Key polynomials

Key polynomials (also known as type A Demazure characters) are defined by applying the divided difference operator \(\pi_\sigma\), where \(\sigma\) is a permutation, to a monomial corresponding to an integer partition \(\mu \vdash n\).

See also

For Demazure characters in other types, see

• sage.combinat.root_system.weyl_characters.WeylCharacterRing.demazure_character()

• sage.categories.classical_crystals.ClassicalCrystals.ParentMethods.demazure_character()

AUTHORS:

• Trevor K. Karn (2022-08-17): initial version

class sage.combinat.key_polynomial.KeyPolynomial

Bases: IndexedFreeModuleElement

A key polynomial.

Key polynomials are polynomials that form a basis for a polynomial ring and are indexed by weak compositions.

Elements should be created by first creating the basis KeyPolynomialBasis and passing a list representing the indexing composition.

EXAMPLES:

Sage

sage: k = KeyPolynomials(QQ)
sage: f = k([4,3,2,1]) + k([1,2,3,4]); f
k[1, 2, 3, 4] + k[4, 3, 2, 1]
sage: f in k
True

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ)
>>> f = k([Integer(4),Integer(3),Integer(2),Integer(1)]) + k([Integer(1),Integer(2),Integer(3),Integer(4)]); f
k[1, 2, 3, 4] + k[4, 3, 2, 1]
>>> f in k
True

divided_difference(w)

Apply the divided difference operator \(\partial_w\) to self.

The convention is to apply from left to right so if w = [w1, w2, ..., wm] then we apply \(\partial_{w_2 \cdots w_m} \circ \partial_{w_1}\)

EXAMPLES:

Sage

sage: k = KeyPolynomials(QQ)
sage: k([3,2,1]).divided_difference(2)
k[3, 1, 1]
sage: k([3,2,1]).divided_difference([2,3])
k[3, 1]

sage: k = KeyPolynomials(QQ, 4)
sage: k([3,2,1,0]).divided_difference(2)
k[3, 1, 1, 0]

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ)
>>> k([Integer(3),Integer(2),Integer(1)]).divided_difference(Integer(2))
k[3, 1, 1]
>>> k([Integer(3),Integer(2),Integer(1)]).divided_difference([Integer(2),Integer(3)])
k[3, 1]

>>> k = KeyPolynomials(QQ, Integer(4))
>>> k([Integer(3),Integer(2),Integer(1),Integer(0)]).divided_difference(Integer(2))
k[3, 1, 1, 0]

expand()

Return self written in the monomial basis (i.e., as an element in the corresponding polynomial ring).

EXAMPLES:

Sage

sage: k = KeyPolynomials(QQ)
sage: f = k([4,3,2,1])
sage: f.expand()
z_3*z_2^2*z_1^3*z_0^4

sage: f = k([1,2,3])
sage: f.expand()
z_2^3*z_1^2*z_0 + z_2^3*z_1*z_0^2 + z_2^2*z_1^3*z_0
 + 2*z_2^2*z_1^2*z_0^2 + z_2^2*z_1*z_0^3 + z_2*z_1^3*z_0^2
 + z_2*z_1^2*z_0^3

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ)
>>> f = k([Integer(4),Integer(3),Integer(2),Integer(1)])
>>> f.expand()
z_3*z_2^2*z_1^3*z_0^4

>>> f = k([Integer(1),Integer(2),Integer(3)])
>>> f.expand()
z_2^3*z_1^2*z_0 + z_2^3*z_1*z_0^2 + z_2^2*z_1^3*z_0
 + 2*z_2^2*z_1^2*z_0^2 + z_2^2*z_1*z_0^3 + z_2*z_1^3*z_0^2
 + z_2*z_1^2*z_0^3

isobaric_divided_difference(w)

Apply the operator \(\pi_w\) to self.

w may be either a Permutation or a list of indices of simple transpositions (1-based).

The convention is to apply from left to right so if w = [w1, w2, ..., wm] then we apply \(\pi_{w_2 \cdots w_m} \circ \pi_{w_1}\)

EXAMPLES:

Sage

sage: k = KeyPolynomials(QQ)
sage: k([3,2,1]).pi(2)
k[3, 1, 2]
sage: k([3,2,1]).pi([2,1])
k[1, 3, 2]
sage: k([3,2,1]).pi(Permutation([3,2,1]))
k[1, 2, 3]
sage: f = k([3,2,1]) + k([3,2,1,1])
sage: f.pi(2)
k[3, 1, 2] + k[3, 1, 2, 1]
sage: k.one().pi(1)
k[]

sage: k([3,2,1,0]).pi(2).pi(2)
k[3, 1, 2]
sage: (-k([3,2,1,0]) + 4*k([3,1,2,0])).pi(2)
3*k[3, 1, 2]

sage: k = KeyPolynomials(QQ, 4)
sage: k([3,2,1,0]).pi(2)
k[3, 1, 2, 0]
sage: k([3,2,1,0]).pi([2,1])
k[1, 3, 2, 0]
sage: k([3,2,1,0]).pi(Permutation([3,2,1,4]))
k[1, 2, 3, 0]
sage: f = k([3,2,1,0]) + k([3,2,1,1])
sage: f.pi(2)
k[3, 1, 2, 0] + k[3, 1, 2, 1]
sage: k.one().pi(1)
k[0, 0, 0, 0]

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ)
>>> k([Integer(3),Integer(2),Integer(1)]).pi(Integer(2))
k[3, 1, 2]
>>> k([Integer(3),Integer(2),Integer(1)]).pi([Integer(2),Integer(1)])
k[1, 3, 2]
>>> k([Integer(3),Integer(2),Integer(1)]).pi(Permutation([Integer(3),Integer(2),Integer(1)]))
k[1, 2, 3]
>>> f = k([Integer(3),Integer(2),Integer(1)]) + k([Integer(3),Integer(2),Integer(1),Integer(1)])
>>> f.pi(Integer(2))
k[3, 1, 2] + k[3, 1, 2, 1]
>>> k.one().pi(Integer(1))
k[]

>>> k([Integer(3),Integer(2),Integer(1),Integer(0)]).pi(Integer(2)).pi(Integer(2))
k[3, 1, 2]
>>> (-k([Integer(3),Integer(2),Integer(1),Integer(0)]) + Integer(4)*k([Integer(3),Integer(1),Integer(2),Integer(0)])).pi(Integer(2))
3*k[3, 1, 2]

>>> k = KeyPolynomials(QQ, Integer(4))
>>> k([Integer(3),Integer(2),Integer(1),Integer(0)]).pi(Integer(2))
k[3, 1, 2, 0]
>>> k([Integer(3),Integer(2),Integer(1),Integer(0)]).pi([Integer(2),Integer(1)])
k[1, 3, 2, 0]
>>> k([Integer(3),Integer(2),Integer(1),Integer(0)]).pi(Permutation([Integer(3),Integer(2),Integer(1),Integer(4)]))
k[1, 2, 3, 0]
>>> f = k([Integer(3),Integer(2),Integer(1),Integer(0)]) + k([Integer(3),Integer(2),Integer(1),Integer(1)])
>>> f.pi(Integer(2))
k[3, 1, 2, 0] + k[3, 1, 2, 1]
>>> k.one().pi(Integer(1))
k[0, 0, 0, 0]

pi(w)

Apply the operator \(\pi_w\) to self.

w may be either a Permutation or a list of indices of simple transpositions (1-based).

The convention is to apply from left to right so if w = [w1, w2, ..., wm] then we apply \(\pi_{w_2 \cdots w_m} \circ \pi_{w_1}\)

EXAMPLES:

Sage

sage: k = KeyPolynomials(QQ)
sage: k([3,2,1]).pi(2)
k[3, 1, 2]
sage: k([3,2,1]).pi([2,1])
k[1, 3, 2]
sage: k([3,2,1]).pi(Permutation([3,2,1]))
k[1, 2, 3]
sage: f = k([3,2,1]) + k([3,2,1,1])
sage: f.pi(2)
k[3, 1, 2] + k[3, 1, 2, 1]
sage: k.one().pi(1)
k[]

sage: k([3,2,1,0]).pi(2).pi(2)
k[3, 1, 2]
sage: (-k([3,2,1,0]) + 4*k([3,1,2,0])).pi(2)
3*k[3, 1, 2]

sage: k = KeyPolynomials(QQ, 4)
sage: k([3,2,1,0]).pi(2)
k[3, 1, 2, 0]
sage: k([3,2,1,0]).pi([2,1])
k[1, 3, 2, 0]
sage: k([3,2,1,0]).pi(Permutation([3,2,1,4]))
k[1, 2, 3, 0]
sage: f = k([3,2,1,0]) + k([3,2,1,1])
sage: f.pi(2)
k[3, 1, 2, 0] + k[3, 1, 2, 1]
sage: k.one().pi(1)
k[0, 0, 0, 0]

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ)
>>> k([Integer(3),Integer(2),Integer(1)]).pi(Integer(2))
k[3, 1, 2]
>>> k([Integer(3),Integer(2),Integer(1)]).pi([Integer(2),Integer(1)])
k[1, 3, 2]
>>> k([Integer(3),Integer(2),Integer(1)]).pi(Permutation([Integer(3),Integer(2),Integer(1)]))
k[1, 2, 3]
>>> f = k([Integer(3),Integer(2),Integer(1)]) + k([Integer(3),Integer(2),Integer(1),Integer(1)])
>>> f.pi(Integer(2))
k[3, 1, 2] + k[3, 1, 2, 1]
>>> k.one().pi(Integer(1))
k[]

>>> k([Integer(3),Integer(2),Integer(1),Integer(0)]).pi(Integer(2)).pi(Integer(2))
k[3, 1, 2]
>>> (-k([Integer(3),Integer(2),Integer(1),Integer(0)]) + Integer(4)*k([Integer(3),Integer(1),Integer(2),Integer(0)])).pi(Integer(2))
3*k[3, 1, 2]

>>> k = KeyPolynomials(QQ, Integer(4))
>>> k([Integer(3),Integer(2),Integer(1),Integer(0)]).pi(Integer(2))
k[3, 1, 2, 0]
>>> k([Integer(3),Integer(2),Integer(1),Integer(0)]).pi([Integer(2),Integer(1)])
k[1, 3, 2, 0]
>>> k([Integer(3),Integer(2),Integer(1),Integer(0)]).pi(Permutation([Integer(3),Integer(2),Integer(1),Integer(4)]))
k[1, 2, 3, 0]
>>> f = k([Integer(3),Integer(2),Integer(1),Integer(0)]) + k([Integer(3),Integer(2),Integer(1),Integer(1)])
>>> f.pi(Integer(2))
k[3, 1, 2, 0] + k[3, 1, 2, 1]
>>> k.one().pi(Integer(1))
k[0, 0, 0, 0]

to_polynomial()

Return self written in the monomial basis (i.e., as an element in the corresponding polynomial ring).

EXAMPLES:

Sage

sage: k = KeyPolynomials(QQ)
sage: f = k([4,3,2,1])
sage: f.expand()
z_3*z_2^2*z_1^3*z_0^4

sage: f = k([1,2,3])
sage: f.expand()
z_2^3*z_1^2*z_0 + z_2^3*z_1*z_0^2 + z_2^2*z_1^3*z_0
 + 2*z_2^2*z_1^2*z_0^2 + z_2^2*z_1*z_0^3 + z_2*z_1^3*z_0^2
 + z_2*z_1^2*z_0^3

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ)
>>> f = k([Integer(4),Integer(3),Integer(2),Integer(1)])
>>> f.expand()
z_3*z_2^2*z_1^3*z_0^4

>>> f = k([Integer(1),Integer(2),Integer(3)])
>>> f.expand()
z_2^3*z_1^2*z_0 + z_2^3*z_1*z_0^2 + z_2^2*z_1^3*z_0
 + 2*z_2^2*z_1^2*z_0^2 + z_2^2*z_1*z_0^3 + z_2*z_1^3*z_0^2
 + z_2*z_1^2*z_0^3

class sage.combinat.key_polynomial.KeyPolynomialBasis(R=None, k=None, poly_ring=None)

Bases: CombinatorialFreeModule

The key polynomial basis for a polynomial ring.

For a full definition, see SymmetricFunctions.com. Key polynomials are indexed by weak compositions with no trailing zeros, and \(\sigma\) is the permutation of shortest length which sorts the indexing composition into a partition.

EXAMPLES:

Key polynomials are a basis, indexed by (weak) compositions, for polynomial rings:

Sage

sage: k = KeyPolynomials(QQ)
sage: k([3,0,1,2])
k[3, 0, 1, 2]
sage: k([3,0,1,2])/2
1/2*k[3, 0, 1, 2]
sage: R = k.polynomial_ring(); R
Infinite polynomial ring in z over Rational Field

sage: K = KeyPolynomials(GF(5)); K
Key polynomial basis over Finite Field of size 5
sage: 2*K([3,0,1,2])
2*k[3, 0, 1, 2]
sage: 5*(K([3,0,1,2]) + K([3,1,1]))
0

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ)
>>> k([Integer(3),Integer(0),Integer(1),Integer(2)])
k[3, 0, 1, 2]
>>> k([Integer(3),Integer(0),Integer(1),Integer(2)])/Integer(2)
1/2*k[3, 0, 1, 2]
>>> R = k.polynomial_ring(); R
Infinite polynomial ring in z over Rational Field

>>> K = KeyPolynomials(GF(Integer(5))); K
Key polynomial basis over Finite Field of size 5
>>> Integer(2)*K([Integer(3),Integer(0),Integer(1),Integer(2)])
2*k[3, 0, 1, 2]
>>> Integer(5)*(K([Integer(3),Integer(0),Integer(1),Integer(2)]) + K([Integer(3),Integer(1),Integer(1)]))
0

We can expand them in the standard monomial basis:

Sage

sage: k([3,0,1,2]).expand()
z_3^2*z_2*z_0^3 + z_3^2*z_1*z_0^3 + z_3*z_2^2*z_0^3
 + 2*z_3*z_2*z_1*z_0^3 + z_3*z_1^2*z_0^3 + z_2^2*z_1*z_0^3
 + z_2*z_1^2*z_0^3

sage: k([0,0,2]).expand()
z_2^2 + z_2*z_1 + z_2*z_0 + z_1^2 + z_1*z_0 + z_0^2

Python

>>> from sage.all import *
>>> k([Integer(3),Integer(0),Integer(1),Integer(2)]).expand()
z_3^2*z_2*z_0^3 + z_3^2*z_1*z_0^3 + z_3*z_2^2*z_0^3
 + 2*z_3*z_2*z_1*z_0^3 + z_3*z_1^2*z_0^3 + z_2^2*z_1*z_0^3
 + z_2*z_1^2*z_0^3

>>> k([Integer(0),Integer(0),Integer(2)]).expand()
z_2^2 + z_2*z_1 + z_2*z_0 + z_1^2 + z_1*z_0 + z_0^2

If we have a polynomial, we can express it in the key basis:

Sage

sage: z = R.gen()
sage: k.from_polynomial(z[2]^2*z[1]*z[0])
k[1, 1, 2] - k[1, 2, 1]

sage: f = z[3]^2*z[2]*z[0]^3 + z[3]^2*z[1]*z[0]^3 + z[3]*z[2]^2*z[0]^3 + \
....: 2*z[3]*z[2]*z[1]*z[0]^3 + z[3]*z[1]^2*z[0]^3 + z[2]^2*z[1]*z[0]^3 + \
....: z[2]*z[1]^2*z[0]^3
sage: k.from_polynomial(f)
k[3, 0, 1, 2]

Python

>>> from sage.all import *
>>> z = R.gen()
>>> k.from_polynomial(z[Integer(2)]**Integer(2)*z[Integer(1)]*z[Integer(0)])
k[1, 1, 2] - k[1, 2, 1]

>>> f = z[Integer(3)]**Integer(2)*z[Integer(2)]*z[Integer(0)]**Integer(3) + z[Integer(3)]**Integer(2)*z[Integer(1)]*z[Integer(0)]**Integer(3) + z[Integer(3)]*z[Integer(2)]**Integer(2)*z[Integer(0)]**Integer(3) + Integer(2)*z[Integer(3)]*z[Integer(2)]*z[Integer(1)]*z[Integer(0)]**Integer(3) + z[Integer(3)]*z[Integer(1)]**Integer(2)*z[Integer(0)]**Integer(3) + z[Integer(2)]**Integer(2)*z[Integer(1)]*z[Integer(0)]**Integer(3) + z[Integer(2)]*z[Integer(1)]**Integer(2)*z[Integer(0)]**Integer(3)
>>> k.from_polynomial(f)
k[3, 0, 1, 2]

Since the ring of key polynomials may be regarded as a different choice of basis for a polynomial ring, it forms an algebra, so we have multiplication:

Sage

sage: k([10,5,2])*k([1,1,1])
k[11, 6, 3]

Python

>>> from sage.all import *
>>> k([Integer(10),Integer(5),Integer(2)])*k([Integer(1),Integer(1),Integer(1)])
k[11, 6, 3]

We can also multiply by polynomials in the monomial basis:

Sage

sage: k([10,9,1])*z[0]
k[11, 9, 1]
sage: z[0] * k([10,9,1])
k[11, 9, 1]
sage: k([10,9,1])*(z[0] + z[3])
k[10, 9, 1, 1] + k[11, 9, 1]

Python

>>> from sage.all import *
>>> k([Integer(10),Integer(9),Integer(1)])*z[Integer(0)]
k[11, 9, 1]
>>> z[Integer(0)] * k([Integer(10),Integer(9),Integer(1)])
k[11, 9, 1]
>>> k([Integer(10),Integer(9),Integer(1)])*(z[Integer(0)] + z[Integer(3)])
k[10, 9, 1, 1] + k[11, 9, 1]

When the sorting permutation is the longest element, the key polynomial agrees with the Schur polynomial:

Sage

sage: s = SymmetricFunctions(QQ).schur()
sage: k([1,2,3]).expand()
z_2^3*z_1^2*z_0 + z_2^3*z_1*z_0^2 + z_2^2*z_1^3*z_0
 + 2*z_2^2*z_1^2*z_0^2 + z_2^2*z_1*z_0^3 + z_2*z_1^3*z_0^2
 + z_2*z_1^2*z_0^3
sage: s[3,2,1].expand(3)
x0^3*x1^2*x2 + x0^2*x1^3*x2 + x0^3*x1*x2^2 + 2*x0^2*x1^2*x2^2
 + x0*x1^3*x2^2 + x0^2*x1*x2^3 + x0*x1^2*x2^3

Python

>>> from sage.all import *
>>> s = SymmetricFunctions(QQ).schur()
>>> k([Integer(1),Integer(2),Integer(3)]).expand()
z_2^3*z_1^2*z_0 + z_2^3*z_1*z_0^2 + z_2^2*z_1^3*z_0
 + 2*z_2^2*z_1^2*z_0^2 + z_2^2*z_1*z_0^3 + z_2*z_1^3*z_0^2
 + z_2*z_1^2*z_0^3
>>> s[Integer(3),Integer(2),Integer(1)].expand(Integer(3))
x0^3*x1^2*x2 + x0^2*x1^3*x2 + x0^3*x1*x2^2 + 2*x0^2*x1^2*x2^2
 + x0*x1^3*x2^2 + x0^2*x1*x2^3 + x0*x1^2*x2^3

The polynomial expansions can be computed using crystals and expressed in terms of the key basis:

Sage

sage: T = crystals.Tableaux(['A',3],shape=[2,1])
sage: f = T.demazure_character([3,2,1])
sage: k.from_polynomial(f)
k[1, 0, 0, 2]

Python

>>> from sage.all import *
>>> T = crystals.Tableaux(['A',Integer(3)],shape=[Integer(2),Integer(1)])
>>> f = T.demazure_character([Integer(3),Integer(2),Integer(1)])
>>> k.from_polynomial(f)
k[1, 0, 0, 2]

The default behavior is to work in a polynomial ring with infinitely many variables. One can work in a specicfied number of variables:

Sage

sage: k = KeyPolynomials(QQ, 4)
sage: k([3,0,1,2]).expand()
z_0^3*z_1^2*z_2 + z_0^3*z_1*z_2^2 + z_0^3*z_1^2*z_3
 + 2*z_0^3*z_1*z_2*z_3 + z_0^3*z_2^2*z_3 + z_0^3*z_1*z_3^2 + z_0^3*z_2*z_3^2

sage: k([0,0,2,0]).expand()
z_0^2 + z_0*z_1 + z_1^2 + z_0*z_2  + z_1*z_2 + z_2^2

sage: k([0,0,2,0]).expand().parent()
Multivariate Polynomial Ring in z_0, z_1, z_2, z_3 over Rational Field

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ, Integer(4))
>>> k([Integer(3),Integer(0),Integer(1),Integer(2)]).expand()
z_0^3*z_1^2*z_2 + z_0^3*z_1*z_2^2 + z_0^3*z_1^2*z_3
 + 2*z_0^3*z_1*z_2*z_3 + z_0^3*z_2^2*z_3 + z_0^3*z_1*z_3^2 + z_0^3*z_2*z_3^2

>>> k([Integer(0),Integer(0),Integer(2),Integer(0)]).expand()
z_0^2 + z_0*z_1 + z_1^2 + z_0*z_2  + z_1*z_2 + z_2^2

>>> k([Integer(0),Integer(0),Integer(2),Integer(0)]).expand().parent()
Multivariate Polynomial Ring in z_0, z_1, z_2, z_3 over Rational Field

If working in a specified number of variables, the length of the indexing composition must be the same as the number of variables:

Sage

sage: k([0,0,2])
Traceback (most recent call last):
 ...
TypeError: do not know how to make x (= [0, 0, 2]) an element of self
 (=Key polynomial basis over Rational Field)

Python

>>> from sage.all import *
>>> k([Integer(0),Integer(0),Integer(2)])
Traceback (most recent call last):
 ...
TypeError: do not know how to make x (= [0, 0, 2]) an element of self
 (=Key polynomial basis over Rational Field)

One can also work in a specified polynomial ring:

Sage

sage: k = KeyPolynomials(QQ['x0', 'x1', 'x2', 'x3'])
sage: k([0,2,0,0])
k[0, 2, 0, 0]
sage: k([4,0,0,0]).expand()
x0^4

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ['x0', 'x1', 'x2', 'x3'])
>>> k([Integer(0),Integer(2),Integer(0),Integer(0)])
k[0, 2, 0, 0]
>>> k([Integer(4),Integer(0),Integer(0),Integer(0)]).expand()
x0^4

If one wishes to use a polynomial ring as coefficients for the key polynomials, pass the keyword argument poly_coeffs=True:

Sage

sage: k = KeyPolynomials(QQ['q'], poly_coeffs=True)
sage: R = k.base_ring(); R
Univariate Polynomial Ring in q over Rational Field
sage: R.inject_variables()
Defining q
sage: (q^2 + q + 1)*k([0,2,2,0,3,2])
(q^2+q+1)*k[0, 2, 2, 0, 3, 2]

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ['q'], poly_coeffs=True)
>>> R = k.base_ring(); R
Univariate Polynomial Ring in q over Rational Field
>>> R.inject_variables()
Defining q
>>> (q**Integer(2) + q + Integer(1))*k([Integer(0),Integer(2),Integer(2),Integer(0),Integer(3),Integer(2)])
(q^2+q+1)*k[0, 2, 2, 0, 3, 2]

Element

alias of KeyPolynomial

degree_on_basis(alpha)

Return the degree of the basis element indexed by alpha.

EXAMPLES:

Sage

sage: k = KeyPolynomials(QQ)
sage: k.degree_on_basis([2,1,0,2])
5

sage: k = KeyPolynomials(QQ, 5)
sage: k.degree_on_basis([2,1,0,2,0])
5

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ)
>>> k.degree_on_basis([Integer(2),Integer(1),Integer(0),Integer(2)])
5

>>> k = KeyPolynomials(QQ, Integer(5))
>>> k.degree_on_basis([Integer(2),Integer(1),Integer(0),Integer(2),Integer(0)])
5

from_polynomial(f)

Expand a polynomial in terms of the key basis.

EXAMPLES:

Sage

sage: k = KeyPolynomials(QQ)
sage: z = k.poly_gens(); z
z_*
sage: p = z[0]^4*z[1]^2*z[2]*z[3] + z[0]^4*z[1]*z[2]^2*z[3]
sage: k.from_polynomial(p)
k[4, 1, 2, 1]

sage: all(k(c) == k.from_polynomial(k(c).expand()) for c in IntegerVectors(n=5, k=4))
True

sage: T = crystals.Tableaux(['A', 4], shape=[4,2,1,1])
sage: k.from_polynomial(T.demazure_character([2]))
k[4, 1, 2, 1]

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ)
>>> z = k.poly_gens(); z
z_*
>>> p = z[Integer(0)]**Integer(4)*z[Integer(1)]**Integer(2)*z[Integer(2)]*z[Integer(3)] + z[Integer(0)]**Integer(4)*z[Integer(1)]*z[Integer(2)]**Integer(2)*z[Integer(3)]
>>> k.from_polynomial(p)
k[4, 1, 2, 1]

>>> all(k(c) == k.from_polynomial(k(c).expand()) for c in IntegerVectors(n=Integer(5), k=Integer(4)))
True

>>> T = crystals.Tableaux(['A', Integer(4)], shape=[Integer(4),Integer(2),Integer(1),Integer(1)])
>>> k.from_polynomial(T.demazure_character([Integer(2)]))
k[4, 1, 2, 1]

from_schubert_polynomial(x)

Expand a Schubert polynomial in the key basis.

EXAMPLES:

Sage

sage: k = KeyPolynomials(ZZ)
sage: X = SchubertPolynomialRing(ZZ)
sage: f = X([2,1,5,4,3])
sage: k.from_schubert_polynomial(f)
k[1, 0, 2, 1] + k[2, 0, 2] + k[3, 0, 0, 1]
sage: k.from_schubert_polynomial(2)
2*k[]
sage: k(f)
k[1, 0, 2, 1] + k[2, 0, 2] + k[3, 0, 0, 1]

sage: k = KeyPolynomials(GF(7), 4)
sage: k.from_schubert_polynomial(f)
k[1, 0, 2, 1] + k[2, 0, 2, 0] + k[3, 0, 0, 1]

Python

>>> from sage.all import *
>>> k = KeyPolynomials(ZZ)
>>> X = SchubertPolynomialRing(ZZ)
>>> f = X([Integer(2),Integer(1),Integer(5),Integer(4),Integer(3)])
>>> k.from_schubert_polynomial(f)
k[1, 0, 2, 1] + k[2, 0, 2] + k[3, 0, 0, 1]
>>> k.from_schubert_polynomial(Integer(2))
2*k[]
>>> k(f)
k[1, 0, 2, 1] + k[2, 0, 2] + k[3, 0, 0, 1]

>>> k = KeyPolynomials(GF(Integer(7)), Integer(4))
>>> k.from_schubert_polynomial(f)
k[1, 0, 2, 1] + k[2, 0, 2, 0] + k[3, 0, 0, 1]

one_basis()

Return the basis element indexing the identity.

EXAMPLES:

Sage

sage: k = KeyPolynomials(QQ)
sage: k.one_basis()
[]

sage: k = KeyPolynomials(QQ, 4)
sage: k.one_basis()
[0, 0, 0, 0]

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ)
>>> k.one_basis()
[]

>>> k = KeyPolynomials(QQ, Integer(4))
>>> k.one_basis()
[0, 0, 0, 0]

poly_gens()

Return the polynomial generators for the polynomial ring associated to self.

EXAMPLES:

Sage

sage: k = KeyPolynomials(QQ)
sage: k.poly_gens()
z_*

sage: k = KeyPolynomials(QQ, 4)
sage: k.poly_gens()
(z_0, z_1, z_2, z_3)

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ)
>>> k.poly_gens()
z_*

>>> k = KeyPolynomials(QQ, Integer(4))
>>> k.poly_gens()
(z_0, z_1, z_2, z_3)

polynomial_ring()

Return the polynomial ring associated to self.

EXAMPLES:

Sage

sage: k = KeyPolynomials(QQ)
sage: k.polynomial_ring()
Infinite polynomial ring in z over Rational Field

sage: k = KeyPolynomials(QQ, 4)
sage: k.polynomial_ring()
Multivariate Polynomial Ring in z_0, z_1, z_2, z_3 over Rational Field

Python

>>> from sage.all import *
>>> k = KeyPolynomials(QQ)
>>> k.polynomial_ring()
Infinite polynomial ring in z over Rational Field

>>> k = KeyPolynomials(QQ, Integer(4))
>>> k.polynomial_ring()
Multivariate Polynomial Ring in z_0, z_1, z_2, z_3 over Rational Field

sage.combinat.key_polynomial.divided_difference(f, i)

Apply the i-th divided difference operator to the polynomial f.

EXAMPLES:

Sage

sage: from sage.combinat.key_polynomial import divided_difference
sage: k = KeyPolynomials(QQ)
sage: z = k.poly_gens()
sage: f = z[1]*z[2]^3 + z[1]*z[2]*z[3]
sage: divided_difference(f, 3)
z_3^2*z_1 + z_3*z_2*z_1 + z_2^2*z_1

sage: k = KeyPolynomials(QQ, 4)
sage: z = k.poly_gens()
sage: f = z[1]*z[2]^3 + z[1]*z[2]*z[3]
sage: divided_difference(f, 3)
z_1*z_2^2 + z_1*z_2*z_3 + z_1*z_3^2

sage: k = KeyPolynomials(QQ)
sage: R = k.polynomial_ring(); R
Infinite polynomial ring in z over Rational Field
sage: z = R.gen()
sage: divided_difference(z[1]*z[2]^3, 2)
-z_2^2*z_1 - z_2*z_1^2
sage: divided_difference(z[1]*z[2]*z[3], 3)
0
sage: divided_difference(z[1]*z[2]*z[3], 4)
z_2*z_1
sage: divided_difference(z[1]*z[2]*z[4], 4)
-z_2*z_1

sage: k = KeyPolynomials(QQ, 5)
sage: z = k.polynomial_ring().gens()
sage: divided_difference(z[1]*z[2]^3, 2)
-z_1^2*z_2 - z_1*z_2^2
sage: divided_difference(z[1]*z[2]*z[3], 3)
0
sage: divided_difference(z[1]*z[2]*z[3], 4)
z_1*z_2
sage: divided_difference(z[1]*z[2]*z[4], 4)
-z_1*z_2

Python

>>> from sage.all import *
>>> from sage.combinat.key_polynomial import divided_difference
>>> k = KeyPolynomials(QQ)
>>> z = k.poly_gens()
>>> f = z[Integer(1)]*z[Integer(2)]**Integer(3) + z[Integer(1)]*z[Integer(2)]*z[Integer(3)]
>>> divided_difference(f, Integer(3))
z_3^2*z_1 + z_3*z_2*z_1 + z_2^2*z_1

>>> k = KeyPolynomials(QQ, Integer(4))
>>> z = k.poly_gens()
>>> f = z[Integer(1)]*z[Integer(2)]**Integer(3) + z[Integer(1)]*z[Integer(2)]*z[Integer(3)]
>>> divided_difference(f, Integer(3))
z_1*z_2^2 + z_1*z_2*z_3 + z_1*z_3^2

>>> k = KeyPolynomials(QQ)
>>> R = k.polynomial_ring(); R
Infinite polynomial ring in z over Rational Field
>>> z = R.gen()
>>> divided_difference(z[Integer(1)]*z[Integer(2)]**Integer(3), Integer(2))
-z_2^2*z_1 - z_2*z_1^2
>>> divided_difference(z[Integer(1)]*z[Integer(2)]*z[Integer(3)], Integer(3))
0
>>> divided_difference(z[Integer(1)]*z[Integer(2)]*z[Integer(3)], Integer(4))
z_2*z_1
>>> divided_difference(z[Integer(1)]*z[Integer(2)]*z[Integer(4)], Integer(4))
-z_2*z_1

>>> k = KeyPolynomials(QQ, Integer(5))
>>> z = k.polynomial_ring().gens()
>>> divided_difference(z[Integer(1)]*z[Integer(2)]**Integer(3), Integer(2))
-z_1^2*z_2 - z_1*z_2^2
>>> divided_difference(z[Integer(1)]*z[Integer(2)]*z[Integer(3)], Integer(3))
0
>>> divided_difference(z[Integer(1)]*z[Integer(2)]*z[Integer(3)], Integer(4))
z_1*z_2
>>> divided_difference(z[Integer(1)]*z[Integer(2)]*z[Integer(4)], Integer(4))
-z_1*z_2

sage.combinat.key_polynomial.isobaric_divided_difference(f, w)

Apply the isobaric divided difference operator \(\pi_w\) to the polynomial \(f\).

w may be either a single index or a list of indices of simple transpositions.

Warning

The simple transpositions should be applied from left to right.

EXAMPLES:

Sage

sage: from sage.combinat.key_polynomial import isobaric_divided_difference as idd
sage: R.<z> = InfinitePolynomialRing(GF(3))
sage: idd(z[1]^4*z[2]^2*z[4], 4)
0

sage: idd(z[1]^4*z[2]^2*z[3]*z[4], 3)
z_4*z_3^2*z_2*z_1^4 + z_4*z_3*z_2^2*z_1^4

sage: idd(z[1]^4*z[2]^2*z[3]*z[4], [3, 4])
z_4^2*z_3*z_2*z_1^4 + z_4*z_3^2*z_2*z_1^4 + z_4*z_3*z_2^2*z_1^4

sage: idd(z[1]^4*z[2]^2*z[3]*z[4], [4, 3])
z_4*z_3^2*z_2*z_1^4 + z_4*z_3*z_2^2*z_1^4

sage: idd(z[1]^2*z[2], [3, 2])
z_3*z_2^2 + z_3*z_2*z_1 + z_3*z_1^2 + z_2^2*z_1 + z_2*z_1^2

Python

>>> from sage.all import *
>>> from sage.combinat.key_polynomial import isobaric_divided_difference as idd
>>> R = InfinitePolynomialRing(GF(Integer(3)), names=('z',)); (z,) = R._first_ngens(1)
>>> idd(z[Integer(1)]**Integer(4)*z[Integer(2)]**Integer(2)*z[Integer(4)], Integer(4))
0

>>> idd(z[Integer(1)]**Integer(4)*z[Integer(2)]**Integer(2)*z[Integer(3)]*z[Integer(4)], Integer(3))
z_4*z_3^2*z_2*z_1^4 + z_4*z_3*z_2^2*z_1^4

>>> idd(z[Integer(1)]**Integer(4)*z[Integer(2)]**Integer(2)*z[Integer(3)]*z[Integer(4)], [Integer(3), Integer(4)])
z_4^2*z_3*z_2*z_1^4 + z_4*z_3^2*z_2*z_1^4 + z_4*z_3*z_2^2*z_1^4

>>> idd(z[Integer(1)]**Integer(4)*z[Integer(2)]**Integer(2)*z[Integer(3)]*z[Integer(4)], [Integer(4), Integer(3)])
z_4*z_3^2*z_2*z_1^4 + z_4*z_3*z_2^2*z_1^4

>>> idd(z[Integer(1)]**Integer(2)*z[Integer(2)], [Integer(3), Integer(2)])
z_3*z_2^2 + z_3*z_2*z_1 + z_3*z_1^2 + z_2^2*z_1 + z_2*z_1^2

sage.combinat.key_polynomial.sorting_word(alpha)

Get a reduced word for the permutation which sorts alpha into a partition.

The result is a list l = [i0, i1, i2, ...] where each ij is a positive integer such that it applies the simple transposition \((i_j, i_j+1)\). The transpositions are applied starting with i0, then i1 is applied, followed by i2, and so on. See sage.combinat.permutation.Permutation.reduced_words() for the convention used.

EXAMPLES:

Sage

sage: IV = IntegerVectors()
sage: from sage.combinat.key_polynomial import sorting_word
sage: list(sorting_word(IV([2,3,2]))[0])
[1]
sage: sorting_word(IV([2,3,2]))[1]
[3, 2, 2]
sage: list(sorting_word(IV([5,6,7]))[0])
[1, 2, 1]
sage: list(sorting_word(IV([0,3,2]))[0])
[2, 1]
sage: list(sorting_word(IV([0,3,0,2]))[0])
[2, 3, 1]
sage: list(sorting_word(IV([3,2,1]))[0])
[]
sage: list(sorting_word(IV([2,3,3]))[0])
[2, 1]

Python

>>> from sage.all import *
>>> IV = IntegerVectors()
>>> from sage.combinat.key_polynomial import sorting_word
>>> list(sorting_word(IV([Integer(2),Integer(3),Integer(2)]))[Integer(0)])
[1]
>>> sorting_word(IV([Integer(2),Integer(3),Integer(2)]))[Integer(1)]
[3, 2, 2]
>>> list(sorting_word(IV([Integer(5),Integer(6),Integer(7)]))[Integer(0)])
[1, 2, 1]
>>> list(sorting_word(IV([Integer(0),Integer(3),Integer(2)]))[Integer(0)])
[2, 1]
>>> list(sorting_word(IV([Integer(0),Integer(3),Integer(0),Integer(2)]))[Integer(0)])
[2, 3, 1]
>>> list(sorting_word(IV([Integer(3),Integer(2),Integer(1)]))[Integer(0)])
[]
>>> list(sorting_word(IV([Integer(2),Integer(3),Integer(3)]))[Integer(0)])
[2, 1]