schubmult.rings.polynomial_algebra.lascoux_poly_basis¶

LascouxPolyBasis: the Lascoux polynomial (K-theoretic key polynomial) basis of PolynomialAlgebra.

LascouxPolyBasis Objects¶

class LascouxPolyBasis(PolynomialBasis)

Lascoux polynomial basis.

Keys are weak compositions. Lascoux polynomials provide a basis that refines Grothendieck polynomials and coarsens monomials, with an efficient combinatorial product rule.

to_monoms¶

def to_monoms(key)

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

dual_basis¶

@classmethod
def dual_basis(cls)

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

expand¶

def expand(dct)

Expand a Lascoux basis dict into a symbolic polynomial expression.

transition_monomial¶

def transition_monomial(dct)

Transition from Lascoux basis to monomial basis.

transition_glide_key¶

def transition_glide_key(key)

Transition a Lascoux key to the glide basis.

transition_glide¶

def transition_glide(dct)

Transition from Lascoux basis to glide basis.

transition¶

def transition(other_basis)

Return a transition function from Lascoux basis to other_basis.

product¶

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

Multiply two Lascoux keys using the Lascoux product rule.