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

## Synopsis

• Usage:
cycleClass C
• Inputs:
• C, , a subvariety of GG$(k, n)$ representing a cycle of pure codimension $m$ in the Grassmannian of $k$-dimensional subspaces of $\mathbb{P}^n$
• Outputs:
• , 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 = GG(ZZ/33331,2,5); o1 : ProjectiveVariety, GG(2,5) i2 : C = schubertCycle({3,2,1},G); o2 : ProjectiveVariety, threefold in PP^19 (subvariety of codimension 6 in G) i3 : cycleClass C o3 = s 3,2,1 o3 : ZZ[s , s , s ] 3,3,0 3,2,1 2,2,2 i4 : C' = C + schubertCycle({2,2,2},G); o4 : ProjectiveVariety, threefold in PP^19 (subvariety of codimension 6 in G) i5 : cycleClass C' o5 = s + s 3,2,1 2,2,2 o5 : ZZ[s , s , s ] 3,3,0 3,2,1 2,2,2