- First step when using any new Sage functionality... ask Sage what to do!
sage: SymmetricFunctions? # not tested
The ring of symmetric functions over the rational numbers:
sage: S = SymmetricFunctions(QQ)
Typing 'objectname.<tab>' gives a lot of information about what you can do with the object:
sage: S. # not tested
The usual bases for symmetric functions:
sage: p = S.powersum(); s = S.schur(); m = S.monomial(); h = S.homogeneous(); e = S.elementary()
The 'forgotten basis' is dual to the elementary basis:
sage: f = e.dual_basis()
Different ways of entering symmetric functions:
sage: p[2,1] == p([2,1]) and p[2,1] == p(Partition([2,1])) True
Changing bases:
sage: p(s[2,1]) 1/3*p[1, 1, 1] - 1/3*p[3]
Sums of different bases are automatically converted to a single basis:
sage: h[3] + s[3] + e[3] + p[3] 2*h[1, 1, 1] - 5*h[2, 1] + 6*h[3]
Littlewood-Richardson coefficients are relatively fast:
sage: timeit('s[10]^4')
5 loops, best of 3:...
Changing bases:
sage: h(s[10]^4); h[10, 10, 10, 10]
We get an arbitrary symmetric function to demonstrate some functionality:
sage: foo = 1/2*h([]) + 3*h([1, 1, 1]) + 2*h([2, 1, 1]) sage: foo 1/2*h[] + 3*h[1, 1, 1] + 2*h[2, 1, 1]
The omega involution:
sage: foo.omega() 1/2*h[] + 3*h[1, 1, 1] + 2*h[1, 1, 1, 1] - 2*h[2, 1, 1] sage: e(foo.omega()) 1/2*e[] + 3*e[1, 1, 1] + 2*e[2, 1, 1]
The Hall scalar product:
sage: foo.scalar(s[3,1]) 4
sage: foo.is_schur_positive() True
sage: foo.skew_by(e[2,1]) 9*h[] + 10*h[1]
We can define skew partition directly:
sage: mu = Partition([3,2])/Partition([2,1]) sage: mu [3, 2] / [2, 1] sage: s(mu) s[1, 1] + s[2]
We can expand a symmetric function in monomials:
sage: s(mu).expand(3) x0^2 + 2*x0*x1 + x1^2 + 2*x0*x2 + 2*x1*x2 + x2^2
Or we can choose our alphabet:
sage: s(mu).expand(3,alphabet=['a','b','c']) a^2 + 2*a*b + b^2 + 2*a*c + 2*b*c + c^2 sage: mu = Partition([32,18,16,4,1])/Partition([14,3,2,1]) sage: la = Partition([33,19,17,4,1])/Partition([15,4,3,1]) sage: (s(la) - s(mu)).is_schur_positive() True sage: foo.kronecker_product(foo) 1/4*h[] + 54*h[1, 1, 1] + 20*h[1, 1, 1, 1] + 8*h[2, 1, 1] sage: foo.plethysm(h[3]) 1/2*h[] + 3*h[3, 3, 3] + 2*h[4, 3, 3, 2] - 2*h[5, 3, 3, 1] + 2*h[6, 3, 3]
sage: foo.inner_plethysm? # not tested
The transition matrix from the Schur basis to the power basis Try s.transition_matrix? for more information:
sage: s.transition_matrix(m,5) [1 1 1 1 1 1 1] [0 1 1 2 2 3 4] [0 0 1 1 2 3 5] [0 0 0 1 1 3 6] [0 0 0 0 1 2 5] [0 0 0 0 0 1 4] [0 0 0 0 0 0 1]
The sum of degree 6 Schur functions whose first part is even:
sage: foo = sum([s[mu] for mu in Partitions(6) if mu[0]%2 == 0]) sage: foo s[2, 1, 1, 1, 1] + s[2, 2, 1, 1] + s[2, 2, 2] + s[4, 1, 1] + s[4, 2] + s[6]
sage: def remove_last_part(mu): ....: r""" Remove the last part from a partition """ ....: return Partition(mu[:-1])
We can apply this map to all the partitions appearing in 'foo':
sage: foo.map_support(remove_last_part) s[] + s[2, 1, 1, 1] + s[2, 2] + s[2, 2, 1] + s[4] + s[4, 1]
Warning! This gives different results depending on the basis in which foo is expressed:
sage: h(foo).map_support(remove_last_part) 3*h[] + h[2, 1, 1, 1] + h[2, 2] - 2*h[2, 2, 1] - 2*h[3, 1, 1] + 2*h[3, 2] - 2*h[4] + 4*h[4, 1] - 4*h[5]
sage: foo.map_support(remove_last_part) == h(foo).map_support(remove_last_part) False
We can easily get specific coefficients:
sage: foo.coefficient([4,2]) 1
There are many forms of symmetric functions in sage. These are the ~H[X;q,t] often called the 'modified Macdonald polynomials':
sage: S = SymmetricFunctions(FractionField(QQ['q','t'])) sage: Ht = S.macdonald().Ht(); Ht Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald Ht basis sage: p = S.powersum(); s = S.schur(); m = S.monomial(); h = S.homogeneous(); e = S.elementary();
sage: s(Ht([3,2])) q^4*t^2*s[1, 1, 1, 1, 1] + (q^4*t+q^3*t^2+q^3*t+q^2*t^2)*s[2, 1, 1, 1] + (q^4+q^3*t+q^2*t^2+q^2*t+q*t^2)*s[2, 2, 1] + (q^3*t+q^3+2*q^2*t+q*t^2+q*t)*s[3, 1, 1] + (q^3+q^2*t+q^2+q*t+t^2)*s[3, 2] + (q^2+q*t+q+t)*s[4, 1] + s[5]
sage: Ht.base_ring() Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field
sage: q Traceback (most recent call last): ... NameError: name 'q' is not defined
The following is a shortcut notation (based on Magma). It defines R to be the polynomial ring in the variables 'q' and 't' over the rational numbers, and makes these variables available for use:
sage: R.<q,t> = Frac(ZZ['q','t']) sage: S = SymmetricFunctions(R) sage: p = S.powersum(); s = S.schur(); m = S.monomial(); h = S.homogeneous(); e = S.elementary(); sage: Ht = S.macdonald().Ht(); Ht Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Integer Ring in the Macdonald Ht basis
sage: s(Ht([3,2])) q^4*t^2*s[1, 1, 1, 1, 1] + (q^4*t+q^3*t^2+q^3*t+q^2*t^2)*s[2, 1, 1, 1] + (q^4+q^3*t+q^2*t^2+q^2*t+q*t^2)*s[2, 2, 1] + (q^3*t+q^3+2*q^2*t+q*t^2+q*t)*s[3, 1, 1] + (q^3+q^2*t+q^2+q*t+t^2)*s[3, 2] + (q^2+q*t+q+t)*s[4, 1] + s[5]
sage: latex(_)
q^{4} t^{2} s_{1,1,1,1,1} + \left(q^{4} t + q^{3} t^{2} + q^{3} t + q^{2} t^{2}\right) s_{2,1,1,1} + \left(q^{4} + q^{3} t + q^{2} t^{2} + q^{2} t + q t^{2}\right) s_{2,2,1} + \left(q^{3} t + q^{3} + 2 q^{2} t + q t^{2} + q t\right) s_{3,1,1} + \left(q^{3} + q^{2} t + q^{2} + q t + t^{2}\right) s_{3,2} + \left(q^{2} + q t + q + t\right) s_{4,1} + s_{5}
sage: s(Ht([3,2])).coefficient([2,1,1,1]).subs({q:q^(-1), t:t^(-1)}) *q^5 * t^5
q^3*t^3 + q^2*t^4 + q^2*t^3 + q*t^4
We can also create the ring of Macdonald Polynomials using different parameters:
sage: A.<a,b> = QQ[] sage: S = SymmetricFunctions(FractionField(A)) sage: sa = S.schur() sage: P = S.macdonald(a,b).P()
sage: sa(P[2,1]) ((a*b-b^2+a-b)/(-a*b^2+1))*s[1, 1, 1] + s[2, 1]
Press <tab> after the following to see the different variants of Macdonald polynomials in sage
sage: Sym = SymmetricFunctions(FractionField(QQ['q',t'])) sage: J = Sym.macdonald() sage: J.TAB # not tested
Press <tab> after the following to see the different variants of Jack polynomials in sage:
sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) sage: J = Sym.jack() sage: J.TAB # not tested
Press <tab> after the following to see the different variants of Hall-Littlewood polynomials in sage
sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) sage: H = Sym.hall_littlewood() sage: HallLittlewood
Now some examples with k-Schur functions:
sage: Sym = SymmetricFunctions(QQ['t']) sage: KB = Sym.kBoundedSubspace(2) sage: ks2 = KB.kschur() sage: s = Sym.schur()
sage: s(ks2[2,2,1]) s[2, 2, 1] + t*s[3, 1, 1] + (t^2+t)*s[3, 2] + (t^3+t^2)*s[4, 1] + t^4*s[5] sage: ks2(s[1]) ks2[1] sage: ks2(s[3]) Traceback (most recent call last): ... ValueError: s[3] is not in the image