schubmult.rings.polynomial_algebra.grothendieck_poly_basis

GrothendieckPolyBasis: the Grothendieck polynomial basis of PolynomialAlgebra, at the specialization beta = 1 (without loss of generality: beta is recovered from the grading, since the degree inv(w) + d part of G_w carries beta^d).

GrothendieckPolyBasis Objects

class GrothendieckPolyBasis(PolynomialBasis)

Grothendieck polynomial basis at beta = 1.

Keys are (Permutation, length) pairs. Grothendieck polynomials form the canonical basis for the polynomial algebra in Grothendieck calculus, dual to the :class:GrothendieckBasis of the free algebra. The beta parameter is set to 1 without loss of generality (see the module docstring).

product

def product(key1, key2, coeff=S.One)

Multiply two Grothendieck keys using the Grothendieck ring multiplication.

transition_schubert

def transition_schubert(dct)

Transition from Grothendieck basis to separated descents basis.

transition_glide_key

def transition_glide_key(key)

Decompose a Grothendieck polynomial into glide polynomials via omega insertion on RC-graphs.

transition_glide

def transition_glide(dct)

Transition a Grothendieck dict to the glide polynomial basis.

to_monoms

def to_monoms(key)

Expand a Grothendieck key into a dict of monomial exponent tuples.

dual_basis

@classmethod
def dual_basis(cls)

Return the dual free algebra basis class (:class:GrothendieckBasis).

transition_grove_key

def transition_grove_key(key)

Decompose a Grothendieck polynomial into grove polynomials via omega insertion on RC-graphs.

transition_grove

def transition_grove(dct)

Transition a Grothendieck dict to the grove polynomial basis.

transition_lascoux_key

def transition_lascoux_key(key)

Decompose a Grothendieck polynomial into Lascoux polynomials via omega insertion on RC-graphs.

transition_lascoux

def transition_lascoux(dct)

Transition a Grothendieck dict to the Lascoux polynomial basis.

transition

def transition(other_basis)

Return a transition function from Grothendieck basis to other_basis.