Struct CSMClass 

pub struct CSMClass {
    pub coefficients: BTreeMap<Partition, i64>,
    pub grassmannian: (usize, usize),
}

CSM class of a constructible set, expanded in the Schubert basis of a Grassmannian.

c_SM(Z) = Σ_λ a_λ · σ_λ

where a_λ are integer coefficients.

Fields

coefficients: BTreeMap<Partition, i64>

Expansion coefficients in the Schubert basis: partition → coefficient

grassmannian: (usize, usize)

Grassmannian parameters

Implementations

impl CSMClass

pub fn of_schubert_cell( partition: &[usize], grassmannian: (usize, usize), ) -> Self

CSM class of a Schubert cell Ω°_λ in Gr(k, n).

For a Schubert cell (open stratum), the CSM class is computed via inclusion-exclusion from the CSM classes of Schubert varieties:

c_SM(Ω°_λ) = Σ_{μ ≤ λ} (-1)^{|μ|-|λ|} c_SM(Ω_μ)

For the top cell (empty partition), c_SM = σ_∅ = [Gr].

Contract

requires: partition fits in k × (n-k) box
ensures: euler_characteristic() == 1 for any cell

pub fn of_schubert_variety( partition: &[usize], grassmannian: (usize, usize), ) -> Self

CSM class of a Schubert variety Ω_λ (closure of the cell).

c_SM(Ω_λ) = Σ_{μ ≥ λ} c_SM(Ω°_μ)

where the sum is over all partitions μ in the Bruhat order above λ.

Contract

requires: partition fits in k × (n-k) box
ensures: result.coefficients is nonempty

pub fn euler_characteristic(&self) -> i64

Euler characteristic: the degree of the CSM class.

Equal to the coefficient of the point class σ_{m^k} (fundamental class of the Grassmannian).

Contract

ensures: for a Schubert cell, result == 1

pub fn csm_intersection(&self, other: &CSMClass) -> CSMClass

Multiply two CSM classes via the intersection pairing.

Uses Littlewood-Richardson coefficients to expand the product in the Schubert basis.

Contract

requires: self.grassmannian == other.grassmannian
ensures: result is the product in the Schubert basis ring

Trait Implementations

impl Clone for CSMClass

fn clone(&self) -> CSMClass

Returns a duplicate of the value.

fn clone_from(&mut self, source: &Self)

Performs copy-assignment from source.

impl Debug for CSMClass

fn fmt(&self, f: &mut Formatter<'_>) -> Result

Formats the value using the given formatter.

impl PartialEq for CSMClass

fn eq(&self, other: &CSMClass) -> bool

Tests for self and other values to be equal, and is used by ==.

fn ne(&self, other: &Rhs) -> bool

Tests for !=. The default implementation is almost always sufficient, and should not be overridden without very good reason.

impl Eq for CSMClass

impl StructuralPartialEq for CSMClass