schubmult.rings.polynomial_algebra.monomial_basis¶
MonomialBasis: the standard monomial basis x^a (keys are exponent tuples) of PolynomialAlgebra.
This is the hub basis: every other PolynomialBasis transitions through it by default.
MonomialBasis Objects¶
class MonomialBasis(PolynomialBasis)
Standard monomial basis for the polynomial algebra.
Keys are tuples of nonnegative integers representing exponent vectors.
This is the fundamental basis through which other bases transition
by default, and is dual to the :class:WordBasis of the free algebra.
coproduct¶
def coproduct(key)
Compute the deconcatenation coproduct on a monomial key.
product¶
def product(key1, key2, coeff=S.One)
Multiply two monomial keys by component-wise addition of exponents.
expand_monom¶
def expand_monom(monom)
Convert an exponent tuple to a monomial expression in the generating set.
expand¶
def expand(dct)
Expand a dict of monomial keys into a symbolic polynomial expression.
transition_slide¶
def transition_slide(dct, other_basis)
Transition a monomial dict to a slide-type basis via triangular inversion.
transition_slide_monom¶
def transition_slide_monom(other_basis, monom, coeff=S.One)
Express a single monomial in a slide-type basis by dominance-order inversion.
transition_anti_schubert¶
def transition_anti_schubert(dct, other_basis)
Transition monomials to the anti-Schubert basis by reversing exponents.
transition_schubert¶
def transition_schubert(dct)
Transition monomials to the Schubert basis by grouping by length.
transition_double_forest¶
def transition_double_forest(dct, other_basis)
Transition monomials to DoubleForestPolyBasis via ForestPolyBasis.
dual_basis¶
@classmethod
def dual_basis(cls)
Return the dual free algebra basis class (:class:WordBasis).
transition¶
def transition(other_basis)
Return a transition function from monomial basis to other_basis.
from_expr¶
def from_expr(expr, length=None)
Parse a symbolic expression into monomial-basis coefficient dict.