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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 = "example02.g"
gap> ### authors Cohen & Gijsbers

gap> ### THIS IS A GAP PACKAGE FOR COMPUTING NON-COMMUTATIVE GROBNER BASES
gap>
gap> ### Last change: August 29 2001.
gap> ### amc

<#GAPDoc Label="Example02">
<Section Label="Example02"><Heading>A truncated Gröbner basis for Leonard pairs</Heading>
To provide Terwilliger with experimental dimension information in low degrees for his theory of Leonard pairs
a truncated  Gröbner basis computation was carried out as follows.
<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 2 (for more information about the info
level, see Chapter <Ref Chap="Info"/>).

<Listing><![CDATA[
gap> LoadPackage("gbnp", false);
true
gap> SetInfoLevel(InfoGBNP,1);
gap> SetInfoLevel(InfoGBNPTime,2);
]]></Listing>

We truncate the example by putting all monomials of degree <M>n</M> in the
ideal by means of the function <C>MkTrLst</C> to be introduced below;
a better way to compute the
result is by means of the truncated GB algorithms (See <Ref
Sect="Example12"/>).
<P/>
We want to truncate at degree 7 so we have fixed <M>n = 8</M>.

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

Now enter the relations in NP form (see <Ref Sect="NP"/>). The function
<Code>MkTrLst</Code> will be introduced, which will return all monomials of
degree <C>n</C>. The list of ideal generators of interest is called <C>I</C>.

<Listing><![CDATA[
gap> sqbr := function(n,q) ; return (q^3-q^-3)/(q-q^(-1)); end;;

gap> c := sqbr(3,5);
651/25

gap> s1 :=[[[1,1,1,2],[1,1,2,1],[1,2,1,1],[2,1,1,1]],[1,-c,c,-1]];;
gap> s2 :=[[[2,2,2,1],[2,2,1,2],[2,1,2,2],[1,2,2,2]],[1,-c,c,-1]];;

gap> MkTrLst := function(l) local ans, h1, h2, a, i;
>    ans := [[1],[2]];
>    for i in [2..l] do
>       h1 := [];
>       h2 := [];
>       for a in ans do
>         Add(h1,Concatenation([1],a));
>         Add(h2,Concatenation([2],a));
>       od;
>       ans := Concatenation(h1,h2);
>    od;
>    return List(ans, a -> [[a],[1]]);
> end;;

gap> I := Concatenation([s1,s2],MkTrLst(n));;
]]></Listing>

To give an impression, we print the first 20 entries of this list:
<Listing><![CDATA[
gap> PrintNPList(I{[1..20]});
a^3b - 651/25a^2ba + 651/25aba^2 - ba^3
b^3a - 651/25b^2ab + 651/25bab^2 - ab^3
a^8
a^7b
a^6ba
a^6b^2
a^5ba^2
a^5bab
a^5b^2a
a^5b^3
a^4ba^3
a^4ba^2b
a^4baba
a^4bab^2
a^4b^2a^2
a^4b^2ab
a^4b^3a
a^4b^4
a^3ba^4
a^3ba^3b
]]></Listing>

We calculate the Gröbner basis with <Ref Func="SGrobner" Style="Text"/>:

<Listing><![CDATA[
gap> GB := SGrobner(I);;
#I  number of entered polynomials is 258
#I  number of polynomials after reduction is 114
#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 176 msecs.
]]></Listing>

Now print the first part of the Gröbner basis with <Ref Func="PrintNPList"
Style="Text"/> (only the first 20 polynomials are printed here, the full
Gröbner basis can be printed with <C>PrintNPList(GB)</C>):

<Listing><![CDATA[
gap> PrintNPList(GB{[1..20]});
ba^3 - 651/25aba^2 + 651/25a^2ba - a^3b
b^3a - 651/25b^2ab + 651/25bab^2 - ab^3
b^2a^2ba - bab^2a^2 - baba^2b + ba^2bab + ab^2aba - abab^2a - aba^2b^2 + a^2b\
^2ab
b^2ab^2a^2 - 651/25b^2ababa + b^2aba^2b + 626/25bab^2aba - bab^2a^2b + babab^\
2a - ba^2b^2ab + ba^2bab^2 - 651/25ab^2ab^2a + ab^2abab + 423176/625abab^2ab -\
423801/625ababab^2 + 626/25aba^2b^3 - 406901/625a^2b^2ab^2 + 423176/625a^2bab\
^3 - 651/25a^3b^4
a^8
a^7b
a^6ba
a^6b^2
a^5ba^2
a^5bab
a^5b^2a
a^5b^3
a^4ba^2b
a^4baba
a^4bab^2
a^4b^2a^2
a^4b^2ab
a^4b^4
a^3ba^2ba
a^3ba^2b^2
]]></Listing>

The truncated quotient algebra is obtained by
factoring out the ideal generated by
the Gröbner basis <C>GB</C> and so
its dimension can be calculated with <Ref
Func="DimQA" Style="Text"/>:

<Listing><![CDATA[
gap> DimQA(GB,2);
#I  The computation took 0 msecs.
157
]]></Listing>

Here is what Paul Terwilliger wrote in reaction to the computation carried
out by this example:

<P/>
  I just wanted to thank you again for the dimension
  data that you gave me after the Durham
  meeting. It ended up having a large impact.
  See the attached paper; joint with Tatsuro
  Ito.<P/>
  I spent several weeks in Japan this past January,
  working with Tatsuro and trying to find a good basis
  for the algebra on two symbols subject to the
  <M>q</M>-Serre relations. After much frustration,
  we thought of feeding your data into Sloane's
  online handbook of integer sequences. We did it
  out of curiosity more than anything; we did not
  expect the handbook data to be particularly useful.
  But it was.<P/>
  The handbook told us that the graded dimension
  generating function, using your
  data for the coefficients, matched the <M>q</M>-series
  for the inverse of the Jacobi theta
  function <M>\vartheta_4</M>; armed with this overwhelming
  hint we were able to prove that the graded
  dimension generating function was indeed given by
  the inverse of <M>\vartheta_4</M>.  With that
  info we were able to get a nice result about
  td pairs.<P/>
  Paul<P/>


</Section>
<#/GAPDoc>

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