# Singular

#### D.15.24.5 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); ==> // characteristic : 0 ==> // number of vars : 2 ==> // block 1 : ordering wp ==> // : names q(1) q(2) ==> // : weights 1 2 ==> // block 2 : ordering C ==> // quotient ring from ideal ... ```