# Schubert Polynomials¶

sage.combinat.schubert_polynomial.SchubertPolynomialRing(R)

Returns the Schubert polynomial ring over R on the X basis.

EXAMPLES:

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]

class sage.combinat.schubert_polynomial.SchubertPolynomialRing_xbasis(R)

EXAMPLES:

sage: X = SchubertPolynomialRing(QQ)
True

Element

alias of SchubertPolynomial_class

class sage.combinat.schubert_polynomial.SchubertPolynomial_class(M, x)

Create a combinatorial module element. This should never be called directly, but only through the parent combinatorial free module’s __call__() method.

TESTS:

sage: F = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: B = F.basis()
sage: f = B['a'] + 3*B['c']; f
B['a'] + 3*B['c']
True

divided_difference(i)

EXAMPLES:

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]

expand()

EXAMPLES:

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())
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
sage: f.expand()
1
sage: f = X([1,2])
sage: type(f.expand())
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>
sage: f = X([1,3,2,4])
sage: type(f.expand())
<type 'sage.rings.polynomial.multi_polynomial_libsingular.MPolynomial_libsingular'>

multiply_variable(i)

Returns the Schubert polynomial obtained by multiplying self by the variable $$x_i$$.

EXAMPLES:

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]

scalar_product(x)

Returns the standard scalar product of self and x.

EXAMPLES:

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 = SymmetricFunctions(ZZ).schur()
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

sage.combinat.schubert_polynomial.is_SchubertPolynomial(x)

Returns True if x is a Schubert polynomial and False otherwise.

EXAMPLES:

sage: from sage.combinat.schubert_polynomial import is_SchubertPolynomial
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


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