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<head><title>pfaffians -- ideal generated by Pfaffians</title>
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<div><h1>pfaffians -- ideal generated by Pfaffians</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>pfaffians(n,M)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>n</tt>, <span>an <a href="___Z__Z.html">integer</a></span>, the size of the Pfaffians</span></li>
<li><span><tt>M</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, which is skew-symmetric, and whose ring is an integral domain</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Ideal.html">ideal</a></span>, the ideal generated by the Pfaffians of the <tt>n</tt> by <tt>n</tt> principal submatrices of <tt>M</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>The determinant of a skew-symmetric matrix <tt>N</tt>, i.e., a matrix for which <tt>transpose N + N == 0</tt>, is always a perfect square whose square root is called the Pfaffian of <tt>N</tt>.<table class="examples"><tr><td><pre>i1 : R = QQ[a..f];</pre>
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<tr><td><pre>i2 : M = genericSkewMatrix(R,a,4)
o2 = | 0 a b c |
| -a 0 d e |
| -b -d 0 f |
| -c -e -f 0 |
4 4
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : pfaffians(2,M)
o3 = ideal (a, b, d, c, e, f)
o3 : Ideal of R</pre>
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<tr><td><pre>i4 : pfaffians(4,M)
o4 = ideal(c*d - b*e + a*f)
o4 : Ideal of R</pre>
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The Pluecker embedding of <tt>Gr(2,6)</tt> and its secant variety:<table class="examples"><tr><td><pre>i5 : S = QQ[y_0..y_14];</pre>
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<tr><td><pre>i6 : M = genericSkewMatrix(S,y_0,6)
o6 = | 0 y_0 y_1 y_2 y_3 y_4 |
| -y_0 0 y_5 y_6 y_7 y_8 |
| -y_1 -y_5 0 y_9 y_10 y_11 |
| -y_2 -y_6 -y_9 0 y_12 y_13 |
| -y_3 -y_7 -y_10 -y_12 0 y_14 |
| -y_4 -y_8 -y_11 -y_13 -y_14 0 |
6 6
o6 : Matrix S <--- S</pre>
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<tr><td><pre>i7 : pluecker = pfaffians(4,M);
o7 : Ideal of S</pre>
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<tr><td><pre>i8 : betti res pluecker
0 1 2 3 4 5 6
o8 = total: 1 15 35 42 35 15 1
0: 1 . . . . . .
1: . 15 35 21 . . .
2: . . . 21 35 15 .
3: . . . . . . 1
o8 : BettiTally</pre>
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<tr><td><pre>i9 : secantvariety = pfaffians(6,M)
o9 = ideal(y y y - y y y - y y y + y y y + y y y - y y y + y y y
4 7 9 3 8 9 4 6 10 2 8 10 3 6 11 2 7 11 4 5 12
------------------------------------------------------------------------
- y y y + y y y - y y y + y y y - y y y + y y y - y y y
1 8 12 0 11 12 3 5 13 1 7 13 0 10 13 2 5 14 1 6 14
------------------------------------------------------------------------
+ y y y )
0 9 14
o9 : Ideal of S</pre>
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Pfaffians of a Moore matrix generate the ideal of a Heisenberg invariant elliptic normal curve in projective Fourspace:<table class="examples"><tr><td><pre>i10 : R = QQ[x_0..x_4]
o10 = R
o10 : PolynomialRing</pre>
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<tr><td><pre>i11 : y = {0,1,13,-13,-1}
o11 = {0, 1, 13, -13, -1}
o11 : List</pre>
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<tr><td><pre>i12 : M = matrix table(5,5, (i,j)-> x_((i+j)%5)*y_((i-j)%5))
o12 = | 0 -x_1 -13x_2 13x_3 x_4 |
| x_1 0 -x_3 -13x_4 13x_0 |
| 13x_2 x_3 0 -x_0 -13x_1 |
| -13x_3 13x_4 x_0 0 -x_2 |
| -x_4 -13x_0 13x_1 x_2 0 |
5 5
o12 : Matrix R <--- R</pre>
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<tr><td><pre>i13 : I = pfaffians(4,M);
o13 : Ideal of R</pre>
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<tr><td><pre>i14 : betti res I
0 1 2 3
o14 = total: 1 5 5 1
0: 1 . . .
1: . 5 5 .
2: . . . 1
o14 : BettiTally</pre>
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<div class="single"><h2>Caveat</h2>
<div>The algorithm used is a modified Gaussian reduction/Bareiss algorithm, which uses division and therefore we must assume that the ring of <tt>M</tt> is an integral domain.<p/>
The skew symmetry of <tt>M</tt> is not checked, but the algorithm proceeds as if it were, with somewhat unpredictable results!</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_determinant.html" title="determinant of a matrix">determinant</a> -- determinant of a matrix</span></li>
<li><span><a href="_matrices.html" title="">matrices</a></span></li>
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<div class="waystouse"><h2>Ways to use <tt>pfaffians</tt> :</h2>
<ul><li>pfaffians(ZZ,Matrix)</li>
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