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 ##########################
<#GAPDoc Label="Example04">
<Section Label="Example04">
<Heading>The icosahedral group</Heading>
In this example
the two-sided relations give the group algebra of
the group with presentation
<M>\langle a,b,c \mid a^2=b^2=c^2=(ab)^3=(bc)^5=(ac)^2=1\rangle</M>,
the icosahedral group of order 120.
This is the Coxeter group of type H<M>_3</M>.
The module under construction is a 3-dimensional reflection representation,
<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"/>).
Create the field containing the golden ratio <C>tau</C>.
<Listing><![CDATA[
gap> x := Indeterminate(Rationals,"x");
x
gap> p := x^2+ x-1;
x^2+x-1
gap> K := AlgebraicExtension(Rationals,p);
<algebraic extension over the Rationals of degree 2>
gap> tau:=RootOfDefiningPolynomial(K);
a
]]></Listing>
Create the free algebra with three generators over this field:
<Listing><![CDATA[
gap> A:=FreeAssociativeAlgebraWithOne(K, "a", "b", "c");
<algebra-with-one over <algebraic extension over the Rationals of degree
2>, with 3 generators>
gap> e:=One(A);; a:=A.a;; b:=A.b;; c:=A.c;;
]]></Listing>
The ideal for a quotient of the icosahedral group algebra over this field,
in which <C>b</C><M>*</M><C>c</C> has a quadratic minimal polynomial
involving <C>tau</C>:
We now give module relations. The first two describe group elements
of a vector stabilizer, the third forces the central element <M>(abc)^5</M>
to be nontrivial.
First the relations are converted into NP format (see <Ref Sect="NP"/>) with
the function <Ref Func="GP2NPList" Style="Text"/>.
Next the function
<Ref Func="SGrobnerModule" Style="Text"/> is called to calculate a Gröbner basis
record (see <Ref Sect="GBR"/>).
<Listing><![CDATA[
gap> GBR:=SGrobnerModule(GP2NPList(Mrels),GP2NPList(Irels));;
#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 16 msecs.
#I number of entered polynomials is 12
#I number of polynomials after reduction is 12
#I End of phase I
#I End of phase II
#I End of phase III
#I End of phase IV
#I The computation took 36 msecs.
gap> PrintNPList(GBR.ts);;
a^2 + !-1
b^2 + !-1
ca + !-1ac
c^2 + !-1
bab + !-1aba
cbc + !-1bcb + -a-1c + a+1b
bcba + !-1acba + !-1abcb + abac + cb + !-1bc + -a-2ba + a+2ab
cbac + !-1acba + !-1abcb + abac + cb + !-1bc + !-1ba + -a-1ac + a+2ab
bacba + abacb + !-1cba + !-1bcb + !-1abc + -a-2aba + c + a+2a
gap> PrintNPList(GBR.p);;
[ b + !-1 ]
[ c + !-1 ]
[ ac + !-1a ]
[ aba + !-1ab ]
[ abc + ab + -aa + -a ]
]]></Listing>
It is now possible to calculate the basis of the quotient algebra
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 (in this case 3), the number of generators of the
free module in which the vectors are chosen (in this case 1),
and a variable <C>maxno</C> for returning
partial quotient algebras (0 means full basis).
<Listing><![CDATA[
gap> B:=BaseQM(GBR,3,1,0);;
gap> PrintNPList(B);
[ !1 ]
[ a ]
[ ab ]
]]></Listing>
Calculate the dimension of the quotient algebra
with the function <Ref Func="DimQM" Style="Text"/>. This function has as
arguments the Gröbner basis record <C>GBR</C>,
the number of generators of
the algebra (in this case 3) and
the number of generators of
the module (in this case 1).
Finally we check the defining relations for the icosahedral group
on the three new matrix generators. This can be done by verifying if the
result is equal to the identity matrix or with the function <Ref
BookName="Reference" Func="IsOne" Style="Text"/>.
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