This document is one of More SageMath Tutorials. You may edit it on github. \(\def\NN{\mathbb{N}}\) \(\def\ZZ{\mathbb{Z}}\) \(\def\QQ{\mathbb{Q}}\) \(\def\RR{\mathbb{R}}\) \(\def\CC{\mathbb{C}}\)
Demonstration: Symmetric functions¶
- First step when using any new Sage functionality… ask Sage what to do!
sage: SymmetricFunctions?
sage: S = SymmetricFunctions(QQ) # The ring of symmetric functions over the rational numbers
sage: # Typing 'objectname.<tab>' gives a lot of information about whant
sage: # you can do with the object
sage: S.
sage: # The usual bases for symmetric functions
sage: p = S.powersum(); s = S.schur(); m = S.monomial(); h = S.homogeneous(); e = S.elementary()
sage: # The 'forgotten basis' is dual to the elementary basis
sage: f = e.dual_basis()
sage: # Different ways of entering symmetric functions
sage: p[2,1] == p([2,1]) and p[2,1] == p(Partition([2,1]))
True
sage: # Changing bases
sage: p(s[2,1])
1/3*p[1, 1, 1] - 1/3*p[3]
sage: # 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]
sage: # Littlewood-Richardson coefficients are relatively fast
sage: timeit('s[10]^4')
5 loops, best of 3:..
sage: # Changing bases
sage: time h(s[10]^4);
h[10, 10, 10, 10]
Time: CPU 1.07 s, Wall: 1.08 s
sage: # We get an arbitrary symmetric function to demonstrate some functionality
sage: foo = h.an_element()
sage: foo
1/2*h[] + 3*h[1, 1, 1] + 2*h[2, 1, 1]
sage: foo.omega() # The omega involution
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]
sage: foo.scalar(s[3,1]) # The Hall scalar product
4
sage: foo.is_schur_positive()
True
sage: foo.skew_by(e[2,1])
9*h[] + 10*h[1]
sage: # 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]
sage: # 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
sage: # 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?
sage: # The transition matrix from the Schur basis to the power basis
sage: # 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]
sage: # 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])
sage: # 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]
sage: # 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
sage: # We can easily get specific coefficients
sage: foo.coefficient([4,2])
1
sage: # There are many forms of symmetric functions in sage.
sage: # They do not (yet) all appear under 'SymmetricFunctions'
sage: # These are the ~H[X;q,t] often called the 'modified Macdonald polynomials'
sage: Ht = MacdonaldPolynomialsHt(QQ)
sage: s(Ht([3,2]))
Traceback (most recent call last):
...
TypeError
sage: Ht.base_ring()
Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field
sage: S.base_ring()
Rational Field
sage: q
Traceback (most recent call last):
...
NameError: name 'q' is not defined
sage: # The following is a shortcut notation (based on Magma).
sage: # It defines R to be the polynomial ring in the variables
sage: # 'q' and 't' over the rational numbers, and makes these variables
sage: # 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 = MacdonaldPolynomialsHt(R)
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
sage: # We can also create the ring of Macdonald Polynomials
sage: # using different parameters
sage: A.<a,b> = QQ[]
sage: P = MacdonaldPolynomialsP(FractionField(A),a,b)
sage: sa = SymmetricFunctions(FractionField(A)).schur()
sage: sa(P[2,1])
((a*b-b^2+a-b)/(-a*b^2+1))*s[1, 1, 1] + s[2, 1]
sage: # Press <tab> after the following to see the different
sage: # variants of Macdonald polynomials in sage
sage: MacdonaldPolynomials
Traceback (most recent call last):
...
NameError: name 'MacdonaldPolynomials' is not defined
sage: # Press <tab> after the following to see the different
sage: # variants of Jack polynomials in sage
sage: JackPolynomials
sage: # Press <tab> after the following to see the different
sage: # variants of Hall-Littlewood polynomials in sage
sage: HallLittlewood
sage: ks2 = kSchurFunctions(R,2,t=R(t))
sage: s = SymmetricFunctions(R).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 space spanned by k-Schur Functions at level 2 over Multivariate Polynomial Ring in q, t over Rational Field.
sage: # Warning: Not well supported yet!
sage: SchubertPolynomialRing
sage: # Warning: Not well supported yet!
sage: LLT