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D.5.15.5 Grassmannian
Procedure from library schubert.lib (see schubert_lib).
- Usage:
- Grassmannian(k,n); k int, n int
- Return:
- variety
- Theory:
- create a Grassmannian G(k,n) as an abstract variety. This abstract
variety has diemnsion k(n-k) and its Chow ring is the quotient ring
of a polynomial ring in n-k variables q(1),...,q(n-k), which are the
Chern classes of tautological quotient bundle on G(k,n), modulo some
ideal generated by n-k polynomials which come from the Giambelli
formula. The monomial ordering of this Chow ring is 'wp' with vector
(1..k,1..n-k). Moreover, we export the Chern characters of
tautological subbundle and quotient bundle on G(k,n)
(say 'subBundle' and 'quotientBundle').
Example:
| LIB "schubert.lib";
variety G24 = Grassmannian(2,4);
G24;
==> A variety of dimension 4
==>
def r = G24.baseRing;
setring r;
subBundle;
==> 1/6*q(1)*q(2)-1/2*q(1)^2+q(2)-q(1)+2
quotientBundle;
==> -1/6*q(1)*q(2)+1/2*q(1)^2-q(2)+q(1)+2
G24.dimension;
==> 4
G24.relations;
==> _[1]=q(1)^3-2*q(1)*q(2)
==> _[2]=q(1)^4-3*q(1)^2*q(2)+q(2)^2
ChowRing(G24);
==> // coefficients: QQ
==> // number of vars : 2
==> // block 1 : ordering wp
==> // : names q(1) q(2)
==> // : weights 1 2
==> // block 2 : ordering C
==> // quotient ring from ideal ...
| See also:
projectiveBundle;
projectiveSpace.
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