Description
The determinant of a skew-symmetric matrix
N, i.e., a matrix for which
transpose N + N == 0, is always a perfect square whose square root is called the Pfaffian of
N.
i1 : R = QQ[a..f];
|
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
|
i3 : pfaffians(2,M)
o3 = ideal (a, b, d, c, e, f)
o3 : Ideal of R
|
i4 : pfaffians(4,M)
o4 = ideal(c*d - b*e + a*f)
o4 : Ideal of R
|
The Plücker embedding of
Gr(2,6) and its secant variety:
i5 : S = QQ[y_0..y_14];
|
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
|
i7 : pluecker = pfaffians(4,M);
o7 : Ideal of S
|
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
|
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
|
Pfaffians of a Moore matrix generate the ideal of a Heisenberg invariant elliptic normal curve in projective Fourspace:
i10 : R = QQ[x_0..x_4]
o10 = R
o10 : PolynomialRing
|
i11 : y = {0,1,13,-13,-1}
o11 = {0, 1, 13, -13, -1}
o11 : List
|
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
|
i13 : I = pfaffians(4,M);
o13 : Ideal of R
|
i14 : betti res I
0 1 2 3
o14 = total: 1 5 5 1
0: 1 . . .
1: . 5 5 .
2: . . . 1
o14 : BettiTally
|
Caveat
The algorithm used is a modified Gaussian reduction/Bareiss algorithm, which uses division and therefore we must assume that the ring of
M is an integral domain.
The skew symmetry of
M is not checked, but the algorithm proceeds as if it were, with somewhat unpredictable results!