Module: sage.combinat.schubert_polynomial
Schubert Polynomials
Module-level Functions
R) |
Returns the Schubert polynomial ring over R on the X basis.
sage: X = SchubertPolynomialRing(ZZ); X Schubert polynomial ring with X basis over Integer Ring sage: X(1) X[1] sage: X([1,2,3])*X([2,1,3]) X[2, 1] sage: X([2,1,3])*X([2,1,3]) X[3, 1, 2] sage: X([2,1,3])+X([3,1,2,4]) X[2, 1] + X[3, 1, 2] sage: a = X([2,1,3])+X([3,1,2,4]) sage: a^2 X[3, 1, 2] + 2*X[4, 1, 2, 3] + X[5, 1, 2, 3, 4]
x) |
Returns True if x is a Schubert polynomial and False otherwise.
sage: X = SchubertPolynomialRing(ZZ) sage: a = 1 sage: is_SchubertPolynomial(a) False sage: b = X(1) sage: is_SchubertPolynomial(b) True sage: c = X([2,1,3]) sage: is_SchubertPolynomial(c) True
Class: SchubertPolynomial_class
Functions: divided_difference,
expand,
multiply_variable,
scalar_product
self, i) |
sage: X = SchubertPolynomialRing(ZZ) sage: a = X([3,2,1]) sage: a.divided_difference(1) X[2, 3, 1] sage: a.divided_difference([3,2,1]) X[1]
self) |
sage: X = SchubertPolynomialRing(ZZ) sage: X([2,1,3]).expand() x0 sage: map(lambda x: x.expand(), [X(p) for p in Permutations(3)]) [1, x0 + x1, x0, x0*x1, x0^2, x0^2*x1]
TESTS: Calling .expand() should always return an element of an MPolynomialRing
sage: X = SchubertPolynomialRing(ZZ) sage: f = X([1]); f X[1] sage: type(f.expand()) <class 'sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict '> sage: f.expand() 1 sage: f = X([1,2]) sage: type(f.expand()) <class 'sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict '> sage: f = X([1,3,2,4]) sage: type(f.expand()) <class 'sage.rings.polynomial.multi_polynomial_element.MPolynomial_polydict '>
self, i) |
Returns the Schubert polynomial obtained by multiplying self by the variable x_i.
sage: X = SchubertPolynomialRing(ZZ) sage: a = X([3,2,4,1]) sage: a.multiply_variable(0) X[4, 2, 3, 1] sage: a.multiply_variable(1) X[3, 4, 2, 1] sage: a.multiply_variable(2) X[3, 2, 5, 1, 4] - X[3, 4, 2, 1] - X[4, 2, 3, 1] sage: a.multiply_variable(3) X[3, 2, 4, 5, 1]
self, x) |
Returns the standard scalar product of self and x.
sage: X = SchubertPolynomialRing(ZZ) sage: a = X([3,2,4,1]) sage: a.scalar_product(a) 0 sage: b = X([4,3,2,1]) sage: b.scalar_product(a) X[1, 3, 4, 6, 2, 5] sage: Permutation([1, 3, 4, 6, 2, 5, 7]).to_lehmer_code() [0, 1, 1, 2, 0, 0, 0] sage: s = SFASchur(ZZ) sage: c = s([2,1,1]) sage: b.scalar_product(a).expand() x0^2*x1*x2 + x0*x1^2*x2 + x0*x1*x2^2 + x0^2*x1*x3 + x0*x1^2*x3 + x0^2*x2*x3 + 3*x0*x1*x2*x3 + x1^2*x2*x3 + x0*x2^2*x3 + x1*x2^2*x3 + x0*x1*x3^2 + x0*x2*x3^2 + x1*x2*x3^2 sage: c.expand(4) x0^2*x1*x2 + x0*x1^2*x2 + x0*x1*x2^2 + x0^2*x1*x3 + x0*x1^2*x3 + x0^2*x2*x3 + 3*x0*x1*x2*x3 + x1^2*x2*x3 + x0*x2^2*x3 + x1*x2^2*x3 + x0*x1*x3^2 + x0*x2*x3^2 + x1*x2*x3^2
Class: SchubertPolynomialRing_xbasis
self, R) |
sage: X = SchubertPolynomialRing(QQ) sage: X == loads(dumps(X)) True
Special Functions: __init__,
_coerce_start,
_multiply_basis
self, x) |
Coerce x into self.
sage: X = SchubertPolynomialRing(QQ) sage: X._coerce_start([2,1,3]) X[2, 1] sage: X._coerce_start(Permutation([2,1,3])) X[2, 1]
sage: R.<x1, x2, x3> = QQ[] sage: X(x1^2*x2) X[3, 2, 1]
self, left, right) |
sage: p1 = Permutation([3,2,1]) sage: p2 = Permutation([2,1,3]) sage: X = SchubertPolynomialRing(QQ) sage: X._multiply_basis(p1,p2) {[4, 2, 1, 3]: 1}