schubertCycle -- take a random Schubert cycle

Synopsis

Usage:

schubertCycle(a,G)
Inputs:
- a, a visible list, a list of integers $a = (a_0,\ldots,a_k)$ with $n-k\geq a_0 \geq \cdots \geq a_k \geq 0$
- G, a Grassmannian variety, which parameterizes the $k$-dimensional subspaces of $\mathbb{P}^n$
Optional inputs:
- Standard => ..., default value false
Outputs:
- an embedded projective variety, the Schubert cycle $\Sigma_a(\mathcal P)\subset\mathbb{G}(k,n)$ associated to a random complete flag $\mathcal P$ of nested projective subspace $\emptyset\subset P_0\subset \cdots \subset P_{n-1} \subset P_{n} = \mathbb{P}^n$ with $dim(P_i)=i$

For the general theory, see e.g. the book 3264 & All That - Intersection Theory in Algebraic Geometry, by D. Eisenbud and J. Harris.

i1 : G = GG(ZZ/33331,1,5);

o1 : ProjectiveVariety, GG(1,5)

i2 : S = schubertCycle({2,1},G)

o2 = S

o2 : ProjectiveVariety, 5-dimensional subvariety of PP^14 (subvariety of codimension 3 in G)

i3 : cycleClass S

o3 = s
      2,1

o3 : ZZ[s   , s   ]
         3,0   2,1