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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 = "example23.g"
gap> ### author Knopper

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

<#GAPDoc Label="Example23">
<Section Label="Example23"><Heading>Generalized Temperley-Lieb algebras</Heading>
This example shows how the dimension of a Generalized Temperley-Lieb
Algebra of type A, D, or E can be calculated. For <M>\textrm{A}_{n-1}</M>
this is the usual Temperley-Lieb Algebra on <M>n</M> strands with dimension
<M>\textrm{dim TL}(A_{n-1})={{2n \choose n}}/{(n+1)}</M>. For more
information see <Cite Key="Graham1995"/>.
<P/>
First load the package and set the standard infolevel <Ref
InfoClass="InfoGBNP" Style="Text"/> to 0 and the time infolevel <Ref
Func="InfoGBNPTime" Style="Text"/> to 1 (for more information about timing;
see Chapter <Ref Chap="Info"/>).

<Listing><![CDATA[
gap> LoadPackage("gbnp", false);
true
gap> SetInfoLevel(InfoGBNP,0);
gap> SetInfoLevel(InfoGBNPTime,1);
]]></Listing>

The relations are generated automatically from the Coxeter diagram.
This example
can be easily adapted by specifying the number of points and the set of edges
describing another Coxeter diagram. First enter the number of points,
<C>numpoints</C>.

<Listing><![CDATA[
gap> numpoints:=8;
8
]]></Listing>

Now define some edges describing the diagrams of <M>\textrm{E}_n</M>,
(these can be easily extended). In this example the dimension of the
Generalized Temperley-Lieb algebra of type <M>\textrm{E}_8</M> will be
calculated. For <M>\textrm{A}_{1\ldots 10}</M> the command
<P/>
<C>edges:=[[1,2],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]];</C>
<P/>
can be used. For <M>\textrm{D}_{1\ldots 10}</M> the command
<P/>
<C>edges:=[[1,3],[2,3],[3,4],[4,5],[5,6],[6,7],[7,8],[8,9],[9,10]];</C> can
<P/>
be used.

<Listing><![CDATA[
gap> edges:=[[1,3],[2,4],[3,4],[4,5],[5,6],[6,7],[7,8]]; # for E6..8
[ [ 1, 3 ], [ 2, 4 ], [ 3, 4 ], [ 4, 5 ], [ 5, 6 ], [ 6, 7 ], [ 7, 8 ] ]
]]></Listing>

Now enter the relations as GAP polynomials. First a free associative algebra
with identity element is created over the Rationals (see also <Ref BookName="Reference"
Label="FreeAssociativeAlgebraWithOne"/>). For convenience the generators are
stored in <C>gens</C>.

<Listing><![CDATA[
gap> A:=FreeAssociativeAlgebraWithOne(Rationals,numpoints,"e");;
gap> e := GeneratorsOfAlgebraWithOne(A);
[ (1)*e.1, (1)*e.2, (1)*e.3, (1)*e.4, (1)*e.5, (1)*e.6, (1)*e.7, (1)*e.8 ]
]]></Listing>

It is possible to print symbols like they are printed in the algebra <C>A</C>
with the function <Ref Func="GBNP.ConfigPrint" Style="Text"/>:

<Listing><![CDATA[
gap> GBNP.ConfigPrint(A);
]]></Listing>

Now the relations are generated automatically. For this we need to make sure
the edges are sorted and converted to a set.

<Listing><![CDATA[
gap> edges:=Set(edges, x->SortedList(x));
[ [ 1, 3 ], [ 2, 4 ], [ 3, 4 ], [ 4, 5 ], [ 5, 6 ], [ 6, 7 ], [ 7, 8 ] ]
]]></Listing>

Now the relations can be generated. The relations are <M>e_i*e_i=e_i</M>, for
all <M>i</M> and <M>e_i*e_j*e_i=e_i</M> for all <M>i</M>,<M>j</M> that are
connected in the Coxeter diagram and <M>e_i*e_j=e_j*e_i</M> for all <M>i</M>,
<M>j</M> that are not connected in the Coxeter diagram.

<Listing><![CDATA[
gap> rels:=[];;
gap> for i in [1..numpoints] do
>   for j in [1..numpoints] do
>     if (i=j) then
>       # if i=j then add e.i*e.i=e.i
>       Add(rels, e[i]*e[i]-e[i]);
>     elif ([i,j] in edges) or ([j,i] in edges) then
>       # if {i,j} is an edge then add e.i*e.j*e.i=e.i
>       Add(rels, e[i]*e[j]*e[i]- e[i]);
>     else
>       # if {i,j} is not an edge then add e.i*e.j=e.j*e.i
>       # (note: this causes double rules, but that's ok)
>       Add(rels, e[i]*e[j]- e[j]*e[i]);
>     fi;
>   od;
> od;
]]></Listing>

Then the relations are converted into NP format (see <Ref Sect="NP"/>) after
which the function <Ref Func="SGrobner" Style="Text"/> is called to
calculate a Gröbner basis.

<Listing><![CDATA[
gap> relsNP:=GP2NPList(rels);;
gap> GB:=SGrobner(relsNP);;
#I  The computation took 184 msecs.
]]></Listing>

It is now possible to calculate the dimension of the quotient algebra
with the function <Ref Func="DimQA" Style="Text"/>. This function has as
arguments the Gröbner basis <C>GB</C> and the number of generators of the
algebra (here this is <C>numpoints</C>). To get the full basis the function
<Ref Func="BaseQA" Style="Text"/> can be used.

<Listing><![CDATA[
gap> DimQA(GB,numpoints);
10846
]]></Listing>
</Section>
<#/GAPDoc>

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