cycleClass -- determine the expression of the class of a cycle as a linear combination of Schubert classes

Synopsis

• Usage:

  cycleClass C

• Inputs:
  • C, an ideal, an ideal in Grass$(k, n)$ representing a cycle of pure codimension $m$ in the Grassmannian of $k$-dimensional subspaces of $\mathbb{P}^n$
• Outputs:
  • a ring element, the expression of the class of the cycle as a linear combination of Schubert classes

Description

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

i1 : G = Grass(2,5,ZZ/33331);

i2 : C = schubertCycle({3,2,1},G);

o2 : Ideal of G

i3 : time cycleClass C
     -- used 1.59114 seconds

o3 = s
      3,2,1

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

i4 : C' = intersect(C,schubertCycle({2,2,2},G));

o4 : Ideal of G

i5 : time cycleClass C'
     -- used 2.07487 seconds

o5 = s      + s
      3,2,1    2,2,2

o5 : ZZ[s     , s     , s     ]
         3,3,0   3,2,1   2,2,2

See also

• schubertCycle -- take a random Schubert cycle