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: http://www.gnu.org/licenses/lgpl.html gap> ########################## END COPYRIGHT MESSAGE ##########################

gap> ### filename = "example18.g" gap> ### author Knopper gap> ### $Id: example18.g,v 1.7 2004/04/16 14:37:38 jwk Exp $

gap> ### THIS IS A GAP PACKAGE FOR COMPUTING NON-COMMUTATIVE GROBNER BASES

<#GAPDoc Label="Example18">
The dihedral group of order 8
In this example (Example 1 from Linton ) the two-sided relations give the group algebra of the group with presentation \langle a,b \mid a^4=b^2=(ab)^2=1\rangle, the dihedral group of order 8. It is possible to construct a permutation module of degree 4, over a field k. In this example k will be the field of rational numbers.

First load the package and set the standard infolevel to 1 and the time infolevel to 1 (for more information about the info level, see Chapter ).

gap> LoadPackage("GBNP","0",false);; gap> SetInfoLevel(InfoGBNP,1); gap> SetInfoLevel(InfoGBNPTime,1);

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 ). For convenience we use the variables a and b for the generators of the algebra and e for the one of the algebra.

gap> A:=FreeAssociativeAlgebraWithOne(Rationals, "a", "b"); <algebra-with-one over Rationals, with 2 generators> gap> a:=A.a;;b:=A.b;;e:=One(A);;

Now the relations are entered:

gap> twosidrels:=[a^4-e,b^2-e,(a*b)^2-e];; gap> prefixrels:=[b-e];;

First the relations are converted into NP format, see Section , after which the function is called to calculate a Gröbner basis record.

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.

The record GBR has two members: the two-sided relations GBR.ts and the prefix relations GBR.p. It is possible to print these using the function :

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 ]

It is now possible to calculate the standard basis of the quotient module with the function . This function has as arguments the Gröbner basis record GBR, the number of generators of the algebra (2), the number of generators of the module (1), and a variable maxno for returning partial bases (0 means full basis).

gap> B:=BaseQM(GBR,2,1,0);; gap> PrintNPList(B); [ 1 ] [ a ] [ a^2 ] [ ab ]

It is also possible to use a module with one generator to enter these relations:

gap> D:=A^1;; gap> gd:=GeneratorsOfLeftModule(D);; gap> prefixrelsdom:=[gd[1]*(b-e)];;

It is possible to use the two-sided Gröbner basis which was already calculated.

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 ]

To compute the image of right multiplication of the basis element B[Length(B)] of the module with the quotient algebra element corresponding to ab we use the function with arguments B[Length(B)], GB2NP(a*b), and GBR We subsequently use to display the result as a 1-dimensional vector with an entry from A.

gap> v := MulQM(B[Length(B)],GP2NP(a*b),GBR); [ [ [ -1 ] ], [ 1 ] ] gap> PrintNP(v); [ 1 ]
<#/GAPDoc>