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<a id="X7A489A5D79DA9E5C" name="X7A489A5D79DA9E5C"></a>
<div class="ChapSects"><a href="chapA.html#X7A489A5D79DA9E5C">A Examples</a>
<div class="ContSect"> <a href="chapA.html#X7DFB63A97E67C0A1">A.1 Introduction</a>

</div>
<div class="ContSect"> <a href="chapA.html#X784586E47E2739E3">A.2 A simple commutative Gröbner basis computation</a>

</div>
<div class="ContSect"> <a href="chapA.html#X7E1B57AA85C2BA70">A.3 A truncated Gröbner basis for Leonard pairs</a>

</div>
<div class="ContSect"> <a href="chapA.html#X79AC59C482A2E4C1">A.4 The truncated variant on two weighted homogeneous polynomials</a>

</div>
<div class="ContSect"> <a href="chapA.html#X7C7742957CEC6E7B">A.5 The order of the Weyl group of type E_6</a>

</div>
<div class="ContSect"> <a href="chapA.html#X7E39C9738509A036">A.6 The gcd of some univariate polynomials</a>

</div>
<div class="ContSect"> <a href="chapA.html#X7F5A6ABA85CDB6E2">A.7 From the Tapas book</a>

</div>
<div class="ContSect"> <a href="chapA.html#X7C2CD4FA838EEE64">A.8 The Birman-Murakami-Wenzl algebra of type A_3</a>

</div>
<div class="ContSect"> <a href="chapA.html#X7B5CA7F379B78CE0">A.9 The Birman-Murakami-Wenzl algebra of type A_2</a>

</div>
<div class="ContSect"> <a href="chapA.html#X83C81C987A4DE15F">A.10 A commutative example by Mora</a>

</div>
<div class="ContSect"> <a href="chapA.html#X7CAB94A37D580C4A">A.11 Tracing an example by Mora</a>

</div>
<div class="ContSect"> <a href="chapA.html#X8599AE8F7E9E0368">A.12 Finiteness of the Weyl group of type E_6</a>

</div>
<div class="ContSect"> <a href="chapA.html#X7B1822C67CF83041">A.13 Preprocessing for Weyl group computations</a>

</div>
<div class="ContSect"> <a href="chapA.html#X7BE4A97886B0930E">A.14 A quotient algebra with exponential growth</a>

</div>
<div class="ContSect"> <a href="chapA.html#X78679D7D80CD8822">A.15 A commutative quotient algebra of polynomial growth</a>

</div>
<div class="ContSect"> <a href="chapA.html#X7CE3005580EF632D">A.16 An algebra over a finite field</a>

</div>
<div class="ContSect"> <a href="chapA.html#X7E4CEC577A18C8ED">A.17 The dihedral group of order 8</a>

</div>
<div class="ContSect"> <a href="chapA.html#X83328C357FB33D17">A.18 The dihedral group of order 8 on another module</a>

</div>
<div class="ContSect"> <a href="chapA.html#X85DBF3967C4DF5FE">A.19 The dihedral group on a non-cyclic module</a>

</div>
<div class="ContSect"> <a href="chapA.html#X78FCAC347D9D607E">A.20 The icosahedral group</a>

</div>
<div class="ContSect"> <a href="chapA.html#X780C4B777FEA9080">A.21 The symmetric inverse monoid for a set of size four</a>

</div>
<div class="ContSect"> <a href="chapA.html#X84C07DC479FBBCD5">A.22 A module of the Hecke algebra of type
A_3 over GF(3)</a>

</div>
<div class="ContSect"> <a href="chapA.html#X78C01D1987FEF3FE">A.23 Generalized Temperley-Lieb algebras</a>

</div>
<div class="ContSect"> <a href="chapA.html#X85A9CEF087F3936B">A.24 The universal enveloping
algebra of a Lie algebra</a>

</div>
<div class="ContSect"> <a href="chapA.html#X8498D69D8160E5FF">A.25 Serre's exercise

</div>
<div class="ContSect"> <a href="chapA.html#X8116448A84D69022">A.26 Baur and Draisma's transformations

</div>
<div class="ContSect"> <a href="chapA.html#X7912E411867E5F8B">A.27 The cola gene puzzle</a>

</div>
</div>

<h3>A Examples</h3>

<a id="X7DFB63A97E67C0A1" name="X7DFB63A97E67C0A1"></a>

<h4>A.1 Introduction</h4>

In this chapter all available commented examples can be found. Those without comments are in the directory <code class="file">gbnp/examples</code>. Timing results are obtained on an Intel(R) Xeon(R) CPU E5-2630 v3 @ 2.40GHz processor running Linux (3.19.0-42-generic #48~14.04.1-Ubuntu SMP Fri Dec 18 10:24:49 UTC 2015) and using GAP 4.6.5.

<ul>
<li><a href="chapA.html#X784586E47E2739E3">A.2</a> <a href="chapA.html#X784586E47E2739E3">A simple commutative Gröbner basis computation</a>

</li>
<li><a href="chapA.html#X7E1B57AA85C2BA70">A.3</a> <a href="chapA.html#X7E1B57AA85C2BA70">A truncated Gröbner basis for Leonard pairs</a>

</li>
<li><a href="chapA.html#X79AC59C482A2E4C1">A.4</a> <a href="chapA.html#X79AC59C482A2E4C1">The truncated variant on two weighted homogeneous polynomials</a>

</li>
<li><a href="chapA.html#X7C7742957CEC6E7B">A.5</a> <a href="chapA.html#X7C7742957CEC6E7B">The order of the Weyl group of type E_6</a>

</li>
<li><a href="chapA.html#X7E39C9738509A036">A.6</a> <a href="chapA.html#X7E39C9738509A036">The gcd of some univariate polynomials</a>

</li>
<li><a href="chapA.html#X7F5A6ABA85CDB6E2">A.7</a> <a href="chapA.html#X7F5A6ABA85CDB6E2">From the Tapas book</a>

</li>
<li><a href="chapA.html#X7C2CD4FA838EEE64">A.8</a> <a href="chapA.html#X7C2CD4FA838EEE64">The Birman-Murakami-Wenzl algebra of type A_3</a>

</li>
<li><a href="chapA.html#X7B5CA7F379B78CE0">A.9</a> <a href="chapA.html#X7B5CA7F379B78CE0">The Birman-Murakami-Wenzl algebra of type A_2</a>

</li>
<li><a href="chapA.html#X83C81C987A4DE15F">A.10</a> <a href="chapA.html#X83C81C987A4DE15F">A commutative example by Mora</a>

</li>
<li><a href="chapA.html#X7CAB94A37D580C4A">A.11</a> <a href="chapA.html#X7CAB94A37D580C4A">Tracing an example by Mora</a>

</li>
<li><a href="chapA.html#X8599AE8F7E9E0368">A.12</a> <a href="chapA.html#X8599AE8F7E9E0368"> Finiteness of the Weyl group of type E_6</a>

This extends Example <a href="chapA.html#X7C7742957CEC6E7B">A.5</a>.

</li>
<li><a href="chapA.html#X7B1822C67CF83041">A.13</a> <a href="chapA.html#X7B1822C67CF83041">Preprocessing for Weyl group computations</a>

This extends two earlier examples <a href="chapA.html#X7C7742957CEC6E7B">A.5</a> and <a href="chapA.html#X8599AE8F7E9E0368">A.12</a>.

</li>
<li><a href="chapA.html#X7BE4A97886B0930E">A.14</a> <a href="chapA.html#X7BE4A97886B0930E">A quotient algebra with exponential growth</a>

</li>
<li><a href="chapA.html#X78679D7D80CD8822">A.15</a> <a href="chapA.html#X78679D7D80CD8822">A commutative quotient algebra of polynomial growth</a>

This extends Example <a href="chapA.html#X7F5A6ABA85CDB6E2">A.7</a>.

</li>
<li><a href="chapA.html#X7CE3005580EF632D">A.16</a> <a href="chapA.html#X7CE3005580EF632D">An algebra over a finite field</a>

</li>
<li><a href="chapA.html#X7E4CEC577A18C8ED">A.17</a> <a href="chapA.html#X7E4CEC577A18C8ED">The dihedral group of order 8</a>

</li>
<li><a href="chapA.html#X83328C357FB33D17">A.18</a> <a href="chapA.html#X83328C357FB33D17">The dihedral group of order 8 on another module</a>

This extends Example <a href="chapA.html#X7E4CEC577A18C8ED">A.17</a>.

</li>
<li><a href="chapA.html#X85DBF3967C4DF5FE">A.19</a> <a href="chapA.html#X85DBF3967C4DF5FE">The dihedral group on a non-cyclic module</a>

This example also extends Example <a href="chapA.html#X7E4CEC577A18C8ED">A.17</a>.

</li>
<li><a href="chapA.html#X78FCAC347D9D607E">A.20</a> <a href="chapA.html#X78FCAC347D9D607E">The icosahedral group</a>

</li>
<li><a href="chapA.html#X780C4B777FEA9080">A.21</a> <a href="chapA.html#X780C4B777FEA9080">The symmetric inverse monoid for a set of size four</a>

</li>
<li><a href="chapA.html#X84C07DC479FBBCD5">A.22</a> <a href="chapA.html#X84C07DC479FBBCD5">A module of the Hecke algebra of type A_3 over GF(3)</a>

</li>
<li><a href="chapA.html#X78C01D1987FEF3FE">A.23</a> <a href="chapA.html#X78C01D1987FEF3FE">Generalized Temperley-Lieb algebras</a>

</li>
<li><a href="chapA.html#X85A9CEF087F3936B">A.24</a> <a href="chapA.html#X85A9CEF087F3936B">The universal enveloping algebra of a Lie algebra</a>

</li>
<li><a href="chapA.html#X8498D69D8160E5FF">A.25</a> <a href="chapA.html#X8498D69D8160E5FF">Serre's exercise

</li>
<li><a href="chapA.html#X8116448A84D69022">A.26</a> <a href="chapA.html#X8116448A84D69022">Baur and Draisma's transformations

</li>
<li><a href="chapA.html#X7912E411867E5F8B">A.27</a> <a href="chapA.html#X7912E411867E5F8B">The cola gene puzzle</a>

</li>
</ul>
<a id="X784586E47E2739E3" name="X784586E47E2739E3"></a>

<h4>A.2 A simple commutative Gröbner basis computation</h4>

In this commutative example the relations are x^2y-1 and xy^2-1; we add xy-yx to enforce that x and y commute. The answer should be {x^3-1, x-y, xy-yx}, as the reduction ordering is total degree first and then lexicographic with x smaller than y.

First load the package and set the standard infolevel <code class="func">InfoGBNP</code> (<a href="chap4.html#X82D40B0E84383BBC">4.2-1</a>) to 2 and the time infolevel <code class="func">InfoGBNPTime</code> (<a href="chap4.html#X7FAE244E80397B9A">4.3-1</a>) to 1 (for more information about the info level, see Chapter <a href="chap4.html#X79C5DF3782576D98">4</a>).

<div class="example"><pre>
gap> LoadPackage("gbnp", false);
true
gap> SetInfoLevel(InfoGBNP,2);
gap> SetInfoLevel(InfoGBNPTime,1);
</pre></div>

Then input the relations in NP format (see Section <a href="chap2.html#X7FDF3E5E7F33D3A2">2.1</a>). They will be put in the list <code class="code">Lnp</code>.

<div class="example"><pre>
gap> Lnp := [ [[[1,2],[2,1]],[1,-1]] ];
[ [ [ [ 1, 2 ], [ 2, 1 ] ], [ 1, -1 ] ] ]
gap> x2y := [[[1,1,2],[]],[1,-1]];
[ [ [ 1, 1, 2 ], [ ] ], [ 1, -1 ] ]
gap> AddSet(Lnp,x2y);
gap> xy2 := [[[1,2,2],[]],[1,-1]];
[ [ [ 1, 2, 2 ], [ ] ], [ 1, -1 ] ]
gap> AddSet(Lnp,xy2);
</pre></div>

The relations can be exhibited with <code class="func">PrintNPList</code> (<a href="chap3.html#X832103DC79A9E9D0">3.2-3</a>):

<div class="example"><pre>
gap> PrintNPList(Lnp);
a^2b - 1
ab - ba
ab^2 - 1
</pre></div>

Let the variables be printed as x and y instead of a and b by means of <code class="func">GBNP.ConfigPrint</code> (<a href="chap3.html#X7F7510A878045D3A">3.2-2</a>)

<div class="example"><pre>
gap> GBNP.ConfigPrint("x","y");
</pre></div>

The Gröbner basis can now be calculated with <code class="func">SGrobner</code> (<a href="chap3.html#X7FEDA29E78B0CEED">3.4-2</a>):

<div class="example"><pre>
gap> GB := SGrobner(Lnp);
#I number of entered polynomials is 3
#I number of polynomials after reduction is 3
#I End of phase I
#I End of phase II
#I length of G =1
#I length of todo is 1
#I length of G =2
#I length of todo is 0
#I List of todo lengths is [ 1, 1, 0 ]
#I End of phase III
#I G: Cleaning finished, 0 polynomials reduced
#I End of phase IV
#I The computation took 4 msecs.
[ [ [ [ 2 ], [ 1 ] ], [ 1, -1 ] ], [ [ [ 1, 1, 1 ], [ ] ], [ 1, -1 ] ] ]
</pre></div>

When printed, it looks like:

<div class="example"><pre>
gap> PrintNPList(GB);
y - x
x^3 - 1
</pre></div>

The dimension of the quotient algebra can be calculated with <code class="func">DimQA</code> (<a href="chap3.html#X81A50EEE7B56C723">3.5-2</a>). The arguments are the Gröbner basis <code class="code">GB</code> and the number of variables is <code class="code">2</code>:

<div class="example"><pre>
gap> DimQA(GB,2);
3
</pre></div>

A basis of this quotient algebra can be calculated with <code class="func">BaseQA</code> (<a href="chap3.html#X7EAA04247B2C6330">3.5-1</a>). The arguments are a Gröbner basis <code class="code">GB</code>, the number of variables <var class="Arg">t</var> (=2) and a variable <var class="Arg">maxno</var> for returning partial quotient algebras (0 means full basis). The calculated basis will be printed as well.

<div class="example"><pre>
gap> B:=BaseQA(GB,2,0);;
gap> PrintNPList(B);
1
x
x^2
</pre></div>

The strong normal form of the element xyxyxyx can be found by use of <code class="func">StrongNormalFormNP</code> (<a href="chap3.html#X8563683E7FA604F8">3.5-6</a>). The arguments are this element and the Gröbner basis <code class="code">GB</code>.

<div class="example"><pre>
gap> f:=[[[1,2,1,2,1,2,1]],[1]];;
gap> PrintNP(f);
xyxyxyx
gap> p:=StrongNormalFormNP(f,GB);;
gap> PrintNP(p);
x
</pre></div>

<a id="X7E1B57AA85C2BA70" name="X7E1B57AA85C2BA70"></a>

<h4>A.3 A truncated Gröbner basis for Leonard pairs</h4>

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.

First load the package and set the standard infolevel <code class="func">InfoGBNP</code> (<a href="chap4.html#X82D40B0E84383BBC">4.2-1</a>) to 1 and the time infolevel <code class="func">InfoGBNPTime</code> (<a href="chap4.html#X7FAE244E80397B9A">4.3-1</a>) to 2 (for more information about the info level, see Chapter <a href="chap4.html#X79C5DF3782576D98">4</a>).

<div class="example"><pre>
gap> LoadPackage("gbnp", false);
true
gap> SetInfoLevel(InfoGBNP,1);
gap> SetInfoLevel(InfoGBNPTime,2);
</pre></div>

We truncate the example by putting all monomials of degree n in the ideal by means of the function <code class="code">MkTrLst</code> to be introduced below; a better way to compute the result is by means of the truncated GB algorithms (See <a href="chapA.html#X85A9CEF087F3936B">A.24</a>).

We want to truncate at degree 7 so we have fixed n = 8.

<div class="example"><pre>
gap> n := 8;;
</pre></div>

Now enter the relations in NP form (see <a href="chap2.html#X7FDF3E5E7F33D3A2">2.1</a>). The function <code class="code">MkTrLst</code> will be introduced, which will return all monomials of degree <code class="code">n</code>. The list of ideal generators of interest is called <code class="code">I</code>.

<div class="example"><pre>
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));;
</pre></div>

To give an impression, we print the first 20 entries of this list:

<div class="example"><pre>
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
</pre></div>

We calculate the Gröbner basis with <code class="func">SGrobner</code> (<a href="chap3.html#X7FEDA29E78B0CEED">3.4-2</a>):

<div class="example"><pre>
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.
</pre></div>

Now print the first part of the Gröbner basis with <code class="func">PrintNPList</code> (<a href="chap3.html#X832103DC79A9E9D0">3.2-3</a>) (only the first 20 polynomials are printed here, the full Gröbner basis can be printed with <code class="code">PrintNPList(GB)</code>):

<div class="example"><pre>
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
</pre></div>

The truncated quotient algebra is obtained by factoring out the ideal generated by the Gröbner basis <code class="code">GB</code> and so its dimension can be calculated with <code class="func">DimQA</code> (<a href="chap3.html#X81A50EEE7B56C723">3.5-2</a>):

<div class="example"><pre>
gap> DimQA(GB,2);
#I The computation took 0 msecs.
157
</pre></div>

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

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.

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 q-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.

The handbook told us that the graded dimension generating function, using your data for the coefficients, matched the q-series for the inverse of the Jacobi theta function ϑ_4; armed with this overwhelming hint we were able to prove that the graded dimension generating function was indeed given by the inverse of ϑ_4. With that info we were able to get a nice result about td pairs.

Paul

<a id="X79AC59C482A2E4C1" name="X79AC59C482A2E4C1"></a>

<h4>A.4 The truncated variant on two weighted homogeneous polynomials</h4>

Here we exhibit a truncated non-commutative homogeneous weighted Gröbner basis computation. This example uses the functions from Section <a href="chap3.html#X7E4E3AD07B2465F9">3.8</a>, the truncation variants (see also Section <a href="chap2.html#X78CF5C44879D34B6">2.6</a>).

The input is a set of polynomials in x and y, which are homogeneous when the weight of x is 2 and of y is 3. The input is {x^3y^2-x^6+y^4,y^2x^3+xyxyx+x^2yxy}. We truncate the computation at degree 16. The truncated Gröbner basis is {y^2x^3+xyxyx+x^2yxy,x^6-x^3y^2-y^4,x^3y^2x-x^4y^2-xy^4} and the dimension of the quotient algebra is 134.

First load the package and set the standard infolevel <code class="func">InfoGBNP</code> (<a href="chap4.html#X82D40B0E84383BBC">4.2-1</a>) to 1 and the time infolevel <code class="func">InfoGBNPTime</code> (<a href="chap4.html#X7FAE244E80397B9A">4.3-1</a>) to 1 (for more information about the info level, see Chapter <a href="chap4.html#X79C5DF3782576D98">4</a>).

<div class="example"><pre>
gap> LoadPackage("gbnp", false);
true
gap> SetInfoLevel(InfoGBNP,1);
gap> SetInfoLevel(InfoGBNPTime,1);
</pre></div>

The variables will be printed as x and y.

<div class="example"><pre>
gap> GBNP.ConfigPrint("x","y");
</pre></div>

The level to truncate at is assigned to n.

<div class="example"><pre>
gap> n := 16;;
</pre></div>

Now enter the relations in NP form (see Section <a href="chap2.html#X7FDF3E5E7F33D3A2">2.1</a>) and the weights.

<div class="example"><pre>
gap> s1 :=[[[1,1,1,2,2],[1,1,1,1,1,1],[2,2,2,2]],[1,-1,1]];;
gap> s2 :=[[[2,2,1,1,1],[1,2,1,2,1],[1,1,2,1,2]],[1,1,1]];;
gap> K := [s1,s2];;
gap> weights:=[2,3];;
</pre></div>

The input can be printed with <code class="func">PrintNPList</code> (<a href="chap3.html#X832103DC79A9E9D0">3.2-3</a>)

<div class="example"><pre>
gap> PrintNPList(K);
x^3y^2 - x^6 + y^4
y^2x^3 + xyxyx + x^2yxy
</pre></div>

Verify whether the list <code class="code">K</code> consists only of polynomials that are homogeneous with respect to <code class="code">weights</code> by means of <code class="func">CheckHomogeneousNPs</code> (<a href="chap3.html#X83C9E598798D5809">3.8-3</a>).

<div class="example"><pre>
gap> CheckHomogeneousNPs(K,weights);
#I Input is homogeneous
[ 12, 12 ]
</pre></div>

Now calculate the truncated Gröbner basis with <code class="func">SGrobnerTrunc</code> (<a href="chap3.html#X7CD043E081BF2302">3.8-2</a>). The output will only contain homogeneous polynomials of degree at most <code class="code">n</code>.

<div class="example"><pre>
gap> G := SGrobnerTrunc(K,n,weights);;
#I number of entered polynomials is 2
#I number of polynomials after reduction is 2
#I End of phase I
#I Input is homogeneous
#I Reached level 16
#I end of the algorithm
#I The computation took 4 msecs.
</pre></div>

The Gröbner basis of the truncated quotient algebra can be printed with <code class="func">PrintNPList</code> (<a href="chap3.html#X832103DC79A9E9D0">3.2-3</a>):

<div class="example"><pre>
gap> PrintNPList(G);
y^2x^3 + xyxyx + x^2yxy
x^6 - x^3y^2 - y^4
x^3y^2x - x^4y^2 + y^4x - xy^4
</pre></div>

The standard basis of the quotient of the free noncommutative algebra on n variables, where n is the length of the vector <code class="code">weights</code>, by the homogeneous ideal generated by <code class="code">K</code> up to degree n is obtained by means of the function <code class="func">BaseQATrunc</code> (<a href="chap3.html#X7E33C064875D95CA">3.8-4</a>) applied to <code class="code">K</code>, <code class="code">n</code>, and <code class="code">weights</code>.

<div class="example"><pre>
gap> B := BaseQATrunc(K,n,weights);;
#I number of entered polynomials is 2
#I number of polynomials after reduction is 2
#I End of phase I
#I Input is homogeneous
#I Reached level 16
#I end of the algorithm
#I The computation took 0 msecs.
gap> i := Length(B);
17
gap> Print("at level ",i-1," the standard monomials are:\n");
at level 16 the standard monomials are:
gap> PrintNPList(List(B[i], qq -> [[qq],[1]]));
yxyx^4
yx^2yx^3
xyxyx^3
yx^3yx^2
xyx^2yx^2
x^2yxyx^2
y^4x^2
yx^4yx
xyx^3yx
x^2yx^2yx
x^3yxyx
y^3xyx
y^2xy^2x
yxy^3x
xy^4x
yx^5y
xyx^4y
x^2yx^3y
x^3yx^2y
y^3x^2y
x^4yxy
y^2xyxy
yxy^2xy
xy^3xy
x^5y^2
y^2x^2y^2
yxyxy^2
xy^2xy^2
yx^2y^3
xyxy^3
x^2y^4
</pre></div>

The same result can be obtained by using the truncated Gröbner basis found for <code class="code">G</code> instead of <code class="code">K</code>.

<div class="example"><pre>
gap> B2 := BaseQATrunc(G,n,weights);;
#I number of entered polynomials is 3
#I number of polynomials after reduction is 3
#I End of phase I
#I Input is homogeneous
#I Reached level 16
#I end of the algorithm
#I The computation took 0 msecs.
gap> B = B2;
true
</pre></div>

Also, the same result can be obtained by using the leading terms of the truncated Gröbner basis found for <code class="code">G</code> instead of <code class="code">K</code>.

<div class="example"><pre>
gap> B3 := BaseQATrunc(List( LMonsNP(G), qq -> [[qq],[1]]),n,weights);;
#I number of entered polynomials is 3
#I number of polynomials after reduction is 3
#I End of phase I
#I Input is homogeneous
#I Reached level 16
#I end of the algorithm
#I The computation took 0 msecs.
gap> B = B3;
true
</pre></div>

A list of dimensions of the homogeneous parts of the quotient algebra up to degree n is obtained by means of <code class="func">DimsQATrunc</code> (<a href="chap3.html#X7C6882DB837A9F5A">3.8-5</a>) with arguments <code class="code">G</code>, <code class="code">n</code>, and <code class="code">weights</code>.

<div class="example"><pre>
gap> DimsQATrunc(G,n,weights);
#I number of entered polynomials is 3
#I number of polynomials after reduction is 3
#I End of phase I
#I Input is homogeneous
#I Reached level 16
#I end of the algorithm
#I The computation took 4 msecs.
[ 1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 10, 16, 17, 24, 31 ]
</pre></div>

Even more detailed information is given by the list of frequencies up to degree <code class="code">n</code>. This is obtained by means of <code class="func">FreqsQATrunc</code> (<a href="chap3.html#X7FBA7F1D79DA883F">3.8-6</a>) with arguments <code class="code">G</code>, <code class="code">n</code>, and <code class="code">weights</code>.

<div class="example"><pre>
gap> FreqsQATrunc(G,n,weights);
#I number of entered polynomials is 3
#I number of polynomials after reduction is 3
#I End of phase I
#I Input is homogeneous
#I Reached level 16
#I end of the algorithm
#I The computation took 0 msecs.
[ [ [ [ ], 1 ] ], [ [ [ 1, 0 ], 1 ] ], [ [ [ 0, 1 ], 1 ] ],
 [ [ [ 2, 0 ], 1 ] ], [ [ [ 1, 1 ], 2 ] ],
 [ [ [ 3, 0 ], 1 ], [ [ 0, 2 ], 1 ] ], [ [ [ 2, 1 ], 3 ] ],
 [ [ [ 4, 0 ], 1 ], [ [ 1, 2 ], 3 ] ], [ [ [ 3, 1 ], 4 ], [ [ 0, 3 ], 1 ] ],
 [ [ [ 5, 0 ], 1 ], [ [ 2, 2 ], 6 ] ], [ [ [ 4, 1 ], 5 ], [ [ 1, 3 ], 4 ] ],
 [ [ [ 3, 2 ], 9 ], [ [ 0, 4 ], 1 ] ], [ [ [ 5, 1 ], 6 ], [ [ 2, 3 ], 10 ] ],
 [ [ [ 4, 2 ], 12 ], [ [ 1, 4 ], 5 ] ],
 [ [ [ 6, 1 ], 5 ], [ [ 3, 3 ], 18 ], [ [ 0, 5 ], 1 ] ],
 [ [ [ 5, 2 ], 16 ], [ [ 2, 4 ], 15 ] ] ]
</pre></div>

<a id="X7C7742957CEC6E7B" name="X7C7742957CEC6E7B"></a>

<h4>A.5 The order of the Weyl group of type E_6</h4>

In order to show how the order of a finite group of manageable size with a manageable presentation can be computed, we determine the order of the Weyl group of type E_6. This number is well known to be 51840.

First load the package and set the standard infolevel <code class="func">InfoGBNP</code> (<a href="chap4.html#X82D40B0E84383BBC">4.2-1</a>) to 1 and the time infolevel <code class="func">InfoGBNPTime</code> (<a href="chap4.html#X7FAE244E80397B9A">4.3-1</a>) to 2 (for more information about the info level, see Chapter <a href="chap4.html#X79C5DF3782576D98">4</a>).

<div class="example"><pre>
gap> LoadPackage("gbnp", false);
true
gap> SetInfoLevel(InfoGBNP,1);
gap> SetInfoLevel(InfoGBNPTime,2);
</pre></div>

Then input the relations in NP format (see <a href="chap2.html#X7FDF3E5E7F33D3A2">2.1</a>). They come from the presentation of the Weyl group as a Coxeter group. This means that there are six variables, one for each generator. We let the corresponding variables be printed as r_1, ..., r_6 by means of <code class="func">GBNP.ConfigPrint</code> (<a href="chap3.html#X7F7510A878045D3A">3.2-2</a>)

<div class="example"><pre>
gap> GBNP.ConfigPrint(6,"r");
</pre></div>

The relations are binomial and represent the group relations, which express that the generators are involutions (that is, have order 2) and that the orders of the products of any two generators is specified by the Coxeter diagram (see any book on Coxeter groups for details). The relations will be assigned to <code class="code">KI</code>.

<div class="example"><pre>
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
> ];;
</pre></div>

The relations can be shown with <code class="func">PrintNPList</code> (<a href="chap3.html#X832103DC79A9E9D0">3.2-3</a>):

<div class="example"><pre>
gap> PrintNPList(KI);
r.1r.3r.1 - r.3r.1r.3
r.4r.3r.4 - r.3r.4r.3
r.4r.2r.4 - r.2r.4r.2
r.4r.5r.4 - r.5r.4r.5
r.6r.5r.6 - r.5r.6r.5
r.1r.2 - r.2r.1
r.1r.4 - r.4r.1
r.1r.5 - r.5r.1
r.1r.6 - r.6r.1
r.2r.3 - r.3r.2
r.2r.5 - r.5r.2
r.2r.6 - r.6r.2
r.3r.5 - r.5r.3
r.3r.6 - r.6r.3
r.4r.6 - r.6r.4
r.1^2 - 1
r.2^2 - 1
r.3^2 - 1
r.4^2 - 1
r.5^2 - 1
r.6^2 - 1
</pre></div>

The Gröbner basis can now be calculated with <code class="func">SGrobner</code> (<a href="chap3.html#X7FEDA29E78B0CEED">3.4-2</a>):

<div class="example"><pre>
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 132 msecs.
gap> PrintNPList(GB);
r.1^2 - 1
r.2r.1 - r.1r.2
r.2^2 - 1
r.3r.2 - r.2r.3
r.3^2 - 1
r.4r.1 - r.1r.4
r.4^2 - 1
r.5r.1 - r.1r.5
r.5r.2 - r.2r.5
r.5r.3 - r.3r.5
r.5^2 - 1
r.6r.1 - r.1r.6
r.6r.2 - r.2r.6
r.6r.3 - r.3r.6
r.6r.4 - r.4r.6
r.6^2 - 1
r.3r.1r.2 - r.2r.3r.1
r.3r.1r.3 - r.1r.3r.1
r.4r.2r.4 - r.2r.4r.2
r.4r.3r.4 - r.3r.4r.3
r.5r.4r.5 - r.4r.5r.4
r.6r.5r.6 - r.5r.6r.5
r.4r.3r.1r.4 - r.3r.4r.3r.1
r.5r.4r.2r.5 - r.4r.5r.4r.2
r.5r.4r.3r.5 - r.4r.5r.4r.3
r.6r.5r.4r.6 - r.5r.6r.5r.4
r.4r.2r.3r.4r.2 - r.3r.4r.2r.3r.4
r.4r.2r.3r.4r.3 - r.2r.4r.2r.3r.4
r.5r.4r.2r.3r.5 - r.4r.5r.4r.2r.3
r.5r.4r.3r.1r.5 - r.4r.5r.4r.3r.1
r.6r.5r.4r.2r.6 - r.5r.6r.5r.4r.2
r.6r.5r.4r.3r.6 - r.5r.6r.5r.4r.3
r.4r.2r.3r.1r.4r.2 - r.3r.4r.2r.3r.1r.4
r.5r.4r.2r.3r.1r.5 - r.4r.5r.4r.2r.3r.1
r.6r.5r.4r.2r.3r.6 - r.5r.6r.5r.4r.2r.3
r.6r.5r.4r.3r.1r.6 - r.5r.6r.5r.4r.3r.1
r.4r.2r.3r.1r.4r.3r.1 - r.2r.4r.2r.3r.1r.4r.3
r.5r.4r.2r.3r.4r.5r.4 - r.4r.5r.4r.2r.3r.4r.5
r.6r.5r.4r.2r.3r.1r.6 - r.5r.6r.5r.4r.2r.3r.1
r.6r.5r.4r.2r.3r.4r.6 - r.5r.6r.5r.4r.2r.3r.4
r.5r.4r.2r.3r.1r.4r.5r.4 - r.4r.5r.4r.2r.3r.1r.4r.5
r.6r.5r.4r.2r.3r.1r.4r.6 - r.5r.6r.5r.4r.2r.3r.1r.4
r.6r.5r.4r.2r.3r.1r.4r.3r.6 - r.5r.6r.5r.4r.2r.3r.1r.4r.3
r.6r.5r.4r.2r.3r.4r.5r.6r.5 - r.5r.6r.5r.4r.2r.3r.4r.5r.6
r.5r.4r.2r.3r.1r.4r.3r.5r.4r.3 - r.4r.5r.4r.2r.3r.1r.4r.3r.5r.4
r.6r.5r.4r.2r.3r.1r.4r.5r.6r.5 - r.5r.6r.5r.4r.2r.3r.1r.4r.5r.6
r.5r.4r.2r.3r.1r.4r.3r.5r.4r.2r.3 - r.4r.5r.4r.2r.3r.1r.4r.3r.5r.4r.2
r.6r.5r.4r.2r.3r.1r.4r.3r.5r.6r.5 - r.5r.6r.5r.4r.2r.3r.1r.4r.3r.5r.6
r.6r.5r.4r.2r.3r.1r.4r.3r.5r.4r.6r.5r.4 - r.5r.6r.5r.4r.2r.3r.1r.4r.3r.5r.4r.\
6r.5
r.6r.5r.4r.2r.3r.1r.4r.3r.5r.4r.2r.6r.5r.4r.2 - r.5r.6r.5r.4r.2r.3r.1r.4r.3r.\
5r.4r.2r.6r.5r.4
</pre></div>

The base of the quotient algebra can be calculated with <code class="func">BaseQA</code> (<a href="chap3.html#X7EAA04247B2C6330">3.5-1</a>), which has as arguments a Gröbner basis <code class="code">GB</code>, a number of symbols <code class="code">6</code> and a maximum terms to be found (here 0 is entered, for a full base) . Since it is very long we will not print it here.

<div class="example"><pre>
gap> B:=BaseQA(GB,6,0);;
</pre></div>

The dimension of the quotient algebra can be calculated with <code class="func">DimQA</code> (<a href="chap3.html#X81A50EEE7B56C723">3.5-2</a>), the arguments are the Gröbner basis <code class="code">GB</code> and the number of symbols <code class="code">6</code>. Since <code class="func">InfoGBNPTime</code> (<a href="chap4.html#X7FAE244E80397B9A">4.3-1</a>) is set to 2, we get timing information from <code class="func">DimQA</code> (<a href="chap3.html#X81A50EEE7B56C723">3.5-2</a>):

<div class="example"><pre>
gap> DimQA(GB,6);
#I The computation took 172 msecs.
51840
</pre></div>

Note that the calculation of the dimension takes almost as long as calculating the base. Since we have already calculated a base <code class="code">B</code> it is much more efficient to calculate the dimension with <code class="code">Length</code>:

<div class="example"><pre>
gap> Length(B);
51840
</pre></div>

<a id="X7E39C9738509A036" name="X7E39C9738509A036"></a>

<h4>A.6 The gcd of some univariate polynomials</h4>

A list of univariate polynomials is generated. The result of the Gröbner basis computation on this list should be a single monic polynomial, their gcd.

First load the package and set the standard infolevel <code class="func">InfoGBNP</code> (<a href="chap4.html#X82D40B0E84383BBC">4.2-1</a>) to 2 and the time infolevel <code class="func">InfoGBNPTime</code> (<a href="chap4.html#X7FAE244E80397B9A">4.3-1</a>) to 1 (for more information about the info level, see Chapter <a href="chap4.html#X79C5DF3782576D98">4</a>).

<div class="example"><pre>
gap> LoadPackage("gbnp", false);
true
gap> SetInfoLevel(InfoGBNP,2);
gap> SetInfoLevel(InfoGBNPTime,1);
</pre></div>

Let the single variable be printed as x by means of <code class="func">GBNP.ConfigPrint</code> (<a href="chap3.html#X7F7510A878045D3A">3.2-2</a>)

<div class="example"><pre>
gap> GBNP.ConfigPrint("x");
</pre></div>

Now input the relations in NP format (see <a href="chap2.html#X7FDF3E5E7F33D3A2">2.1</a>). They will be assigned to <code class="code">KI</code>.

<div class="example"><pre>
gap> p0 := [[[1,1,1],[1,1],[1],[]],[1,2,2,1]];;
gap> p1 := [[[1,1,1,1],[1,1],[]],[1,1,1]];;
gap> KI := [p0,p1];;

gap> for i in [2..12] do
> h := AddNP(AddNP(KI[i],KI[i-1],1,3),
> AddNP(BimulNP([1],KI[i],[]),KI[i-1],2,1),3,-5);
> Add(KI,h);
> od;
</pre></div>

The relations can be shown with <code class="func">PrintNPList</code> (<a href="chap3.html#X832103DC79A9E9D0">3.2-3</a>):

<div class="example"><pre>
gap> PrintNPList(KI);
x^3 + 2x^2 + 2x + 1
x^4 + x^2 + 1
- 10x^5 + 3x^4 - 6x^3 + 11x^2 - 2x + 7
100x^6 - 60x^5 + 73x^4 - 128x^3 + 57x^2 - 76x + 25
- 1000x^7 + 900x^6 - 950x^5 + 1511x^4 - 978x^3 + 975x^2 - 486x + 103
10000x^8 - 12000x^7 + 12600x^6 - 18200x^5 + 14605x^4 - 13196x^3 + 8013x^2 - 2\
792x + 409
- 100000x^9 + 150000x^8 - 166000x^7 + 223400x^6 - 204450x^5 + 181819x^4 - 123\
630x^3 + 55859x^2 - 14410x + 1639
1000000x^10 - 1800000x^9 + 2150000x^8 - 2780000x^7 + 2765100x^6 - 2504340x^5 \
+ 1840177x^4 - 982264x^3 + 343729x^2 - 70788x + 6553
- 10000000x^11 + 21000000x^10 - 27300000x^9 + 34850000x^8 - 36655000x^7 + 342\
32300x^6 - 26732590x^5 + 16070447x^4 - 6878602x^3 + 1962503x^2 - 335534x + 262\
15
100000000x^12 - 240000000x^11 + 340000000x^10 - 437600000x^9 + 479700000x^8 -\
463408000x^7 + 381083200x^6 - 250919600x^5 + 124358069x^4 - 44189892x^3 + 106\
17765x^2 - 1551904x + 104857
- 1000000000x^13 + 2700000000x^12 - 4160000000x^11 + 5480000000x^10 - 6219000\
000x^9 + 6212580000x^8 - 5347676000x^7 + 3789374800x^6 - 2103269850x^5 + 87925\
4915x^4 - 266261734x^3 + 55222347x^2 - 7046418x + 419431
10000000000x^14 - 30000000000x^13 + 50100000000x^12 - 68240000000x^11 + 79990\
000000x^10 - 82533200000x^9 + 74033300000x^8 - 55790408000x^7 + 33925155700x^6\
- 16106037100x^5 + 5797814361x^4 - 1527768240x^3 + 278602281x^2 - 31541180x +\
1677721
- 100000000000x^15 + 330000000000x^14 - 595000000000x^13 + 843500000000x^12 -\
1021260000000x^11 + 1087222000000x^10 - 1012808600000x^9 + 804854300000x^8 - \
528013485000x^7 + 277993337300x^6 - 114709334310x^5 + 36188145143x^4 - 8434374\
466x^3 + 1372108031x^2 - 139586422x + 6710887
gap> Length(KI);
13
</pre></div>

The Gröbner basis can now be calculated with <code class="func">SGrobner</code> (<a href="chap3.html#X7FEDA29E78B0CEED">3.4-2</a>):

<div class="example"><pre>
gap> GB := SGrobner(KI);;
#I number of entered polynomials is 13
#I number of polynomials after reduction is 1
#I End of phase I
#I End of phase II
#I List of todo lengths is [ 0 ]
#I End of phase III
#I G: Cleaning finished, 0 polynomials reduced
#I End of phase IV
#I The computation took 0 msecs.
</pre></div>

Printed it looks like:

<div class="example"><pre>
gap> PrintNPList(GB);
x^2 + x + 1
</pre></div>

<a id="X7F5A6ABA85CDB6E2" name="X7F5A6ABA85CDB6E2"></a>

<h4>A.7 From the Tapas book</h4>

This example is a standard commutative Gröbner basis computation from the book Some Tapas of Computer Algebra <a href="chapBib.html#biBCohenCuypersSterk1999">[CCS99]</a>, page 339. There are six variables, named a, ... , f by default. We work over the rationals and study the ideal generated by the twelve polynomials occurring on the middle of page 339 of the Tapas book in a project by De Boer and Pellikaan on the ternary cyclic code of length 11. Below these are named <code class="code">p1</code>, ..., <code class="code">p12</code>. The result should be the union of {a,b} and the set of 6 homogeneous binomials (that is, polynomials with two terms) of degree 2 forcing commuting between c, d, e, and f.

First load the package and set the standard infolevel <code class="func">InfoGBNP</code> (<a href="chap4.html#X82D40B0E84383BBC">4.2-1</a>) to 2 and the time infolevel <code class="func">InfoGBNPTime</code> (<a href="chap4.html#X7FAE244E80397B9A">4.3-1</a>) to 1 (for more information about the info level, see Chapter <a href="chap4.html#X79C5DF3782576D98">4</a>).

<div (* Title: HOL/TLA/Memory/MemClerk.thy Author: Stephan Merz, University of
gap> LoadPackage("gbnp", false);
true
gap> SetInfoLevel(InfoGBNP,2);
gap> SetInfoLevel(InfoGBNPTime,1);
</pre></div>

Now define some functions which will help in the construction of relations. The function <code class="code">powermon(g, exp)</code> will return the monomial g^exp. The function <code class="code">comm(a, b)</code> will return a relation forcing commutativity between its two arguments <code class="code">a</code> and <code class="code">b</code>.

<div class="example"><pre>
gap> powermon := function(base, exp)
> local ans,i;
> ans := [];
> for i in [1..exp] do ans := Concatenation(ans,[base]); od;
> return ans;
> end;;

gap> comm := function(a,b)
> return [[[a,b],[b,a]],[1,-1]];
> end;;
</pre></div>

Now the relations are entered.

<div class="example"><pre>
gap> p1 := [[[5,1]],[1]];;
gap> p2 := [[powermon(1,3),[6,1]],[1,1]];;
gap> p3 := [[powermon(1,9),Concatenation([3],powermon(1,3))],[1,1]];;
gap> p4 := [[powermon(1,81),Concatenation([3],powermon(1,9)),
> Concatenation([4],powermon(1,3))],[1,1,1]];;
gap> p5 := [[Concatenation([3],powermon(1,81)),Concatenation([4],powermon(1,9)),
> Concatenation([5],powermon(1,3))],[1,1,1]];;
gap> p6 := [[powermon(1,27),Concatenation([4],powermon(1,81)),Concatenation([5],
> powermon(1,9)),Concatenation([6],powermon(1,3))],[1,1,1,1]];;
gap> p7 := [[powermon(2,1),Concatenation([3],powermon(1,27)),Concatenation([5],
> powermon(1,81)),Concatenation([6],powermon(1,9))],[1,1,1,1]];;
gap> p8 := [[Concatenation([3],powermon(2,1)),Concatenation([4],powermon(1,27)),
> Concatenation([6],powermon(1,81))],[1,1,1]];;
gap> p9 := [[Concatenation([],powermon(1,1)),Concatenation([4],powermon(2,1)),
> Concatenation([5],powermon(1,27))],[1,1,1]];;
gap> p10 := [[Concatenation([3],powermon(1,1)),Concatenation([5],powermon(2,1)),
> Concatenation([6],powermon(1,27))],[1,1,1]];;
gap> p11 := [[Concatenation([4],powermon(1,1)),Concatenation([6],powermon(2,1))],
> [1,1]];;
gap> p12 := [[Concatenation([],powermon(2,3)),Concatenation([],powermon(2,1))],
> [1,-1]];;
gap> KI := [p1,p2,p3,p4,p5,p6,p7,p8,p9,p10,p11,p12];;
gap> for i in [1..5] do
> for j in [i+1..6] do
> Add(KI,comm(i,j));
> od;
> od;
</pre></div>

The relations can be shown with <code class="func">PrintNPList</code> (<a href="chap3.html#X832103DC79A9E9D0">3.2-3</a>):

<div class="example"><pre>
gap> PrintNPList(KI);
ea
a^3 + fa
a^9 + ca^3
a^81 + ca^9 + da^3
ca^81 + da^9 + ea^3
a^27 + da^81 + ea^9 + fa^3
b + ca^27 + ea^81 + fa^9
cb + da^27 + fa^81
a + db + ea^27
ca + eb + fa^27
da + fb
b^3 - b
ab - ba
ac - ca
ad - da
ae - ea
af - fa
bc - cb
bd - db
be - eb
bf - fb
cd - dc
ce - ec
cf - fc
de - ed
df - fd
ef - fe
gap> Length(KI);
27
</pre></div>

It is sometimes easier to enter the relations as elements of a free algebra and then use the function <code class="func">GP2NP</code> (<a href="chap3.html#X7B0EBCBC7857F1AE">3.1-1</a>) or the function <code class="func">GP2NPList</code> (<a href="chap3.html#X7CF0ED937DDA5A7E">3.1-2</a>) to convert them. This will be demonstrated below. More about converting can be read in Section <a href="chap3.html#X81ABB91B79E00229">3.1</a>.

<div class="example"><pre>
gap> F:=Rationals;;
gap> A:=FreeAssociativeAlgebraWithOne(F,"a","b","c","d","e","f");;
gap> a:=A.a;; b:=A.b;; c:=A.c;; d:=A.d;; e:=A.e;; f:=A.f;;
gap> KI_gp:=[e*a, #p1
> a^3 + f*a, #p2
> a^9 + c*a^3, #p3
> a^81 + c*a^9 + d*a^3, #p4
> c*a^81 + d*a^9 + e*a^3, #p5
> a^27 + d*a^81 + e*a^9 + f*a^3, #p6
> b + c*a^27 + e*a^81 + f*a^9, #p7
> c*b + d*a^27 + f*a^81, #p8
> a + d*b + e*a^27, #p9
> c*a + e*b + f*a^27, #p10
> d*a + f*b, #p11
> b^3 - b];; #p12
</pre></div>

These relations can be converted to NP form (see <a href="chap2.html#X7FDF3E5E7F33D3A2">2.1</a>) with <code class="func">GP2NPList</code> (<a href="chap3.html#X7CF0ED937DDA5A7E">3.1-2</a>). For use in a Gröbner basis computation we have to order the NP polynomials in <code class="code">KI</code>. This can be done with <code class="func">CleanNP</code> (<a href="chap3.html#X855F3D4C783000E3">3.3-7</a>).

<div class="example"><pre>
gap> KI_np:=GP2NPList(KI_gp);;
gap> Apply(KI,x->CleanNP(x));;
gap> KI_np=KI{[1..12]};
true
</pre></div>

The Gröbner basis can now be calculated with <code class="func">SGrobner</code> (<a href="chap3.html#X7FEDA29E78B0CEED">3.4-2</a>) and printed with <code class="func">PrintNPList</code> (<a href="chap3.html#X832103DC79A9E9D0">3.2-3</a>).

<div class="example"><pre>
gap> GB := SGrobner(KI);;
#I number of entered polynomials is 27
#I number of polynomials after reduction is 8
#I End of phase I
#I End of phase II
#I List of todo lengths is [ 0 ]
#I End of phase III
#I G: Cleaning finished, 0 polynomials reduced
#I End of phase IV
#I The computation took 748 msecs.
gap> PrintNPList(GB);
a
b
dc - cd
ec - ce
ed - de
fc - cf
fd - df
fe - ef
</pre></div>

<a id="X7C2CD4FA838EEE64" name="X7C2CD4FA838EEE64"></a>

<h4>A.8 The Birman-Murakami-Wenzl algebra of type A_3</h4>

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 <a href="chapBib.html#biBMR2124811">[CGW05]</a>.

First load the package and set the standard infolevel <code class="func">InfoGBNP</code> (<a href="chap4.html#X82D40B0E84383BBC">4.2-1</a>) to 1 and the time infolevel <code class="func">InfoGBNPTime</code> (<a href="chap4.html#X7FAE244E80397B9A">4.3-1</a>) to 1 (for more information about the info level, see Chapter <a href="chap4.html#X79C5DF3782576D98">4</a>).

<div class="example"><pre>
gap> LoadPackage("gbnp", false);
true
gap> SetInfoLevel(InfoGBNP,1);
gap> SetInfoLevel(InfoGBNPTime,1);
</pre></div>

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 <code class="func">GBNP.ConfigPrint</code> (<a href="chap3.html#X7F7510A878045D3A">3.2-2</a>)

<div class="example"><pre>
gap> GBNP.ConfigPrint("g1","g2","g3","e1","e2","e3");
</pre></div>

Now enter the relations. This will be done in NP form (see <a href="chap2.html#X7FDF3E5E7F33D3A2">2.1</a>). 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.

<div class="example"><pre>
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