1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
|
-- Bernd Sturmfels and Josephine Yu
needs "matrix1.m2"
--signOfPermu := P -> (
-- sign := 1;
-- scan(#P-1, a->
-- scan(a+1..#P-1, b ->
-- if P#a > P#b then
-- sign = -1 * sign
-- else if P#a == P#b then
-- sign = 0
-- ));
-- sign
-- )
signAndShuffle := (a,b) -> (
ct := 0;
i := 0; m := #a;
j := 0; n := #b;
sh := while a#?i or b#?j list (
t := if a#?i then a#i;
u := if b#?j then b#j;
if t === u then return (0,);
if t === null then (j += 1; u)
else if u === null or t < u then (i += 1; t)
else (ct = ct + m-i; j += 1; u));
((-1)^ct, toSequence sh));
signOfShuffle := (a,b) -> (
ct := 0;
i := 0; m := #a;
j := 0; n := #b;
while i<m and j<n do (
if a#i == b#j then return 0;
if a#i < b#j then i += 1
else (ct = ct + m-i; j += 1));
(-1)^ct);
fixupw = w -> if instance(w,String) then getSymbol w else w
Grassmannian = method(
TypicalValue => Ideal,
Options => {
CoefficientRing => ZZ,
Variable => monoidDefaults.VariableBaseName -- this is a string, so use fixupw
});
Schubert = method(TypicalValue => Ideal, Options => options Grassmannian);
Grassmannian(ZZ,ZZ):= o -> (k,n) -> Schubert(k,n,n-k..n,o)
Grassmannian(ZZ,ZZ,PolynomialRing) := o -> (k,n,R) -> (
I := Grassmannian(k,n,o);
S := ring I;
if numgens R < numgens S then error ("expected a ring with at least ",toString numgens S," generators");
f := map(R,S,apply(numgens S, i -> R_i));
f I)
Schubert(ZZ, ZZ, VisibleList) := o -> (k,n,sigma) -> (
L := toSequence \ subsets(n+1,k+1);
R := o.CoefficientRing (monoid [apply(L, i -> new IndexedVariable from {baseName fixupw o.Variable,unsequence i})]);
vr := new HashTable from apply(#L, i -> L#i => R_i);
higher := apply( select( L, s -> any(s, sigma, (a,b) -> a>b)), s -> vr#s );
G := flatten for i from 0 to #L-1 list for j from i+1 to #L-1 list (
r := L#i;
s := L#j;
up := position(r, s, (a,b) -> a>b);
if up === null then continue;
snake := s_{0..up} | r_{up..k};
sum for s1 in subsets(snake,up+1) list (
s2 := snake - set s1;
(sgn1,c) := signAndShuffle(r_{0..up-1}, s2);
if sgn1 == 0 then continue;
(sgn2,d) := signAndShuffle(s1, s_{up+1..k});
if sgn2 == 0 then continue;
sgn := sgn1 * sgn2 * signOfShuffle(s1,s2);
sgn * vr#c * vr#d));
g := generators forceGB matrix(R,{(G|higher)});
forceGB g;
ideal g)
-- Local Variables:
-- compile-command: "make -C $M2BUILDDIR/Macaulay2/m2 "
-- End:
|
