Enumerative Geometry 05 Schubert Polynomial

Projection to Grassmannian
- What is the image of Schubert class?
- Permutation is Grassmanian if it has only one descent.
Given partitions, how to determine the Schubert structure coefficitents?
- Prieri’s Rule Analogue- Monk’s Formula.
  - Coefficient not positive combinatorial description.
  - AJS formula not positive (Open quest).

Equivariant Cohomology
- Space with symmetry (action of group) then invariance which accounts for symmetry (action).
  - \(EG\) contractible space with free G-Action. The quotient space \(EG \times X\) identifying action points. Cohomology of this space.
    - Example: Let \(G= \mathbb{C*}\). Infinite sphere is contractible.
- Approximate infinite by finite.
- Properties shared with ordinary cohomology.
  - Functorial for equivariant maps.
- Unique to Equivariant setting
  - Restricts to ordinary cohomology.
  - Determined at fixed locus.
    - Torus: invariant \(k\)-dimensional subspaces are generated by basis choose \(k\).
Equivariant Schubert Class
- Schubert Cell with position indexed by \(k\) element subset is T-invariant subvariety.