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 = "exampleNoah.g" gap> ### authors Cohen & Wales

gap> ### THIS IS A GAP PACKAGE FOR COMPUTING NON-COMMUTATIVE GROBNER BASES gap> gap> ### Last change: Aug 12 2008, amc. gap> ### dahg gap> ## [A.M. Cohen, D.A.H. Gijsbers D.B. Wales, BMW Algebras of simply laced type, J. Algebra, 286 (2005) 107--153].

<#GAPDoc Label="Example07">
The Birman-Murakami-Wenzl algebra of type A_3
We study the Birman-Murakami-Wenzl algebra of type A_3 as an algebra given by generators and relations. A reference for the relations used is .

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);

The variables are g_1, g_2, g_3, e_1, e_2, e_3, in this order. In order to have the results printed out with these symbols, we invoke gap> GBNP.ConfigPrint("g1","g2","g3","e1","e2","e3");

Now enter the relations. This will be done in NP form (see ). The inderminates m and l in the coefficient ring of the Birman-Murakami-Wenzl algebra are specialized to 7 and 11 in order to make the computations more efficient. gap> m:= 7;; gap> l:= 11;; gap> #relations Theorem 1.1 gap> k1 := [[[4],[1,1],[1],[]],[1,-l/m,-l,l/m]];; gap> k2 := [[[5],[2,2],[2],[]],[1,-l/m,-l,l/m]];; gap> k3 := [[[6],[3,3],[3],[]],[1,-l/m,-l,l/m]];; gap> #relations B1 gap> #empty set here gap> #relations B2: gap> k4 := [[[1,2,1],[2,1,2]],[1,-1]];; gap> k5 := [[[2,3,2],[3,2,3]],[1,-1]];; gap> k6 := [[[1,3],[3,1]],[1,-1]];; gap> #relations R1 gap> kr1 := [[[1,4],[4]],[1,-1/l]];; gap> kr2 := [[[2,5],[5]],[1,-1/l]];; gap> kr3 := [[[3,6],[6]],[1,-1/l]];; gap> #relations R2: gap> kr4 := [[[4,2,4],[4]],[1,-l]];; gap> kr5 := [[[5,1,5],[5]],[1,-l]];; gap> kr6 := [[[5,3,5],[5]],[1,-l]];; gap> kr7 := [[[6,2,6],[6]],[1,-l]];; gap> #relations R2' gap> km1 := [[[4,5,4],[4]],[1,-1]];; gap> km2 := [[[5,4,5],[5]],[1,-1]];; gap> km3 := [[[5,6,5],[5]],[1,-1]];; gap> km4 := [[[6,5,6],[6]],[1,-1]];; gap> KI := [k1,k2,k3,k4,k5,k6,kr1,kr2,kr3,kr4,kr5,kr6,kr7,km1,km2,km3,km4];;

Now print the relations with :

gap> PrintNPList(KI); e1 - 11/7g1^2 - 11g1 + 11/7 e2 - 11/7g2^2 - 11g2 + 11/7 e3 - 11/7g3^2 - 11g3 + 11/7 g1g2g1 - g2g1g2 g2g3g2 - g3g2g3 g1g3 - g3g1 g1e1 - 1/11e1 g2e2 - 1/11e2 g3e3 - 1/11e3 e1g2e1 - 11e1 e2g1e2 - 11e2 e2g3e2 - 11e2 e3g2e3 - 11e3 e1e2e1 - e1 e2e1e2 - e2 e2e3e2 - e2 e3e2e3 - e3 gap> Length(KI); 17

Now calculate the Gröbner basis with :

gap> GB := SGrobner(KI);; #I number of entered polynomials is 17 #I number of polynomials after reduction is 17 #I End of phase I #I End of phase II #I End of phase III #I End of phase IV #I The computation took 76 msecs. gap> PrintNPList(GB); g1^2 - 7/11e1 + 7g1 - 1 g1e1 - 1/11e1 g2^2 - 7/11e2 + 7g2 - 1 g2e2 - 1/11e2 g3g1 - g1g3 g3^2 - 7/11e3 + 7g3 - 1 g3e3 - 1/11e3 e1g1 - 1/11e1 e1g3 - g3e1 e1^2 + 43/77e1 e2g2 - 1/11e2 e2^2 + 43/77e2 e3g1 - g1e3 e3g3 - 1/11e3 e3e1 - e1e3 e3^2 + 43/77e3 g1g2e1 - e2e1 g1g3e1 - 1/11g3e1 g1e2e1 + 7e2e1 - g2e1 - 7e1 g2g1g2 - g1g2g1 g2g1e2 - e1e2 g2g3e2 - e3e2 g2e1g2 - g1e2g1 - 7e2g1 + 7e1g2 + 7g2e1 - 7g1e2 - 49e2 + 49e1 g2e1e2 + 7e1e2 - g1e2 - 7e2 g2e3e2 + 7e3e2 - g3e2 - 7e2 g3g2g3 - g2g3g2 g3g2e3 - e2e3 g3e1e3 - 1/11e1e3 g3e2g3 - g2e3g2 - 7e3g2 + 7e2g3 + 7g3e2 - 7g2e3 - 49e3 + 49e2 g3e2e3 + 7e2e3 - g2e3 - 7e3 e1g2g1 - e1e2 e1g2e1 - 11e1 e1e2g1 + 7e1e2 - e1g2 - 7e1 e1e2e1 - e1 e2g1g2 - e2e1 e2g1e2 - 11e2 e2g3g2 - e2e3 e2g3e2 - 11e2 e2e1g2 + 7e2e1 - e2g1 - 7e2 e2e1e2 - e2 e2e3g2 + 7e2e3 - e2g3 - 7e2 e2e3e2 - e2 e3g2g3 - e3e2 e3g2e3 - 11e3 e3e2g3 + 7e3e2 - e3g2 - 7e3 e3e2e3 - e3 g1g2g3e1 - e2g3e1 g1g3g2e1 - g3e2e1 g1g3e2e1 + 7g3e2e1 - g3g2e1 - 7g3e1 g1e2g3e1 + 7e2g3e1 - g2g3e1 - 7g3e1 g1e3g2e1 - e3e2e1 g1e3e2e1 + 7e3e2e1 - e3g2e1 - 7e1e3 g3g2g1g3 - g2g3g2g1 g3g2g1e3 - e2g1e3 g3g2e1e3 - e2e1e3 g3e1g2e3 - e1e2e3 g3e1e2e3 + 7e1e2e3 - e1g2e3 - 7e1e3 g3e2g1g3 - g2e3g2g1 - 7e3g2g1 + 7e2g1g3 + 7g3e2g1 - 7g2g1e3 + 49e2g1 - 49g1e3\

g3e2g1e3 + 7e2g1e3 - g2g1e3 - 7g1e3 g3e2e1e3 + 7e2e1e3 - g2e1e3 - 7e1e3 e1g2g3g2 - g3e1g2g3 e1g2g3e1 - 11g3e1 e1g2e3g2 - g3e1e2g3 + 7e1e3g2 - 7e1e2g3 + 7e1g2e3 - 7g3e1e2 + 49e1e3 - 49e1e2\

e1e2g3e1 - g3e1 e1e3g2g1 - e1e3e2 e1e3g2e1 - 11e1e3 e1e3e2g1 + 7e1e3e2 - e1e3g2 - 7e1e3 e1e3e2e1 - e1e3 e2g3e1e2 - e2g1e3e2 e2e1e3e2 + 7e2g1e3e2 - e2g1g3e2 - 77e2 e3g2g1g3 - e3e2g1 e3g2g1e3 - 11g1e3 e3g2e1e3 - 11e1e3 e3e2g1g3 + 7e3e2g1 - e3g2g1 - 7g1e3 e3e2g1e3 - g1e3 e3e2e1e3 - e1e3 g1g2g1g3e2 - g2g1e3e2 g1g2g1e3e2 + 7g2g1e3e2 - g2g1g3e2 - 7e1e2 g1g2g3g2e1 - g2e3g2e1 - 7e3g2e1 + 7e2g3e1 + 7g3e2e1 - 7g2e1e3 + 49e2e1 - 49e1\ e3 g1g2e3g2e1 + 7g2e3g2e1 - g2g3g2e1 + 7e3e2e1 + 49e3g2e1 + 7e2e1e3 - 7g3g2e1 + \ 49g2e1e3 - 7g2g3e1 + 343e1e3 - 49g3e1 - 49g2e1 - 350e1 g1e2g1g3e2 + 7e2g1g3e2 - g2e1e3e2 - 7g2g3e1e2 - 7e1e3e2 - 49g3e1e2 + 77g1e2 +\ 539e2 g1e2g1e3e2 + 7e2g1e3e2 - g2g3e1e2 - 7g3e1e2 g2g3e1g2g3 - g1e2g1g3g2 - 7e2g1g3g2 + 7g3e1g2g3 + 7g2g3e1g2 + 49g3e1g2 - 7g1e\ 2e3 - 49e2e3 g2g3e1e2g3 - g1e2g1e3g2 - 7e2g1e3g2 + 7g3e1e2g3 + 7g2g3e1e2 - 7g1e2g1e3 - 49e\ 2g1e3 + 49g3e1e2 e2g1g3g2g1 - e2g1e3g2 e2g1g3e2g1 - e2e1e3g2 - 7e2g3e1g2 + 7e2g1g3e2 - 7e2e1e3 - 49e2g3e1 + 77e2g1 +\ 539e2 e2g1e3g2g1 + 7e2g1e3g2 - e2g1g3g2 - 7e2e1 e2g1e3e2g1 - e2g3e1g2 + 7e2g1e3e2 - 7e2g3e1 e2g3e1g2g3 + 7e2g3e1g2 - e2g1g3g2 - 7e2e3

Now calculate the dimension of the quotient algebra with (the second argument is the number of symbols):

gap> DimQA(GB,6); 105

The conclusion is that the BMW algebra of type A3 has dimension 105.
<#/GAPDoc>