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 = "example13.g" gap> ### authors Cohen & Gijsbers & Krook gap> ### This example was added by Chris Krook.

gap> ### THIS IS A GAP PACKAGE GBNP gap> ### FOR COMPUTING WITH NON-COMMUTATIVE POLYNOMIALS gap> ### ADD-ON: STUDY GROWTH OF FACTOR ALGEBRA gap> ### $Id: example13.g,v 1.8 2004/04/16 14:37:38 jwk Exp $

<#GAPDoc Label="Example13">
Finiteness of the Weyl group of type E_6

This example extends , which computes the order of the Weyl group of type E_6.

Here, before the dimension is calculated, it is checked whether the quotient algebra is finite dimensional or infinite dimensional. The function is used for this computation. For the use of to speed up the check, see Example .

First load the package and set the standard infolevel to 1 and the time infolevel to 2 (for more information about the info level, see Chapter ).

gap> LoadPackage("GBNP","0",false);; gap> SetInfoLevel(InfoGBNP,1); gap> SetInfoLevel(InfoGBNPTime,2);

Then input the relations in NP format (see Section ). They will be assigned to KI. These relations are the same as those in Example 3.

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 > ];;

The Gröbner basis can now be calculated with :

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 96 msecs.

We will check whether the quotient algebra is finite dimensional or infinite dimensional. The function exists for this purpose. Its first argument is the list of leading monomials of a Gröbner basis and its second argument the number of symbols. The leading monomials can be calculated with .

gap> L:=LMonsNP(GB);; gap> FinCheckQA(L,6); true gap> time; 60

If a quotient algebra is finite dimensional, the dimension can be calculated with , the arguments are the Gröbner basis GB and the number of symbols 6. Since is set to 2, we get timing information from :

gap> dim := DimQA(GB,6); #I The computation took 144 msecs. 51840
<#/GAPDoc>