Module: sage.combinat.sf.jack
Jack Polynomials
Module-level Functions
R, [t=None]) |
Returns the algebra of Jack polynomials in the J basis.
If t is not specified, then the base ring will be obtained by making the univariate polynomial ring over R with the variable t and taking its fraction field.
sage: JackPolynomialsJ(QQ) Jack polynomials in the J basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field sage: JackPolynomialsJ(QQ,t=-1) Jack polynomials in the J basis with t=-1 over Rational Field
At t = 1, the Jack polynomials in the J basis are scalar multiples of the Schur functions with the scalar given by a Partition's hook_product method at 1.
sage: J = JackPolynomialsJ(QQ, t=1) sage: s = SFASchur(J.base_ring()) sage: p = Partition([3,2,1,1]) sage: s(J(p)) == p.hook_product(1)*s(p) True
At t = 2, the Jack polynomials on the J basis are scalar multiples of the zonal polynomials with the scalar given by a Partition's hook_product method at 1.
sage: t = 2 sage: J = JackPolynomialsJ(QQ,t=t) sage: Z = ZonalPolynomials(J.base_ring()) sage: p = Partition([2,2,1]) sage: Z(J(p)) == p.hook_product(t)*Z(p) True
R, [t=None]) |
Returns the algebra of Jack polynomials in the P basis.
If t is not specified, then the base ring will be obtained by making the univariate polynomial ring over R with the variable t and taking its fraction field.
sage: JackPolynomialsP(QQ) Jack polynomials in the P basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field sage: JackPolynomialsP(QQ,t=-1) Jack polynomials in the P basis with t=-1 over Rational Field
At t = 1, the Jack polynomials on the P basis are the Schur symmetric functions.
sage: P = JackPolynomialsP(QQ,1) sage: s = SFASchur(QQ) sage: P([2,1])^2 JackP[2, 2, 1, 1] + JackP[2, 2, 2] + JackP[3, 1, 1, 1] + 2*JackP[3, 2, 1] + JackP[3, 3] + JackP[4, 1, 1] + JackP[4, 2] sage: s([2,1])^2 s[2, 2, 1, 1] + s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[3, 3] + s[4, 1, 1] + s[4, 2]
At t = 2, the Jack polynomials on the P basis are the zonal polynomials.
sage: P = JackPolynomialsP(QQ,2) sage: Z = ZonalPolynomials(QQ) sage: P([2])^2 64/45*JackP[2, 2] + 16/21*JackP[3, 1] + JackP[4] sage: Z([2])^2 64/45*Z[2, 2] + 16/21*Z[3, 1] + Z[4] sage: Z(P([2,1])) Z[2, 1] sage: P(Z([2,1])) JackP[2, 1]
R, [t=None]) |
Returns the algebra of Jack polynomials in the Q basis.
If t is not specified, then the base ring will be obtained by making the univariate polynomial ring over R with the variable t and taking its fraction field.
sage: JackPolynomialsQ(QQ) Jack polynomials in the Q basis over Fraction Field of Univariate Polynomial Ring in t over Rational Field sage: JackPolynomialsQ(QQ,t=-1) Jack polynomials in the Q basis with t=-1 over Rational Field
R, [t=None]) |
Returns the algebra of Jack polynomials in the Qp, which is dual to the P basis with respect to the standard scalar product.
If t is not specified, then the base ring will be obtained by making the univariate polynomial ring over R with the variable t and taking its fraction field.
sage: Qp = JackPolynomialsQp(QQ) sage: P = JackPolynomialsP(QQ) sage: a = Qp([2]) sage: a.scalar(P([2])) 1 sage: a.scalar(P([1,1])) 0 sage: P(Qp([2])) ((t-1)/(t+1))*JackP[1, 1] + JackP[2]
R) |
Returns the algebra of zonal polynomials.
sage: Z = ZonalPolynomials(QQ) sage: a = Z([2]) sage: Z([2])^2 64/45*Z[2, 2] + 16/21*Z[3, 1] + Z[4]
part, t) |
sage: from sage.combinat.sf.jack import c1 sage: t = QQ['t'].gen() sage: [c1(p,t) for p in Partitions(3)] [2*t^2 + 3*t + 1, t + 2, 6]
part, t) |
sage: from sage.combinat.sf.jack import c2 sage: t = QQ['t'].gen() sage: [c2(p,t) for p in Partitions(3)] [6*t^3, 2*t^3 + t^2, t^3 + 3*t^2 + 2*t]
part1, part2, t) |
Returns the Jack scalar product between p(part1) and p(part2) where p is the power-sum basis.
sage: Q.<t> = QQ[] sage: from sage.combinat.sf.jack import scalar_jack sage: matrix([[scalar_jack(p1,p2,t) for p1 in Partitions(4)] for p2 in Partitions(4)]) [ 4*t 0 0 0 0] [ 0 3*t^2 0 0 0] [ 0 0 8*t^2 0 0] [ 0 0 0 4*t^3 0] [ 0 0 0 0 24*t^4]
Class: JackPolynomial_generic
Functions: scalar_jack
self, x) |
sage: P = JackPolynomialsP(QQ) sage: Q = JackPolynomialsQ(QQ) sage: p = Partitions(3).list() sage: matrix([[P(a).scalar_jack(Q(b)) for a in p] for b in p]) [1 0 0] [0 1 0] [0 0 1]
Class: JackPolynomial_j
Class: JackPolynomial_p
Functions: scalar_jack
self, x) |
sage: P = JackPolynomialsP(QQ) sage: l = [P(p) for p in Partitions(3)] sage: matrix([[a.scalar_jack(b) for a in l] for b in l]) [ 6*t^3/(2*t^2 + 3*t + 1) 0 0] [ 0 (2*t^3 + t^2)/(t + 2) 0] [ 0 0 1/6*t^3 + 1/2*t^2 + 1/3*t]
Class: JackPolynomial_q
Class: JackPolynomials_generic
self, R, [t=None]) |
sage: JackPolynomialsJ(QQ).base_ring() Fraction Field of Univariate Polynomial Ring in t over Rational Field sage: JackPolynomialsJ(QQ, t=2).base_ring() Rational Field
Special Functions: __init__,
_normalize_coefficients
self, c) |
If our coefficient ring is the field of fractions over a univariate polynomial ring over the rationals, then we should clear both the numerator and denominator of the denominators of their coefficients.
sage: P = JackPolynomialsP(QQ) sage: t = P.base_ring().gen() sage: a = 2/(1/2*t+1/2) sage: P._normalize_coefficients(a) 4/(t + 1) sage: a = 1/(1/3+1/6*t) sage: P._normalize_coefficients(a) 6/(t + 2) sage: a = 24/(4*t^2 + 12*t + 8) sage: P._normalize_coefficients(a) 6/(t^2 + 3*t + 2)
Class: JackPolynomials_j
self, R, [t=None]) |
sage: J = JackPolynomialsJ(QQ) sage: J == loads(dumps(J)) True
Special Functions: __init__,
_coerce_start,
_multiply
self, x) |
sage: J = JackPolynomialsJ(QQ) sage: P = JackPolynomialsP(QQ) sage: J(sum(P(p) for p in Partitions(3))) 1/6*JackJ[1, 1, 1] + (1/(t+2))*JackJ[2, 1] + (1/(2*t^2+3*t+1))*JackJ[3]
sage: s = SFASchur(J.base_ring()) sage: J(s([3])) # indirect doctest ((t^2-3*t+2)/(6*t^2+18*t+12))*JackJ[1, 1, 1] + ((2*t-2)/(2*t^2+5*t+2))*JackJ[2, 1] + (1/(2*t^2+3*t+1))*JackJ[3] sage: J(s([2,1])) ((t-1)/(3*t+6))*JackJ[1, 1, 1] + (1/(t+2))*JackJ[2, 1] sage: J(s([1,1,1])) 1/6*JackJ[1, 1, 1]
self, left, right) |
sage: J = JackPolynomialsJ(QQ) sage: J([1])^2 #indirect doctest (t/(t+1))*JackJ[1, 1] + (1/(t+1))*JackJ[2] sage: J([2])^2 (2*t^2/(2*t^2+3*t+1))*JackJ[2, 2] + (4*t/(3*t^2+4*t+1))*JackJ[3, 1] + ((t+1)/(6*t^2+5*t+1))*JackJ[4]
Class: JackPolynomials_p
self, R, [t=None]) |
sage: P = JackPolynomialsP(QQ) sage: P == loads(dumps(P)) True
Special Functions: __init__,
_coerce_start,
_m_cache,
_multiply,
_to_m
self, x) |
Coerce things into the Jack polynomials P basis.
sage: Q = JackPolynomialsQ(QQ) sage: P = JackPolynomialsP(QQ) sage: J = JackPolynomialsJ(QQ)
sage: P(Q([2,1])) # indirect doctest ((t+2)/(2*t^3+t^2))*JackP[2, 1] sage: P(Q([3])) ((2*t^2+3*t+1)/(6*t^3))*JackP[3] sage: P(Q([1,1,1])) (6/(t^3+3*t^2+2*t))*JackP[1, 1, 1]
sage: P(J([3])) (2*t^2+3*t+1)*JackP[3] sage: P(J([2,1])) (t+2)*JackP[2, 1] sage: P(J([1,1,1])) 6*JackP[1, 1, 1]
sage: s = SFASchur(QQ) sage: P(s([2,1])) ((2*t-2)/(t+2))*JackP[1, 1, 1] + JackP[2, 1] sage: s(_) s[2, 1]
self, n) |
Computes the change of basis between the Jack polynomials in the P basis and the monomial symmetric functions. This uses Gram-Schmidt to go to the monomials, and then that matrix is simply inverted.
sage: P = JackPolynomialsP(QQ) sage: l = lambda c: [ (i[0],[j for j in sorted(i[1].items())]) for i in sorted(c.items())] sage: P._m_cache(2) sage: l(P._self_to_m_cache[2]) [([1, 1], [([1, 1], 1)]), ([2], [([1, 1], 2/(t + 1)), ([2], 1)])] sage: l(P._m_to_self_cache[2]) [([1, 1], [([1, 1], 1)]), ([2], [([1, 1], -2/(t + 1)), ([2], 1)])] sage: P._m_cache(3) sage: l(P._m_to_self_cache[3]) [([1, 1, 1], [([1, 1, 1], 1)]), ([2, 1], [([1, 1, 1], -6/(t + 2)), ([2, 1], 1)]), ([3], [([1, 1, 1], 6/(t^2 + 3*t + 2)), ([2, 1], -3/(2*t + 1)), ([3], 1)])] sage: l(P._self_to_m_cache[3]) [([1, 1, 1], [([1, 1, 1], 1)]), ([2, 1], [([1, 1, 1], 6/(t + 2)), ([2, 1], 1)]), ([3], [([1, 1, 1], 6/(2*t^2 + 3*t + 1)), ([2, 1], 3/(2*t + 1)), ([3], 1)])]
self, left, right) |
sage: P = JackPolynomialsP(QQ) sage: P([1])^2 # indirect doctest (2*t/(t+1))*JackP[1, 1] + JackP[2] sage: P._m(_) 2*m[1, 1] + m[2] sage: P = JackPolynomialsP(QQ, 2) sage: P([2,1])^2 125/63*JackP[2, 2, 1, 1] + 25/12*JackP[2, 2, 2] + 25/18*JackP[3, 1, 1, 1] + 12/5*JackP[3, 2, 1] + 4/3*JackP[3, 3] + 4/3*JackP[4, 1, 1] + JackP[4, 2] sage: P._m(_) 45*m[1, 1, 1, 1, 1, 1] + 51/2*m[2, 1, 1, 1, 1] + 29/2*m[2, 2, 1, 1] + 33/4*m[2, 2, 2] + 9*m[3, 1, 1, 1] + 5*m[3, 2, 1] + 2*m[3, 3] + 2*m[4, 1, 1] + m[4, 2]
self, part) |
Return a function that takes in a partition lambda that returns the coefficient of lambda in the expansion of self(part) in the monomial basis.
This assumes that the cache from the Jack polynomials in the P basis to the monomial symmetric functions has already been computed.
sage: P = JackPolynomialsP(QQ) sage: P._m_cache(2) sage: p2 = Partition([2]) sage: p11 = Partition([1,1]) sage: f = P._to_m(p2) sage: f(p11) 2/(t + 1) sage: f(p2) 1 sage: f = P._to_m(p11) sage: f(p2) 0 sage: f(p11) 1
Class: JackPolynomials_q
self, R, [t=None]) |
sage: Q = JackPolynomialsQ(QQ) sage: Q == loads(dumps(Q)) True
Special Functions: __init__,
_coerce_start,
_multiply
self, x) |
sage: Q = JackPolynomialsQ(QQ) sage: P = JackPolynomialsP(QQ) sage: Q(sum(P(p) for p in Partitions(3))) (1/6*t^3+1/2*t^2+1/3*t)*JackQ[1, 1, 1] + ((2*t^3+t^2)/(t+2))*JackQ[2, 1] + (6*t^3/(2*t^2+3*t+1))*JackQ[3]
sage: s = SFASchur(Q.base_ring()) sage: Q(s([3])) # indirect doctest (1/6*t^3-1/2*t^2+1/3*t)*JackQ[1, 1, 1] + ((2*t^3-2*t^2)/(t+2))*JackQ[2, 1] + (6*t^3/(2*t^2+3*t+1))*JackQ[3] sage: Q(s([2,1])) (1/3*t^3-1/3*t)*JackQ[1, 1, 1] + ((2*t^3+t^2)/(t+2))*JackQ[2, 1] sage: Q(s([1,1,1])) (1/6*t^3+1/2*t^2+1/3*t)*JackQ[1, 1, 1]
self, left, right) |
sage: Q = JackPolynomialsQ(QQ) sage: Q([1])^2 # indirect doctest JackQ[1, 1] + (2/(t+1))*JackQ[2] sage: Q([2])^2 JackQ[2, 2] + (2/(t+1))*JackQ[3, 1] + ((6*t+6)/(6*t^2+5*t+1))*JackQ[4]