fix some doctests in demo-symmetric-functions

Author: fchapoton

demo-number-theory.rst

@@ -4,6 +4,8 @@
 Demonstration: Sage combines the power of multiple software
 ===========================================================
 
+.. linkall
+
 (taken from a talk from William Stein)
 
 Construct an elliptic curve using John Cremona's table::

demo-symmetric-functions.rst

@@ -11,82 +11,67 @@ Demonstration: Symmetric functions
 
 ::
 
-    sage: SymmetricFunctions?
+    sage: SymmetricFunctions?              # not tested
 
-::
+The ring of symmetric functions over the rational numbers::
 
-    sage: S = SymmetricFunctions(QQ) # 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
 
-    sage: # Typing 'objectname.<tab>' gives a lot of information about what
-    sage: # you can do with the object
-    sage: S.
+The usual bases for symmetric functions::
 
-::
-
-    sage: # 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: # The 'forgotten basis' is dual to the elementary basis
     sage: f = e.dual_basis()
 
+Different ways of entering symmetric functions::
 
-::
-
-    sage: # 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: # 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: # 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: # Littlewood-Richardson coefficients are relatively fast
     sage: timeit('s[10]^4')
-    5 loops, best of 3:..
+    5 loops, best of 3:...
 
-::
+Changing bases::
 
-    sage: # Changing bases
-    sage: time h(s[10]^4);
+    sage: h(s[10]^4);
     h[10, 10, 10, 10]
-    Time: CPU 1.07 s, Wall: 1.08 s
 
-::
+We get an arbitrary symmetric function to demonstrate some functionality::
 
-    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
     1/2*h[] + 3*h[1, 1, 1] + 2*h[2, 1, 1]
 
-::
+The omega involution::
 
-    sage: foo.omega() # 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]) # The Hall scalar product
+    sage: foo.scalar(s[3,1])
     4
 
 ::
@@ -99,60 +84,44 @@ Demonstration: Symmetric functions
     sage: foo.skew_by(e[2,1])
     9*h[] + 10*h[1]
 
-::
+We can define skew partition directly::
 
-    sage: # We can define skew partition directly
     sage: mu = Partition([3,2])/Partition([2,1])
     sage: mu
-    [[3, 2], [2, 1]]
-
-::
+    [3, 2] / [2, 1]
 
     sage: s(mu)
     s[1, 1] + s[2]
 
-::
+We can expand a symmetric function in monomials::
 
-    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
 
-::
+Or we can choose our alphabet::
 
-    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: foo.inner_plethysm?  # not tested
 
-::
+The transition matrix from the Schur basis to the power basis
+Try s.transition_matrix? for more information::
 
-    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]
@@ -162,9 +131,8 @@ Demonstration: Symmetric functions
     [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: # 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]
@@ -175,16 +143,13 @@ Demonstration: Symmetric functions
     ....:     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: # 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: # 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]
 
@@ -193,62 +158,47 @@ Demonstration: Symmetric functions
     sage: foo.map_support(remove_last_part) == h(foo).map_support(remove_last_part)
     False
 
-::
+We can easily get specific coefficients::
 
-    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)
-
+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]))
-    Traceback (most recent call last):
-    ...
-    TypeError
+    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: S.base_ring()
-    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: # 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: 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
 
 ::
 
@@ -258,80 +208,62 @@ Demonstration: Symmetric functions
 ::
 
     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}
+    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: # 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: 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]
 
-::
-
-    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
+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
 
-    sage: # Press <tab> after the following to see the different
-    sage: # variants of Jack polynomials in sage
-    sage: JackPolynomials
+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: # Press <tab> after the following to see the different
-    sage: # 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: ks2 = kSchurFunctions(R,2,t=R(t))
-    sage: s = SymmetricFunctions(R).schur()
-
+    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 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
+    ValueError: s[3] is not in the image