(2)^0)*<identity ...>+(Z(2)^0)*a^2, (Z(2)^0)*<identity ...>+(Z(2)^0)*b^2,
  (Z(2)^0)*<identity ...>+(Z(2)^0)*(a*b)^3 ]
gap> KI:=GP2NPList(KI_GP);;
gap> PrintNPList(KI);
a^2 + Z(2)^0
b^2 + Z(2)^0
ababab + Z(2)^0
]]></Listing>

Now calculate the Gröbner basis with <Ref Func="SGrobner" Style="Text"/> and
print it with <Ref Func="PrintNPList" Style="Text"/>:

<Listing><![CDATA[
gap> GB:=SGrobner(KI);;
#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  length of G =3
#I  length of todo is 2
#I  length of G =3
#I  length of todo is 1
#I  length of G =3
#I  length of todo is 0
#I  List of todo lengths is [ 2, 2, 1, 0 ]
#I  End of phase III
#I  G: Cleaning finished, 0 polynomials reduced
#I  End of phase IV
#I  The computation took 0 msecs.
gap> PrintNPList(GB);
a^2 + Z(2)^0
b^2 + Z(2)^0
bab + aba
]]></Listing>

Now calculate the dimension of the quotient algebra with <Ref Func="DimQA"
Style="Text"/> (2 symbols) and a base with <Ref Func="BaseQA" Style="Text"/>
(2 symbols, 0 for whole base) and print the base.
This will give a list of elements of the group.

<Listing><![CDATA[
gap> DimQA(GB,2);
6
gap> B:=BaseQA(GB,2,0);;
gap> PrintNPList(B);
Z(2)^0
a
b
ab
ba
aba
]]></Listing>

We can print the Gröbner basis and the basis of the quotient algebra,
converted back to GAP polynomials with <Ref Func="NP2GPList" Style="Text"/>.
The functions used to convert the polynomials also require the algebra as an
argument. The result is useful for further computations in <M>A</M>.

<Listing><![CDATA[
gap> NP2GPList(GB,A);
[ (Z(2)^0)*a^2+(Z(2)^0)*<identity ...>, (Z(2)^0)*b^2+(Z(2)^0)*<identity ...>,
  (Z(2)^0)*b*a*b+(Z(2)^0)*a*b*a ]
gap> NP2GPList(B,A);
[ (Z(2)^0)*<identity ...>, (Z(2)^0)*a, (Z(2)^0)*b, (Z(2)^0)*a*b,
  (Z(2)^0)*b*a, (Z(2)^0)*a*b*a ]
]]></Listing>

The matrix of right multiplication with the image of the
first variable
can be computed by <Ref Func="MatrixQA" Style="Text"/>.
<Listing><![CDATA[
gap> Display(MatrixQA(1,B,GB));
. 1 . . . .
1 . . . . .
. . . . 1 .
. . . . . 1
. . 1 . . .
. . . 1 . .
]]></Listing>

</Section>
<#/GAPDoc>

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