Non-Commutative Symmetric Functions¶

class sage.combinat.ncsf_qsym.ncsf.NonCommutativeSymmetricFunctions(R)¶

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

The abstract algebra of non-commutative symmetric functions.

We construct the abstract algebra of non-commutative symmetric functions over the rational numbers:

sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: NCSF
Non-Commutative Symmetric Functions over the Rational Field
sage: S = NCSF.complete()
sage: R = NCSF.ribbon()
sage: S[2,1]*R[1,2]
S[2, 1, 1, 2] - S[2, 1, 3]

NCSF is the unique free (non-commutative!) graded connected algebra with one generator in each degree:

sage: NCSF.category()
Join of Category of hopf algebras over Rational Field
    and Category of graded algebras over Rational Field
    and Category of monoids with realizations
    and Category of graded coalgebras over Rational Field
    and Category of coalgebras over Rational Field with realizations
    and Category of cocommutative coalgebras over Rational Field

sage: [S[i].degree() for i in range(10)]
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

We use the Sage standard renaming idiom to get shorter outputs:

sage: NCSF.rename("NCSF")
sage: NCSF
NCSF

NCSF has many representations as a concrete algebra. Each of them has a distinguished basis, and its elements are expanded in this basis. Here is the \(\Psi\) (Psi) representation:

sage: Psi = NCSF.Psi()
sage: Psi
NCSF in the Psi basis

Elements of Psi are linear combinations of basis elements indexed by compositions:

sage: Psi.an_element()
2*Psi[] + 2*Psi[1] + 3*Psi[1, 1]

The basis itself is accessible through:

sage: Psi.basis()
Lazy family (Term map from Compositions of non-negative integers...
sage: Psi.basis().keys()
Compositions of non-negative integers

To construct an element one can therefore do:

sage: Psi.basis()[Composition([2,1,3])]
Psi[2, 1, 3]

As this is rather cumbersome, the following abuses of notation are allowed:

sage: Psi[Composition([2, 1, 3])]
Psi[2, 1, 3]
sage: Psi[[2, 1, 3]]
Psi[2, 1, 3]
sage: Psi[2, 1, 3]
Psi[2, 1, 3]

or even:

sage: Psi[(i for i in [2, 1, 3])]
Psi[2, 1, 3]

Unfortunately, due to a limitation in Python syntax, one cannot use:

sage: Psi[]       # not implemented

Instead, you can use:

sage: Psi[[]]
Psi[]

Now, we can construct linear combinations of basis elements:

sage: Psi[2,1,3] + 2 * (Psi[4] + Psi[2,1])
2*Psi[2, 1] + Psi[2, 1, 3] + 2*Psi[4]

Algebra structure

To start with, Psi is a graded algebra, the grading being induced by the size of compositions. The one is the basis element indexed by the empty composition:

sage: Psi.one()
Psi[]
sage: S.one()
S[]
sage: R.one()
R[]

As we have seen above, the Psi basis is multiplicative; that is multiplication is induced by linearity from the concatenation of compositions:

sage: Psi[1,3] * Psi[2,1]
Psi[1, 3, 2, 1]
sage: (Psi.one() + 2 * Psi[1,3]) * Psi[2, 4]
2*Psi[1, 3, 2, 4] + Psi[2, 4]

Hopf algebra structure

Psi is further endowed with a coalgebra structure. The coproduct is an algebra morphism, and therefore determined by its values on the generators; those are primitive:

sage: Psi[1].coproduct()
Psi[] # Psi[1] + Psi[1] # Psi[]
sage: Psi[2].coproduct()
Psi[] # Psi[2] + Psi[2] # Psi[]

The coproduct, being cocommutative on the generators, is cocommutative everywhere:

sage: Psi[1,2].coproduct()
Psi[] # Psi[1, 2] + Psi[1] # Psi[2] + Psi[1, 2] # Psi[] + Psi[2] # Psi[1]

The algebra and coalgebra structures on Psi combine to form a bialgebra structure, which cooperates with the grading to form a connected graded bialgebra. Thus, as any connected graded bialgebra, Psi is a Hopf algebra. Over QQ (or any other \(\QQ\)-algebra), this Hopf algebra Psi is isomorphic to the tensor algebra of its space of primitive elements.

The antipode is an anti-algebra morphism; in the Psi basis, it sends the generators to their opposites and changes their sign if they are of odd degree:

sage: Psi[3].antipode()
-Psi[3]
sage: Psi[1,3,2].antipode()
-Psi[2, 3, 1]
sage: Psi[1,3,2].coproduct().apply_multilinear_morphism(lambda be,ga: Psi(be)*Psi(ga).antipode())
0

The counit is defined by sending all elements of positive degree to zero:

sage: S[3].degree(), S[3,1,2].degree(), S.one().degree()
(3, 6, 0)
sage: S[3].counit()
0
sage: S[3,1,2].counit()
0
sage: S.one().counit()
1
sage: (S[3] - 2*S[3,1,2] + 7).counit()
7
sage: (R[3] - 2*R[3,1,2] + 7).counit()
7

It is possible to change the prefix used to display the basis elements using the method print_options(). Say that for instance one wanted to display the Complete basis as having a prefix H instead of the default S:

sage: H = NCSF.complete()
sage: H.an_element()
2*S[] + 2*S[1] + 3*S[1, 1]
sage: H.print_options(prefix='H')
sage: H.an_element()
2*H[] + 2*H[1] + 3*H[1, 1]
sage: H.print_options(prefix='S') #restore to 'S'

Concrete representations

NCSF admits the concrete realizations defined in [NCSF1]:

sage: Phi        = NCSF.Phi()
sage: Psi        = NCSF.Psi()
sage: ribbon     = NCSF.ribbon()
sage: complete   = NCSF.complete()
sage: elementary = NCSF.elementary()

To change from one basis to another, one simply does:

sage: Phi(Psi[1])
Phi[1]
sage: Phi(Psi[3])
-1/4*Phi[1, 2] + 1/4*Phi[2, 1] + Phi[3]

In general, one can mix up different bases in computations:

sage: Phi[1] * Psi[1]
Phi[1, 1]

Some of the changes of basis are easy to guess:

sage: ribbon(complete[1,3,2])
R[1, 3, 2] + R[1, 5] + R[4, 2] + R[6]

This is the sum of all fatter compositions. Using the usual Möbius function for the boolean lattice, the inverse change of basis is given by the alternating sum of all fatter compositions:

sage: complete(ribbon[1,3,2])
S[1, 3, 2] - S[1, 5] - S[4, 2] + S[6]

The analogue of the elementary basis is the sum over all finer compositions than the ‘complement’ of the composition in the ribbon basis:

sage: Composition([1,3,2]).complement()
[2, 1, 2, 1]
sage: ribbon(elementary([1,3,2]))
R[1, 1, 1, 1, 1, 1] + R[1, 1, 1, 2, 1] + R[2, 1, 1, 1, 1] + R[2, 1, 2, 1]

By Möbius inversion on the composition poset, the ribbon basis element corresponding to a composition \(I\) is then the alternating sum over all compositions fatter than the complement composition of \(I\) in the elementary basis:

sage: elementary(ribbon[2,1,2,1])
L[1, 3, 2] - L[1, 5] - L[4, 2] + L[6]

The \(\Phi\) (Phi) and \(\Psi\) bases are computed by changing to and from the Complete basis. The expansion of \(\Psi\) basis is given in Proposition 4.5 of [NCSF1] by the formulae

\[S^I = \sum_{J \geq I} \frac{1}{\pi_u(J,I)} \Psi^J\]

and

\[\Psi^I = \sum_{J \geq I} (-1)^{\ell(J)-\ell(I)} lp(J,I) S^J\]

where the coefficients \(\pi_u(J,I)\) and \(lp(J,I)\) are coefficients in the methods coeff_pi() and coeff_lp() respectively. For example:

sage: Psi(complete[3])
1/6*Psi[1, 1, 1] + 1/3*Psi[1, 2] + 1/6*Psi[2, 1] + 1/3*Psi[3]
sage: complete(Psi[3])
S[1, 1, 1] - 2*S[1, 2] - S[2, 1] + 3*S[3]

The Phi basis is another analogue of the power sum basis from the algebra of symmetric functions and the expansion in the Complete basis is given in Proposition 4.9 of [NCSF1] by the formulae

\[S^I = \sum_{J \geq I} \frac{1}{sp(J,I)} \Phi^J\]

and

\[\Phi^I = \sum_{J \geq I} (-1)^{\ell(J)-\ell(I)} \frac{\prod_i I_i}{\ell(J,I)} S^J\]

where the coefficients \(sp(J,I)\) and \(\ell(J,I)\) are coefficients in the methods coeff_sp() and coeff_ell() respectively. For example:

sage: Phi(complete[3])
1/6*Phi[1, 1, 1] + 1/4*Phi[1, 2] + 1/4*Phi[2, 1] + 1/3*Phi[3]
sage: complete(Phi[3])
S[1, 1, 1] - 3/2*S[1, 2] - 3/2*S[2, 1] + 3*S[3]

Here is how to fetch the conversion morphisms:

sage: f = complete.coerce_map_from(elementary); f
Generic morphism:
  From: NCSF in the Elementary basis
  To:   NCSF in the Complete basis
sage: g = elementary.coerce_map_from(complete); g
Generic morphism:
  From: NCSF in the Complete basis
  To:   NCSF in the Elementary basis
sage: f.category()
Category of homsets of unital magmas and right modules over Rational Field and
  left modules over Rational Field
sage: f(elementary[1,2,2])
S[1, 1, 1, 1, 1] - S[1, 1, 1, 2] - S[1, 2, 1, 1] + S[1, 2, 2]
sage: g(complete[1,2,2])
L[1, 1, 1, 1, 1] - L[1, 1, 1, 2] - L[1, 2, 1, 1] + L[1, 2, 2]
sage: h = f*g; h
Composite map:
  From: NCSF in the Complete basis
  To:   NCSF in the Complete basis
  Defn:   Generic morphism:
          From: NCSF in the Complete basis
          To:   NCSF in the Elementary basis
        then
          Generic morphism:
          From: NCSF in the Elementary basis
          To:   NCSF in the Complete basis
sage: h(complete[1,3,2])
S[1, 3, 2]

Additional concrete representations

NCSF has some additional bases which appear in the literature:

sage: Monomial                 = NCSF.Monomial()
sage: Immaculate               = NCSF.Immaculate()
sage: dualQuasisymmetric_Schur = NCSF.dualQuasisymmetric_Schur()

The Monomial basis was introduced in [Tev2007] and the Immaculate basis was introduced in [BBSSZ2012]. The Quasisymmetric_Schur were defined in [QSCHUR] and the dual basis is implemented here as dualQuasisymmetric_Schur. Refer to the documentation for the use and definition of these bases.

Todo

implement fundamental, forgotten, and simple (coming from the simple modules of HS_n) bases.

We revert back to the original name from our custom short name NCSF:

sage: NCSF
NCSF
sage: NCSF.rename()
sage: NCSF
Non-Commutative Symmetric Functions over the Rational Field

class Bases(parent_with_realization)¶

Bases: sage.categories.realizations.Category_realization_of_parent

Category of bases of non-commutative symmetric functions.

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: N.Bases()
Category of bases of Non-Commutative Symmetric Functions over the Rational Field
sage: R = N.Ribbon()
sage: R in N.Bases()
True

class ElementMethods¶

Bases: object

bernstein_creation_operator(n)¶

Return the image of self under the \(n\)-th Bernstein creation operator.

Let \(n\) be an integer. The \(n\)-th Bernstein creation operator \(\mathbb{B}_n\) is defined as the endomorphism of the space \(NSym\) of noncommutative symmetric functions which sends every \(f\) to

\[\sum_{i \geq 0} (-1)^i H_{n+i} F_{1^i}^\perp,\]

where usual notations are in place (the letter \(H\) stands for the complete basis of \(NSym\), the letter \(F\) stands for the fundamental basis of the algebra \(QSym\) of quasisymmetric functions, and \(F_{1^i}^\perp\) means skewing (skew_by()) by \(F_{1^i}\)). Notice that \(F_{1^i}\) is nothing other than the elementary symmetric function \(e_i\).

This has been introduced in [BBSSZ2012], section 3.1, in analogy to the Bernstein creation operators on the symmetric functions (bernstein_creation_operator()), and studied further in [BBSSZ2012], mainly in the context of immaculate functions (Immaculate). In fact, if \((\alpha_1, \alpha_2, \ldots, \alpha_m)\) is an \(m\)-tuple of integers, then

\[\mathbb{B}_n I_{(\alpha_1, \alpha_2, \ldots, \alpha_m)} = I_{(n, \alpha_1, \alpha_2, \ldots, \alpha_m)},\]

where \(I_{(\alpha_1, \alpha_2, \ldots, \alpha_m)}\) is the immaculate function associated to the \(m\)-tuple \((\alpha_1, \alpha_2, \ldots, \alpha_m)\) (see immaculate_function()).

EXAMPLES:

We get the immaculate functions by repeated application of Bernstein creation operators:

sage: NSym = NonCommutativeSymmetricFunctions(ZZ)
sage: I = NSym.I()
sage: S = NSym.S()
sage: def immaculate_by_bernstein(xs):
....:     # immaculate function corresponding to integer
....:     # tuple ``xs``, computed by iterated application
....:     # of Bernstein creation operators.
....:     res = S.one()
....:     for i in reversed(xs):
....:         res = res.bernstein_creation_operator(i)
....:     return res
sage: import itertools
sage: all( immaculate_by_bernstein(p) == I.immaculate_function(p)
....:      for p in itertools.product(range(-1, 3), repeat=3))
True

Some examples:

sage: S[3,2].bernstein_creation_operator(-2)
S[2, 1]
sage: S[3,2].bernstein_creation_operator(-1)
S[1, 2, 1] - S[2, 2] - S[3, 1]
sage: S[3,2].bernstein_creation_operator(0)
-S[1, 2, 2] - S[1, 3, 1] + S[2, 2, 1] + S[3, 2]
sage: S[3,2].bernstein_creation_operator(1)
S[1, 3, 2] - S[2, 2, 2] - S[2, 3, 1] + S[3, 2, 1]
sage: S[3,2].bernstein_creation_operator(2)
S[2, 3, 2] - S[3, 2, 2] - S[3, 3, 1] + S[4, 2, 1]

chi()¶

Return the commutative image of a non-commutative symmetric function.

OUTPUT:

The commutative image of self. This will be a symmetric function.

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: R = N.ribbon()
sage: x = R.an_element(); x
2*R[] + 2*R[1] + 3*R[1, 1]
sage: x.to_symmetric_function()
2*s[] + 2*s[1] + 3*s[1, 1]
sage: y = N.Phi()[1,3]
sage: y.to_symmetric_function()
h[1, 1, 1, 1] - 3*h[2, 1, 1] + 3*h[3, 1]

expand(n, alphabet='x')¶

Expand the noncommutative symmetric function into an element of a free algebra in n indeterminates of an alphabet, which by default is 'x'.

INPUT:

n – a nonnegative integer; the number of variables in the expansion
alphabet – (default: 'x'); the alphabet in which self is to be expanded

OUTPUT:

An expansion of self into the n variables specified by alphabet.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: S = NSym.S()
sage: S[3].expand(3)
x0^3 + x0^2*x1 + x0^2*x2 + x0*x1^2 + x0*x1*x2
 + x0*x2^2 + x1^3 + x1^2*x2 + x1*x2^2 + x2^3
sage: L = NSym.L()
sage: L[3].expand(3)
x2*x1*x0
sage: L[2].expand(3)
x1*x0 + x2*x0 + x2*x1
sage: L[3].expand(4)
x2*x1*x0 + x3*x1*x0 + x3*x2*x0 + x3*x2*x1
sage: Psi = NSym.Psi()
sage: Psi[2, 1].expand(3)
x0^3 + x0^2*x1 + x0^2*x2 + x0*x1*x0 + x0*x1^2 + x0*x1*x2
 + x0*x2*x0 + x0*x2*x1 + x0*x2^2 - x1*x0^2 - x1*x0*x1
 - x1*x0*x2 + x1^2*x0 + x1^3 + x1^2*x2 + x1*x2*x0
 + x1*x2*x1 + x1*x2^2 - x2*x0^2 - x2*x0*x1 - x2*x0*x2
 - x2*x1*x0 - x2*x1^2 - x2*x1*x2 + x2^2*x0 + x2^2*x1 + x2^3

One can use a different set of variables by adding an optional argument alphabet=...:

sage: L[3].expand(4, alphabet="y")
y2*y1*y0 + y3*y1*y0 + y3*y2*y0 + y3*y2*y1

Todo

So far this is only implemented on the elementary basis, and everything else goes through coercion. Maybe it is worth shortcutting some of the other bases?

left_padded_kronecker_product(x)¶

Return the left-padded Kronecker product of self and x in the basis of self.

The left-padded Kronecker product is a bilinear map mapping two non-commutative symmetric functions to another, not necessarily preserving degree. It can be defined as follows: Let \(*\) denote the internal product (internal_product()) on the space of non-commutative symmetric functions. For any composition \(I\), let \(S^I\) denote the complete homogeneous symmetric function indexed by \(I\). For any compositions \(\alpha\), \(\beta\), \(\gamma\), let \(g^{\gamma}_{\alpha, \beta}\) denote the coefficient of \(S^{\gamma}\) in the internal product \(S^{\alpha} * S^{\beta}\). For every composition \(I = (i_1, i_2, \ldots, i_k)\) and every integer \(n > \left\lvert I \right\rvert\), define the `n`-completion of `I` to be the composition \((n - \left\lvert I \right\rvert, i_1, i_2, \ldots, i_k)\); this \(n\)-completion is denoted by \(I[n]\). Then, for any compositions \(\alpha\) and \(\beta\) and every integer \(n > \left\lvert \alpha \right\rvert + \left\lvert\beta\right\rvert\), we can write the internal product \(S^{\alpha[n]} * S^{\beta[n]}\) in the form

\[S^{\alpha[n]} * S^{\beta[n]} = \sum_{\gamma} g^{\gamma[n]}_{\alpha[n], \beta[n]} S^{\gamma[n]}\]

with \(\gamma\) ranging over all compositions. The coefficients \(g^{\gamma[n]}_{\alpha[n], \beta[n]}\) are independent on \(n\). These coefficients \(g^{\gamma[n]}_{\alpha[n], \beta[n]}\) are denoted by \(\widetilde{g}^{\gamma}_{\alpha, \beta}\), and the non-commutative symmetric function

\[\sum_{\gamma} \widetilde{g}^{\gamma}_{\alpha, \beta} S^{\gamma}\]

is said to be the left-padded Kronecker product of \(S^{\alpha}\) and \(S^{\beta}\). By bilinearity, this extends to a definition of a left-padded Kronecker product of any two non-commutative symmetric functions.

The left-padded Kronecker product on the non-commutative symmetric functions lifts the left-padded Kronecker product on the symmetric functions. More precisely: Let \(\pi\) denote the canonical projection (to_symmetric_function()) from the non-commutative symmetric functions to the symmetric functions. Then, any two non-commutative symmetric functions \(f\) and \(g\) satisfy

\[\pi(f \overline{*} g) = \pi(f) \overline{*} \pi(g),\]

where the \(\overline{*}\) on the left-hand side denotes the left-padded Kronecker product on the non-commutative symmetric functions, and the \(\overline{*}\) on the right-hand side denotes the left-padded Kronecker product on the symmetric functions.

INPUT:

x – element of the ring of non-commutative symmetric functions over the same base ring as self

OUTPUT:

the left-padded Kronecker product of self with x (an element of the ring of non-commutative symmetric functions in the same basis as self)

AUTHORS:

Darij Grinberg (15 Mar 2014)

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: S = NSym.S()
sage: S[2,1].left_padded_kronecker_product(S[3])
S[1, 1, 1, 1] + S[1, 2, 1] + S[2, 1] + S[2, 1, 1, 1] + S[2, 2, 1] + S[3, 2, 1]
sage: S[2,1].left_padded_kronecker_product(S[1])
S[1, 1, 1] + S[1, 2, 1] + S[2, 1]
sage: S[1].left_padded_kronecker_product(S[2,1])
S[1, 1, 1] + S[2, 1] + S[2, 1, 1]
sage: S[1,1].left_padded_kronecker_product(S[2])
S[1, 1] + 2*S[1, 1, 1] + S[2, 1, 1]
sage: S[1].left_padded_kronecker_product(S[1,2,1])
S[1, 1, 1, 1] + S[1, 2, 1] + S[1, 2, 1, 1] + S[2, 1, 1]
sage: S[2].left_padded_kronecker_product(S[3])
S[1, 2] + S[2, 1, 1] + S[3, 2]

Taking the left-padded Kronecker product with \(1 = S^{\empty}\) is the identity map on the ring of non-commutative symmetric functions:

sage: all( S[Composition([])].left_padded_kronecker_product(S[lam])
....:      == S[lam].left_padded_kronecker_product(S[Composition([])])
....:      == S[lam] for i in range(4)
....:      for lam in Compositions(i) )
True

Here is a rule for the left-padded Kronecker product of \(S_1\) (this is the same as \(S^{(1)}\)) with any complete homogeneous function: Let \(I\) be a composition. Then, the left-padded Kronecker product of \(S_1\) and \(S^I\) is \(\sum_K a_K S^K\), where the sum runs over all compositions \(K\), and the coefficient \(a_K\) is defined as the number of ways to obtain \(K\) from \(I\) by one of the following two operations:

Insert a \(1\) at the end of \(I\).
Subtract \(1\) from one of the entries of \(I\) (and remove the entry if it thus becomes \(0\)), and insert a \(1\) at the end of \(I\).

We check this for compositions of size \(\leq 4\):

sage: def mults1(I):
....:     # Left left-padded Kronecker multiplication by S[1].
....:     res = S[I[:] + [1]]
....:     for k in range(len(I)):
....:         I2 = I[:]
....:         if I2[k] == 1:
....:             I2 = I2[:k] + I2[k+1:]
....:         else:
....:             I2[k] -= 1
....:         res += S[I2 + [1]]
....:     return res
sage: all( mults1(I) == S[1].left_padded_kronecker_product(S[I])
....:      for i in range(5) for I in Compositions(i) )
True

A similar rule can be made for the left-padded Kronecker product of any complete homogeneous function with \(S_1\): Let \(I\) be a composition. Then, the left-padded Kronecker product of \(S^I\) and \(S_1\) is \(\sum_K b_K S^K\), where the sum runs over all compositions \(K\), and the coefficient \(b_K\) is defined as the number of ways to obtain \(K\) from \(I\) by one of the following two operations:

Insert a \(1\) at the front of \(I\).
Subtract \(1\) from one of the entries of \(I\) (and remove the entry if it thus becomes \(0\)), and insert a \(1\) right after this entry.

We check this for compositions of size \(\leq 4\):

sage: def mults2(I):
....:     # Left left-padded Kronecker multiplication by S[1].
....:     res = S[[1] + I[:]]
....:     for k in range(len(I)):
....:         I2 = I[:]
....:         i2k = I2[k]
....:         if i2k != 1:
....:             I2 = I2[:k] + [i2k-1, 1] + I2[k+1:]
....:         res += S[I2]
....:     return res
sage: all( mults2(I) == S[I].left_padded_kronecker_product(S[1])
....:      for i in range(5) for I in Compositions(i) )
True

Checking the \(\pi(f \overline{*} g) = \pi(f) \overline{*} \pi(g)\) equality:

sage: NSym = NonCommutativeSymmetricFunctions(ZZ)
sage: R = NSym.R()
sage: def testpi(n):
....:     for I in Compositions(n):
....:         for J in Compositions(n):
....:             a = R[I].to_symmetric_function()
....:             b = R[J].to_symmetric_function()
....:             x = a.left_padded_kronecker_product(b)
....:             y = R[I].left_padded_kronecker_product(R[J])
....:             y = y.to_symmetric_function()
....:             if x != y:
....:                 return False
....:     return True
sage: testpi(3)
True

omega_involution()¶

Return the image of the noncommutative symmetric function self under the omega involution.

The omega involution is defined as the algebra antihomomorphism \(NCSF \to NCSF\) which, for every positive integer \(n\), sends the \(n\)-th complete non-commutative symmetric function \(S_n\) to the \(n\)-th elementary non-commutative symmetric function \(\Lambda_n\). This omega involution is denoted by \(\omega\). It can be shown that every composition \(I\) satisfies

\[\omega(S^I) = \Lambda^{I^r}, \quad \omega(\Lambda^I) = S^{I^r}, \quad \omega(R_I) = R_{I^t}, \quad \omega(\Phi^I) = (-1)^{|I|-\ell(I)} \Phi^{I^r}, \omega(\Psi^I) = (-1)^{|I|-\ell(I)} \Psi^{I^r},\]

where \(I^r\) denotes the reversed composition of \(I\), and \(I^t\) denotes the conjugate composition of \(I\), and \(\ell(I)\) denotes the length of the composition \(I\), and standard notations for classical bases of \(NCSF\) are being used (\(S\) for the complete basis, \(\Lambda\) for the elementary basis, \(R\) for the ribbon basis, \(\Phi\) for that of the power-sums of the second kind, and \(\Psi\) for that of the power-sums of the first kind). More generally, if \(f\) is a homogeneous element of \(NCSF\) of degree \(n\), then

\[\omega(f) = (-1)^n S(f),\]

where \(S\) denotes the antipode of \(NCSF\).

The omega involution \(\omega\) is an involution and a coalgebra automorphism of \(NCSF\). It is an automorphism of the graded vector space \(NCSF\). If \(\pi\) denotes the projection from \(NCSF\) to the ring of symmetric functions (to_symmetric_function()), then \(\pi(\omega(f)) = \omega(\pi(f))\) for every \(f \in NCSF\), where the \(\omega\) on the right hand side denotes the omega automorphism of \(Sym\).

The omega involution on \(NCSF\) is adjoint to the omega involution on \(QSym\) by the standard adjunction between \(NCSF\) and \(QSym\).

The omega involution has been denoted by \(\omega\) in [LMvW13], section 3.6. See [NCSF1], section 3.1 for the properties of this map.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(ZZ)
sage: S = NSym.S()
sage: L = NSym.L()
sage: L(S[3,2].omega_involution())
L[2, 3]
sage: L(S[6,3].omega_involution())
L[3, 6]
sage: L(S[1,3].omega_involution())
L[3, 1]
sage: L((S[9,1] - S[8,2] + 2*S[6,4] - 3*S[3] + 4*S[[]]).omega_involution()) # long time
4*L[] + L[1, 9] - L[2, 8] - 3*L[3] + 2*L[4, 6]
sage: L((S[3,3] - 2*S[2]).omega_involution())
-2*L[2] + L[3, 3]
sage: L(S([4,2]).omega_involution())
L[2, 4]
sage: R = NSym.R()
sage: R([4,2]).omega_involution()
R[1, 2, 1, 1, 1]
sage: R.zero().omega_involution()
0
sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: Phi = NSym.Phi()
sage: Phi([2,1]).omega_involution()
-Phi[1, 2]
sage: Psi = NSym.Psi()
sage: Psi([2,1]).omega_involution()
-Psi[1, 2]
sage: Psi([3,1]).omega_involution()
Psi[1, 3]

Testing the \(\pi(\omega(f)) = \omega(\pi(f))\) relation noticed above:

sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: R = NSym.R()
sage: all( R(I).omega_involution().to_symmetric_function()
....:      == R(I).to_symmetric_function().omega_involution()
....:      for I in Compositions(4) )
True

The omega involution on \(QSym\) is adjoint to the omega involution on \(NSym\) with respect to the duality pairing:

sage: QSym = QuasiSymmetricFunctions(QQ)
sage: M = QSym.M()
sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: S = NSym.S()
sage: all( all( M(I).omega_involution().duality_pairing(S(J))
....:           == M(I).duality_pairing(S(J).omega_involution())
....:           for I in Compositions(2) )
....:      for J in Compositions(3) )
True

psi_involution()¶

Return the image of the noncommutative symmetric function self under the involution \(\psi\).

The involution \(\psi\) is defined as the linear map \(NCSF \to NCSF\) which, for every composition \(I\), sends the complete noncommutative symmetric function \(S^I\) to the elementary noncommutative symmetric function \(\Lambda^I\). It can be shown that every composition \(I\) satisfies

\[\psi(R_I) = R_{I^c}, \quad \psi(S^I) = \Lambda^I, \quad \psi(\Lambda^I) = S^I, \quad \psi(\Phi^I) = (-1)^{|I| - \ell(I)} \Phi^I\]

where \(I^c\) denotes the complement of the composition \(I\), and \(\ell(I)\) denotes the length of \(I\), and where standard notations for classical bases of \(NCSF\) are being used (\(S\) for the complete basis, \(\Lambda\) for the elementary basis, \(\Phi\) for the basis of the power sums of the second kind, and \(R\) for the ribbon basis). The map \(\psi\) is an involution and a graded Hopf algebra automorphism of \(NCSF\). If \(\pi\) denotes the projection from \(NCSF\) to the ring of symmetric functions (to_symmetric_function()), then \(\pi(\psi(f)) = \omega(\pi(f))\) for every \(f \in NCSF\), where the \(\omega\) on the right hand side denotes the omega automorphism of \(Sym\).

The involution \(\psi\) of \(NCSF\) is adjoint to the involution \(\psi\) of \(QSym\) by the standard adjunction between \(NCSF\) and \(QSym\).

The involution \(\psi\) has been denoted by \(\psi\) in [LMvW13], section 3.6.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(ZZ)
sage: R = NSym.R()
sage: R[3,2].psi_involution()
R[1, 1, 2, 1]
sage: R[6,3].psi_involution()
R[1, 1, 1, 1, 1, 2, 1, 1]
sage: (R[9,1] - R[8,2] + 2*R[2,4] - 3*R[3] + 4*R[[]]).psi_involution()
4*R[] - 3*R[1, 1, 1] + R[1, 1, 1, 1, 1, 1, 1, 1, 2] - R[1, 1, 1, 1, 1, 1, 1, 2, 1] + 2*R[1, 2, 1, 1, 1]
sage: (R[3,3] - 2*R[2]).psi_involution()
-2*R[1, 1] + R[1, 1, 2, 1, 1]
sage: R([2,1,1]).psi_involution()
R[1, 3]
sage: S = NSym.S()
sage: S([2,1]).psi_involution()
S[1, 1, 1] - S[2, 1]
sage: S.zero().psi_involution()
0
sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: Phi = NSym.Phi()
sage: Phi([2,1]).psi_involution()
-Phi[2, 1]
sage: Phi([3,1]).psi_involution()
Phi[3, 1]

The Psi basis doesn’t behave as nicely:

sage: Psi = NSym.Psi()
sage: Psi([2,1]).psi_involution()
-Psi[2, 1]
sage: Psi([3,1]).psi_involution()
1/2*Psi[1, 2, 1] - 1/2*Psi[2, 1, 1] + Psi[3, 1]

The involution \(\psi\) commutes with the antipode:

sage: all( R(I).psi_involution().antipode()
....:      == R(I).antipode().psi_involution()
....:      for I in Compositions(4) )
True

Testing the \(\pi(\psi(f)) = \omega(\pi(f))\) relation noticed above:

sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: R = NSym.R()
sage: all( R(I).psi_involution().to_symmetric_function()
....:      == R(I).to_symmetric_function().omega()
....:      for I in Compositions(4) )
True

The involution \(\psi\) of \(QSym\) is adjoint to the involution \(\psi\) of \(NSym\) with respect to the duality pairing:

sage: QSym = QuasiSymmetricFunctions(QQ)
sage: M = QSym.M()
sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: S = NSym.S()
sage: all( all( M(I).psi_involution().duality_pairing(S(J))
....:           == M(I).duality_pairing(S(J).psi_involution())
....:           for I in Compositions(2) )
....:      for J in Compositions(3) )
True

star_involution()¶

Return the image of the noncommutative symmetric function self under the star involution.

The star involution is defined as the algebra antihomomorphism \(NCSF \to NCSF\) which, for every positive integer \(n\), sends the \(n\)-th complete non-commutative symmetric function \(S_n\) to \(S_n\). Denoting by \(f^{\ast}\) the image of an element \(f \in NCSF\) under this star involution, it can be shown that every composition \(I\) satisfies

\[(S^I)^{\ast} = S^{I^r}, \quad (\Lambda^I)^{\ast} = \Lambda^{I^r}, \quad R_I^{\ast} = R_{I^r}, \quad (\Phi^I)^{\ast} = \Phi^{I^r},\]

where \(I^r\) denotes the reversed composition of \(I\), and standard notations for classical bases of \(NCSF\) are being used (\(S\) for the complete basis, \(\Lambda\) for the elementary basis, \(R\) for the ribbon basis, and \(\Phi\) for that of the power-sums of the second kind). The star involution is an involution and a coalgebra automorphism of \(NCSF\). It is an automorphism of the graded vector space \(NCSF\). Under the canonical isomorphism between the \(n\)-th graded component of \(NCSF\) and the descent algebra of the symmetric group \(S_n\) (see to_descent_algebra()), the star involution (restricted to the \(n\)-th graded component) corresponds to the automorphism of the descent algebra given by \(x \mapsto \omega_n x \omega_n\), where \(\omega_n\) is the permutation \((n, n-1, \ldots, 1) \in S_n\) (written here in one-line notation). If \(\pi\) denotes the projection from \(NCSF\) to the ring of symmetric functions (to_symmetric_function()), then \(\pi(f^{\ast}) = \pi(f)\) for every \(f \in NCSF\).

The star involution on \(NCSF\) is adjoint to the star involution on \(QSym\) by the standard adjunction between \(NCSF\) and \(QSym\).

The star involution has been denoted by \(\rho\) in [LMvW13], section 3.6. See [NCSF2], section 2.3 for the properties of this map.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(ZZ)
sage: S = NSym.S()
sage: S[3,2].star_involution()
S[2, 3]
sage: S[6,3].star_involution()
S[3, 6]
sage: (S[9,1] - S[8,2] + 2*S[6,4] - 3*S[3] + 4*S[[]]).star_involution()
4*S[] + S[1, 9] - S[2, 8] - 3*S[3] + 2*S[4, 6]
sage: (S[3,3] - 2*S[2]).star_involution()
-2*S[2] + S[3, 3]
sage: S([4,2]).star_involution()
S[2, 4]
sage: R = NSym.R()
sage: R([4,2]).star_involution()
R[2, 4]
sage: R.zero().star_involution()
0
sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: Phi = NSym.Phi()
sage: Phi([2,1]).star_involution()
Phi[1, 2]

The Psi basis doesn’t behave as nicely:

sage: Psi = NSym.Psi()
sage: Psi([2,1]).star_involution()
Psi[1, 2]
sage: Psi([3,1]).star_involution()
1/2*Psi[1, 1, 2] - 1/2*Psi[1, 2, 1] + Psi[1, 3]

The star involution commutes with the antipode:

sage: all( R(I).star_involution().antipode()
....:      == R(I).antipode().star_involution()
....:      for I in Compositions(4) )
True

Checking the relation with the descent algebra described above:

sage: def descent_test(n):
....:     DA = DescentAlgebra(QQ, n)
....:     NSym = NonCommutativeSymmetricFunctions(QQ)
....:     S = NSym.S()
....:     DAD = DA.D()
....:     w_n = DAD(set(range(1, n)))
....:     for I in Compositions(n):
....:         if not (S[I].star_involution()
....:                 == w_n * S[I].to_descent_algebra(n) * w_n):
....:             return False
....:         return True
sage: all( descent_test(i) for i in range(4) )
True
sage: all( descent_test(i) for i in range(6) ) # long time
True

Testing the \(\pi(f^{\ast}) = \pi(f)\) relation noticed above:

sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: R = NSym.R()
sage: all( R(I).star_involution().to_symmetric_function()
....:      == R(I).to_symmetric_function()
....:      for I in Compositions(4) )
True

The star involution on \(QSym\) is adjoint to the star involution on \(NSym\) with respect to the duality pairing:

sage: QSym = QuasiSymmetricFunctions(QQ)
sage: M = QSym.M()
sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: S = NSym.S()
sage: all( all( M(I).star_involution().duality_pairing(S(J))
....:           == M(I).duality_pairing(S(J).star_involution())
....:           for I in Compositions(2) )
....:      for J in Compositions(3) )
True

to_descent_algebra(n=None)¶

Return the image of the n-th degree homogeneous component of self in the descent algebra of \(S_n\) over the same base ring as self.

This is based upon the canonical isomorphism from the \(n\)-th degree homogeneous component of the algebra of noncommutative symmetric functions to the descent algebra of \(S_n\). This isomorphism maps the inner product of noncommutative symmetric functions either to the product in the descent algebra of \(S_n\) or to its opposite (depending on how the latter is defined).

If n is not specified, it will be taken to be the highest homogeneous component of self.

OUTPUT:

The image of the n-th homogeneous component of self under the isomorphism into the descent algebra of \(S_n\) over the same base ring as self.

EXAMPLES:

sage: S = NonCommutativeSymmetricFunctions(ZZ).S()
sage: S[2,1].to_descent_algebra(3)
B[2, 1]
sage: (S[1,2,1] - 3 * S[1,1,2]).to_descent_algebra(4)
-3*B[1, 1, 2] + B[1, 2, 1]
sage: S[2,1].to_descent_algebra(2)
0
sage: S[2,1].to_descent_algebra()
B[2, 1]
sage: S.zero().to_descent_algebra().parent()
Descent algebra of 0 over Integer Ring in the subset basis
sage: (S[1,2,1] - 3 * S[1,1,2]).to_descent_algebra(1)
0

to_fqsym()¶

Return the image of the non-commutative symmetric function self under the morphism \(\iota : NSym \to FQSym\).

This morphism is the injective algebra homomorphism \(NSym \to FQSym\) that sends each Complete generator \(S_n\) to \(F_{[1, 2, \ldots, n]}\). It is the inclusion map, if we regard both \(NSym\) and \(FQSym\) as rings of noncommutative power series.

See also

FreeQuasisymmetricFunctions for a definition of \(FQSym\).

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: R = N.ribbon()
sage: x = 2*R[[]] + 2*R[1] + 3*R[2]
sage: x.to_fqsym()
2*F[] + 2*F[1] + 3*F[1, 2]
sage: R[2,1].to_fqsym()
F[1, 3, 2] + F[3, 1, 2]
sage: x = R.an_element(); x
2*R[] + 2*R[1] + 3*R[1, 1]
sage: x.to_fqsym()
2*F[] + 2*F[1] + 3*F[2, 1]

sage: y = N.Phi()[1,2]
sage: y.to_fqsym()
F[1, 2, 3] - F[1, 3, 2] + F[2, 1, 3] + F[2, 3, 1]
 - F[3, 1, 2] - F[3, 2, 1]

sage: S = NonCommutativeSymmetricFunctions(QQ).S()
sage: S[2].to_fqsym()
F[1, 2]
sage: S[1,2].to_fqsym()
F[1, 2, 3] + F[2, 1, 3] + F[2, 3, 1]
sage: S[2,1].to_fqsym()
F[1, 2, 3] + F[1, 3, 2] + F[3, 1, 2]
sage: S[1,2,1].to_fqsym()
F[1, 2, 3, 4] + F[1, 2, 4, 3] + F[1, 4, 2, 3]
 + F[2, 1, 3, 4] + F[2, 1, 4, 3] + F[2, 3, 1, 4]
 + F[2, 3, 4, 1] + F[2, 4, 1, 3] + F[2, 4, 3, 1]
 + F[4, 1, 2, 3] + F[4, 2, 1, 3] + F[4, 2, 3, 1]

to_fsym()¶

Return the image of self under the natural map to \(FSym\).

There is an injective Hopf algebra morphism from \(NSym\) to \(FSym\) (see FreeSymmetricFunctions), which maps the ribbon \(R_\alpha\) indexed by a composition \(\alpha\) to the sum of all tableaux whose descent composition is \(\alpha\). If we regard \(NSym\) as a Hopf subalgebra of \(FQSym\) via the morphism \(\iota : NSym \to FQSym\) (implemented as to_fqsym()), then this injective morphism is just the inclusion map.

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: R = N.ribbon()
sage: x = 2*R[[]] + 2*R[1] + 3*R[2]
sage: x.to_fsym()
2*G[] + 2*G[1] + 3*G[12]
sage: R[2,1].to_fsym()
G[12|3]
sage: R[1,2].to_fsym()
G[13|2]
sage: R[2,1,2].to_fsym()
G[12|35|4] + G[125|3|4]
sage: x = R.an_element(); x
2*R[] + 2*R[1] + 3*R[1, 1]
sage: x.to_fsym()
2*G[] + 2*G[1] + 3*G[1|2]

sage: y = N.Phi()[1,2]
sage: y.to_fsym()
-G[1|2|3] - G[12|3] + G[123] + G[13|2]

sage: S = NonCommutativeSymmetricFunctions(QQ).S()
sage: S[2].to_fsym()
G[12]
sage: S[2,1].to_fsym()
G[12|3] + G[123]

to_ncsym()¶

Return the image of self under the injective algebra homomorphism \(\kappa : NSym \to NCSym\) that fixes the symmetric functions.

As usual, \(NCSym\) denotes the ring of symmetric functions in non-commuting variables. Let \(S_n\) denote a generator of the complete basis. The algebra homomorphism \(\kappa : NSym \to NCSym\) is defined by

\[S_n \mapsto \sum_{A \vdash [n]} \frac{\lambda(A)! \lambda(A)^!}{n!} \mathbf{m}_A .\]

It has the property that the canonical maps \(\chi : NCSym \to Sym\) and \(\rho : NSym \to Sym\) satisfy \(\chi \circ \kappa = \rho\).

Note

A remark in [BRRZ08] makes it clear that the embedding of \(NSym\) into \(NCSym\) that preserves the projection into the symmetric functions is not unique. While this seems to be a natural embedding, any free set of algebraic generators of \(NSym\) can be sent to a set of free elements in \(NCSym\) to form another embedding.

See also

NonCommutativeSymmetricFunctions for a definition of \(NCSym\).

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: S = N.complete()
sage: S[2].to_ncsym()
1/2*m{{1}, {2}} + m{{1, 2}}
sage: S[1,2,1].to_ncsym()
1/2*m{{1}, {2}, {3}, {4}} + 1/2*m{{1}, {2}, {3, 4}} + m{{1}, {2, 3}, {4}}
 + m{{1}, {2, 3, 4}} + 1/2*m{{1}, {2, 4}, {3}} + 1/2*m{{1, 2}, {3}, {4}}
 + 1/2*m{{1, 2}, {3, 4}} + m{{1, 2, 3}, {4}} + m{{1, 2, 3, 4}}
 + 1/2*m{{1, 2, 4}, {3}} + 1/2*m{{1, 3}, {2}, {4}} + 1/2*m{{1, 3}, {2, 4}}
 + 1/2*m{{1, 3, 4}, {2}} + 1/2*m{{1, 4}, {2}, {3}} + m{{1, 4}, {2, 3}}
sage: S[1,2].to_ncsym()
1/2*m{{1}, {2}, {3}} + m{{1}, {2, 3}} + 1/2*m{{1, 2}, {3}}
 + m{{1, 2, 3}} + 1/2*m{{1, 3}, {2}}
sage: S[[]].to_ncsym()
m{}

sage: R = N.ribbon()
sage: x = R.an_element(); x
2*R[] + 2*R[1] + 3*R[1, 1]
sage: x.to_ncsym()
2*m{} + 2*m{{1}} + 3/2*m{{1}, {2}}
sage: R[2,1].to_ncsym()
1/3*m{{1}, {2}, {3}} + 1/6*m{{1}, {2, 3}}
 + 2/3*m{{1, 2}, {3}} + 1/6*m{{1, 3}, {2}}

sage: Phi = N.Phi()
sage: Phi[1,2].to_ncsym()
m{{1}, {2, 3}} + m{{1, 2, 3}}
sage: Phi[1,3].to_ncsym()
-1/4*m{{1}, {2}, {3, 4}} - 1/4*m{{1}, {2, 3}, {4}} + m{{1}, {2, 3, 4}}
 + 1/2*m{{1}, {2, 4}, {3}} - 1/4*m{{1, 2}, {3, 4}} - 1/4*m{{1, 2, 3}, {4}}
 + m{{1, 2, 3, 4}} + 1/2*m{{1, 2, 4}, {3}} + 1/2*m{{1, 3}, {2, 4}}
 - 1/4*m{{1, 3, 4}, {2}} - 1/4*m{{1, 4}, {2, 3}}

to_symmetric_function()¶

Return the commutative image of a non-commutative symmetric function.

OUTPUT:

The commutative image of self. This will be a symmetric function.

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: R = N.ribbon()
sage: x = R.an_element(); x
2*R[] + 2*R[1] + 3*R[1, 1]
sage: x.to_symmetric_function()
2*s[] + 2*s[1] + 3*s[1, 1]
sage: y = N.Phi()[1,3]
sage: y.to_symmetric_function()
h[1, 1, 1, 1] - 3*h[2, 1, 1] + 3*h[3, 1]

to_symmetric_group_algebra()¶

Return the image of a non-commutative symmetric function into the symmetric group algebra where the ribbon basis element indexed by a composition is associated with the sum of all permutations which have descent set equal to said composition. In compliance with the anti- isomorphism between the descent algebra and the non-commutative symmetric functions, we index descent positions by the reversed composition.

OUTPUT:

The image of self under the embedding of the \(n\)-th degree homogeneous component of the non-commutative symmetric functions in the symmetric group algebra of \(S_n\). This can behave unexpectedly when self is not homogeneous.

EXAMPLES:

sage: R=NonCommutativeSymmetricFunctions(QQ).R()
sage: R[2,1].to_symmetric_group_algebra()
[1, 3, 2] + [2, 3, 1]
sage: R([]).to_symmetric_group_algebra()
[]

verschiebung(n)¶

Return the image of the noncommutative symmetric function self under the \(n\)-th Verschiebung operator.

The \(n\)-th Verschiebung operator \(\mathbf{V}_n\) is defined to be the map from the \(\mathbf{k}\)-algebra of noncommutative symmetric functions to itself that sends the complete function \(S^I\) indexed by a composition \(I = (i_1, i_2, \ldots , i_k)\) to \(S^{(i_1/n, i_2/n, \ldots , i_k/n)}\) if all of the numbers \(i_1, i_2, \ldots, i_k\) are divisible by \(n\), and to \(0\) otherwise. This operator \(\mathbf{V}_n\) is a Hopf algebra endomorphism. For every positive integer \(r\) with \(n \mid r\), it satisfies

\[\mathbf{V}_n(S_r) = S_{r/n}, \quad \mathbf{V}_n(\Lambda_r) = (-1)^{r - r/n} \Lambda_{r/n}, \quad \mathbf{V}_n(\Psi_r) = n \Psi_{r/n}, \quad \mathbf{V}_n(\Phi_r) = n \Phi_{r/n}\]

(where \(S_r\) denotes the \(r\)-th complete non-commutative symmetric function, \(\Lambda_r\) denotes the \(r\)-th elementary non-commutative symmetric function, \(\Psi_r\) denotes the \(r\)-th power-sum non-commutative symmetric function of the first kind, and \(\Phi_r\) denotes the \(r\)-th power-sum non-commutative symmetric function of the second kind). For every positive integer \(r\) with \(n \nmid r\), it satisfes

\[\mathbf{V}_n(S_r) = \mathbf{V}_n(\Lambda_r) = \mathbf{V}_n(\Psi_r) = \mathbf{V}_n(\Phi_r) = 0.\]

The \(n\)-th Verschiebung operator is also called the \(n\)-th Verschiebung endomorphism.

It is a lift of the \(n\)-th Verschiebung operator on the ring of symmetric functions (verschiebung()) to the ring of noncommutative symmetric functions.

The action of the \(n\)-th Verschiebung operator can also be described on the ribbon Schur functions. Namely, every composition \(I\) of size \(n \ell\) satisfies

\[\mathbf{V}_n ( R_I ) = (-1)^{\ell(I) - \ell(J)} \cdot R_{J / n},\]

where \(J\) denotes the meet of the compositions \(I\) and \((\underbrace{n, n, \ldots, n}_{|I|/n \mbox{ times}})\), where \(\ell(I)\) is the length of \(I\), and where \(J / n\) denotes the composition obtained by dividing every entry of \(J\) by \(n\). For a composition \(I\) of size not divisible by \(n\), we have \(\mathbf{V}_n( R_I ) = 0\).

INPUT:

n – a positive integer

OUTPUT:

The result of applying the \(n\)-th Verschiebung operator (on the ring of noncommutative symmetric functions) to self.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(ZZ)
sage: S = NSym.S()
sage: S[3,2].verschiebung(2)
0
sage: S[6,4].verschiebung(2)
S[3, 2]
sage: (S[9,1] - S[8,2] + 2*S[6,4] - 3*S[3] + 4*S[[]]).verschiebung(2)
4*S[] + 2*S[3, 2] - S[4, 1]
sage: (S[3,3] - 2*S[2]).verschiebung(3)
S[1, 1]
sage: S([4,2]).verschiebung(1)
S[4, 2]
sage: R = NSym.R()
sage: R([4,2]).verschiebung(2)
R[2, 1]

Being Hopf algebra endomorphisms, the Verschiebung operators commute with the antipode:

sage: all( R(I).verschiebung(2).antipode()
....:      == R(I).antipode().verschiebung(2)
....:      for I in Compositions(4) )
True

They lift the Verschiebung operators of the ring of symmetric functions:

sage: all( S(I).verschiebung(2).to_symmetric_function()
....:      == S(I).to_symmetric_function().verschiebung(2)
....:      for I in Compositions(4) )
True

The Frobenius operators on \(QSym\) are adjoint to the Verschiebung operators on \(NSym\) with respect to the duality pairing:

sage: QSym = QuasiSymmetricFunctions(ZZ)
sage: M = QSym.M()
sage: all( all( M(I).frobenius(3).duality_pairing(S(J))
....:           == M(I).duality_pairing(S(J).verschiebung(3))
....:           for I in Compositions(2) )
....:      for J in Compositions(3) )
True

class ParentMethods¶

Bases: object

immaculate_function(xs)¶

Return the immaculate function corresponding to the integer vector xs, written in the basis self.

If \(\alpha\) is any integer vector – i.e., an element of \(\ZZ^m\) for some \(m \in \NN\) –, the immaculate function corresponding to \(\alpha\) is a non-commutative symmetric function denoted by \(\mathfrak{S}_{\alpha}\). One way to define this function is by setting

\[\mathfrak{S}_{\alpha} = \sum_{\sigma \in S_m} (-1)^{\sigma} S_{\alpha_1 + \sigma(1) - 1} S_{\alpha_2 + \sigma(2) - 2} \cdots S_{\alpha_m + \sigma(m) - m},\]

where \(\alpha\) is written in the form \((\alpha_1, \alpha_2, \ldots, \alpha_m)\), and where \(S\) stands for the complete basis (Complete).

The immaculate function \(\mathfrak{S}_{\alpha}\) first appeared in [BBSSZ2012] (where it was defined differently, but the definition we gave above appeared as Theorem 3.27).

The immaculate functions \(\mathfrak{S}_{\alpha}\) for \(\alpha\) running over all compositions (i.e., finite sequences of positive integers) form a basis of \(NCSF\). This is the immaculate basis (Immaculate).

INPUT:

xs – list (or tuple or any iterable – possibly a composition) of integers

OUTPUT:

The immaculate function \(\mathfrak{S}_{xs}\) written in the basis self.

EXAMPLES:

Let us first check that, for xs a composition, we get the same as the result of self.realization_of().I()[xs]:

sage: def test_comp(xs):
....:     NSym = NonCommutativeSymmetricFunctions(QQ)
....:     I = NSym.I()
....:     return I[xs] == I.immaculate_function(xs)
sage: def test_allcomp(n):
....:     return all( test_comp(c) for c in Compositions(n) )
sage: test_allcomp(1)
True
sage: test_allcomp(2)
True
sage: test_allcomp(3)
True

Now some examples of non-composition immaculate functions:

sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: I = NSym.I()
sage: I.immaculate_function([0, 1])
0
sage: I.immaculate_function([0, 2])
-I[1, 1]
sage: I.immaculate_function([-1, 1])
-I[]
sage: I.immaculate_function([2, -1])
0
sage: I.immaculate_function([2, 0])
I[2]
sage: I.immaculate_function([2, 0, 1])
0
sage: I.immaculate_function([1, 0, 2])
-I[1, 1, 1]
sage: I.immaculate_function([2, 0, 2])
-I[2, 1, 1]
sage: I.immaculate_function([0, 2, 0, 2])
I[1, 1, 1, 1] + I[1, 2, 1]
sage: I.immaculate_function([2, 0, 2, 0, 2])
I[2, 1, 1, 1, 1] + I[2, 1, 2, 1]

to_symmetric_function()¶

Morphism to the algebra of symmetric functions.

This is constructed by extending the computation on the basis or by coercion to the complete basis.

OUTPUT:

The module morphism from the basis self to the symmetric functions which corresponds to taking a commutative image.

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: R = N.ribbon()
sage: x = R.an_element(); x
2*R[] + 2*R[1] + 3*R[1, 1]
sage: R.to_symmetric_function(x)
2*s[] + 2*s[1] + 3*s[1, 1]
sage: nM = N.Monomial()
sage: nM.to_symmetric_function(nM[3,1])
h[1, 1, 1, 1] - 7/2*h[2, 1, 1] + h[2, 2] + 7/2*h[3, 1] - 2*h[4]

to_symmetric_function_on_basis(I)¶

The image of the basis element indexed by I under the map to the symmetric functions.

This default implementation does a change of basis and computes the image in the complete basis.

INPUT:

I – a composition

OUTPUT:

The image of the non-commutative basis element of self indexed by the composition I under the map from non-commutative symmetric functions to the symmetric functions. This will be a symmetric function.

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: I = N.Immaculate()
sage: I.to_symmetric_function(I[1,3])
-h[2, 2] + h[3, 1]
sage: I.to_symmetric_function(I[1,2])
0
sage: Phi = N.Phi()
sage: Phi.to_symmetric_function_on_basis([3,1,2])==Phi.to_symmetric_function(Phi[3,1,2])
True
sage: Phi.to_symmetric_function_on_basis([])
h[]

super_categories()¶

Return the super categories of the category of bases of the non-commutative symmetric functions.

OUTPUT:

list

class Complete(NCSF)¶

Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass

The Hopf algebra of non-commutative symmetric functions in the Complete basis.

The Complete basis is defined in Definition 3.4 of [NCSF1], where it is denoted by \((S^I)_I\). It is a multiplicative basis, and is connected to the elementary generators \(\Lambda_i\) of the ring of non-commutative symmetric functions by the following relation: Define a non-commutative symmetric function \(S_n\) for every nonnegative integer \(n\) by the power series identity

\[\sum_{k \geq 0} t^k S_k = \left( \sum_{k \geq 0} (-t)^k \Lambda_k \right)^{-1},\]

with \(\Lambda_0\) denoting \(1\). For every composition \((i_1, i_2, \ldots, i_k)\), we have \(S^{(i_1, i_2, \ldots, i_k)} = S_{i_1} S_{i_2} \cdots S_{i_k}\).

EXAMPLES:

sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: S = NCSF.Complete(); S
Non-Commutative Symmetric Functions over the Rational Field in the Complete basis
sage: S.an_element()
2*S[] + 2*S[1] + 3*S[1, 1]

The following are aliases for this basis:

sage: NCSF.complete()
Non-Commutative Symmetric Functions over the Rational Field in the Complete basis
sage: NCSF.S()
Non-Commutative Symmetric Functions over the Rational Field in the Complete basis

class Element¶

Bases: sage.modules.with_basis.indexed_element.IndexedFreeModuleElement

An element in the Complete basis.

psi_involution()¶

Return the image of the noncommutative symmetric function self under the involution \(\psi\).

\[\psi(R_I) = R_{I^c}, \quad \psi(S^I) = \Lambda^I, \quad \psi(\Lambda^I) = S^I, \quad \psi(\Phi^I) = (-1)^{|I| - \ell(I)} \Phi^I\]

The involution \(\psi\) of \(NCSF\) is adjoint to the involution \(\psi\) of \(QSym\) by the standard adjunction between \(NCSF\) and \(QSym\).

The involution \(\psi\) has been denoted by \(\psi\) in [LMvW13], section 3.6.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(ZZ)
sage: S = NSym.S()
sage: L = NSym.L()
sage: S[3,1].psi_involution()
S[1, 1, 1, 1] - S[1, 2, 1] - S[2, 1, 1] + S[3, 1]
sage: L(S[3,1].psi_involution())
L[3, 1]
sage: S[[]].psi_involution()
S[]
sage: S[1,1].psi_involution()
S[1, 1]
sage: (S[2,1] - 2*S[2]).psi_involution()
-2*S[1, 1] + S[1, 1, 1] + 2*S[2] - S[2, 1]

The implementation at hand is tailored to the complete basis. It is equivalent to the generic implementation via the ribbon basis:

sage: R = NSym.R()
sage: all( R(S[I].psi_involution()) == R(S[I]).psi_involution()
....:      for I in Compositions(4) )
True

dual()¶

Return the dual basis to the complete basis of non-commutative symmetric functions. This is the Monomial basis of quasi-symmetric functions.

OUTPUT:

The Monomial basis of quasi-symmetric functions.

EXAMPLES:

sage: S = NonCommutativeSymmetricFunctions(QQ).S()
sage: S.dual()
Quasisymmetric functions over the Rational Field in the Monomial basis

internal_product_on_basis(I, J)¶

The internal product of two non-commutative symmetric complete functions.

See internal_product() for a thorough documentation of this operation.

INPUT:

I, J – compositions

OUTPUT:

The internal product of the complete non-commutative symmetric function basis elements indexed by I and J, expressed in the complete basis.

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: S = N.complete()
sage: S.internal_product_on_basis([2,2],[1,2,1])
2*S[1, 1, 1, 1] + S[1, 1, 2] + S[2, 1, 1]
sage: S.internal_product_on_basis([2,2],[1,2])
0

to_symmetric_function()¶

Morphism to the algebra of symmetric functions.

This is constructed by extending the computation on the complete basis.

OUTPUT:

The module morphism from the basis self to the symmetric functions which corresponds to taking a commutative image.

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: S = N.complete()
sage: S.to_symmetric_function(S[3,1,2])
h[3, 2, 1]
sage: S.to_symmetric_function(S[[]])
h[]

to_symmetric_function_on_basis(I)¶

The commutative image of a complete element

The commutative image of a basis element is obtained by sorting the indexing composition of the basis element and the output is in the complete basis of the symmetric functions.

INPUT:

I – a composition

OUTPUT:

The commutative image of the complete basis element indexed by I. The result is the complete symmetric function indexed by the partition obtained by sorting I.

EXAMPLES:

sage: S=NonCommutativeSymmetricFunctions(QQ).complete()
sage: S.to_symmetric_function_on_basis([2,1,3])
h[3, 2, 1]
sage: S.to_symmetric_function_on_basis([])
h[]

class Elementary(NCSF)¶

Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass

The Hopf algebra of non-commutative symmetric functions in the Elementary basis.

The Elementary basis is defined in Definition 3.4 of [NCSF1], where it is denoted by \((\Lambda^I)_I\). It is a multiplicative basis, and is obtained from the elementary generators \(\Lambda_i\) of the ring of non-commutative symmetric functions through the formula \(\Lambda^{(i_1, i_2, \ldots, i_k)} = \Lambda_{i_1} \Lambda_{i_2} \cdots \Lambda_{i_k}\) for every composition \((i_1, i_2, \ldots, i_k)\).

EXAMPLES:

sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: L = NCSF.Elementary(); L
Non-Commutative Symmetric Functions over the Rational Field in the Elementary basis
sage: L.an_element()
2*L[] + 2*L[1] + 3*L[1, 1]

The following are aliases for this basis:

sage: NCSF.elementary()
Non-Commutative Symmetric Functions over the Rational Field in the Elementary basis
sage: NCSF.L()
Non-Commutative Symmetric Functions over the Rational Field in the Elementary basis

class Element¶

Bases: sage.modules.with_basis.indexed_element.IndexedFreeModuleElement

psi_involution()¶

Return the image of the noncommutative symmetric function self under the involution \(\psi\).

\[\psi(R_I) = R_{I^c}, \quad \psi(S^I) = \Lambda^I, \quad \psi(\Lambda^I) = S^I, \quad \psi(\Phi^I) = (-1)^{|I| - \ell(I)} \Phi^I\]

The involution \(\psi\) of \(NCSF\) is adjoint to the involution \(\psi\) of \(QSym\) by the standard adjunction between \(NCSF\) and \(QSym\).

The involution \(\psi\) has been denoted by \(\psi\) in [LMvW13], section 3.6.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: S = NSym.S()
sage: L = NSym.L()
sage: L[3,1].psi_involution()
L[1, 1, 1, 1] - L[1, 2, 1] - L[2, 1, 1] + L[3, 1]
sage: S(L[3,1].psi_involution())
S[3, 1]
sage: L[[]].psi_involution()
L[]
sage: L[1,1].psi_involution()
L[1, 1]
sage: (L[2,1] - 2*L[2]).psi_involution()
-2*L[1, 1] + L[1, 1, 1] + 2*L[2] - L[2, 1]

The implementation at hand is tailored to the elementary basis. It is equivalent to the generic implementation via the ribbon basis:

sage: R = NSym.R()
sage: all( R(L[I].psi_involution()) == R(L[I]).psi_involution()
....:      for I in Compositions(3) )
True
sage: all( R(L[I].psi_involution()) == R(L[I]).psi_involution()
....:      for I in Compositions(4) )
True

star_involution()¶

Return the image of the noncommutative symmetric function self under the star involution.

\[(S^I)^{\ast} = S^{I^r}, \quad (\Lambda^I)^{\ast} = \Lambda^{I^r}, \quad R_I^{\ast} = R_{I^r}, \quad (\Phi^I)^{\ast} = \Phi^{I^r},\]

The star involution on \(NCSF\) is adjoint to the star involution on \(QSym\) by the standard adjunction between \(NCSF\) and \(QSym\).

The star involution has been denoted by \(\rho\) in [LMvW13], section 3.6. See [NCSF2], section 2.3 for the properties of this map.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(ZZ)
sage: L = NSym.L()
sage: L[3,3,2,3].star_involution()
L[3, 2, 3, 3]
sage: L[6,3,3].star_involution()
L[3, 3, 6]
sage: (L[1,9,1] - L[8,2] + 2*L[6,4] - 3*L[3] + 4*L[[]]).star_involution()
4*L[] + L[1, 9, 1] - L[2, 8] - 3*L[3] + 2*L[4, 6]
sage: (L[3,3] - 2*L[2]).star_involution()
-2*L[2] + L[3, 3]
sage: L([4,1]).star_involution()
L[1, 4]

The implementation at hand is tailored to the elementary basis. It is equivalent to the generic implementation via the complete basis:

sage: S = NSym.S()
sage: all( S(L[I].star_involution()) == S(L[I]).star_involution()
....:      for I in Compositions(4) )
True

verschiebung(n)¶

Return the image of the noncommutative symmetric function self under the \(n\)-th Verschiebung operator.

\[\mathbf{V}_n(S_r) = S_{r/n}, \quad \mathbf{V}_n(\Lambda_r) = (-1)^{r - r/n} \Lambda_{r/n}, \quad \mathbf{V}_n(\Psi_r) = n \Psi_{r/n}, \quad \mathbf{V}_n(\Phi_r) = n \Phi_{r/n}\]

\[\mathbf{V}_n(S_r) = \mathbf{V}_n(\Lambda_r) = \mathbf{V}_n(\Psi_r) = \mathbf{V}_n(\Phi_r) = 0.\]

The \(n\)-th Verschiebung operator is also called the \(n\)-th Verschiebung endomorphism.

It is a lift of the \(n\)-th Verschiebung operator on the ring of symmetric functions (verschiebung()) to the ring of noncommutative symmetric functions.

The action of the \(n\)-th Verschiebung operator can also be described on the ribbon Schur functions. Namely, every composition \(I\) of size \(n \ell\) satisfies

\[\mathbf{V}_n ( R_I ) = (-1)^{\ell(I) - \ell(J)} \cdot R_{J / n},\]

INPUT:

n – a positive integer

OUTPUT:

The result of applying the \(n\)-th Verschiebung operator (on the ring of noncommutative symmetric functions) to self.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(ZZ)
sage: L = NSym.L()
sage: L([4,2]).verschiebung(2)
-L[2, 1]
sage: L([2,4]).verschiebung(2)
-L[1, 2]
sage: L([6]).verschiebung(2)
-L[3]
sage: L([2,1]).verschiebung(3)
0
sage: L([3]).verschiebung(2)
0
sage: L([]).verschiebung(2)
L[]
sage: L([5, 1]).verschiebung(3)
0
sage: L([5, 1]).verschiebung(6)
0
sage: L([5, 1]).verschiebung(2)
0
sage: L([1, 2, 3, 1]).verschiebung(7)
0
sage: L([7]).verschiebung(7)
L[1]
sage: L([1, 2, 3, 1]).verschiebung(5)
0
sage: (L[1] - L[2] + 2*L[3]).verschiebung(1)
L[1] - L[2] + 2*L[3]

I¶: alias of NonCommutativeSymmetricFunctions.Immaculate

class Immaculate(NCSF)¶

Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass

The immaculate basis of the non-commutative symmetric functions.

The immaculate basis first appears in Berg, Bergeron, Saliola, Serrano and Zabrocki’s [BBSSZ2012]. It can be described as the family \((\mathfrak{S}_{\alpha})\), where \(\alpha\) runs over all compositions, and \(\mathfrak{S}_{\alpha}\) denotes the immaculate function corresponding to \(\alpha\) (see immaculate_function()).

If \(\alpha\) is a composition \((\alpha_1, \alpha_2, \ldots, \alpha_m)\), then

\[\mathfrak{S}_{\alpha} = \sum_{\sigma \in S_m} (-1)^{\sigma} S_{\alpha_1 + \sigma(1) - 1} S_{\alpha_2 + \sigma(2) - 2} \cdots S_{\alpha_m + \sigma(m) - m}.\]

Warning

This basis contains only the immaculate functions indexed by compositions (i.e., finite sequences of positive integers). To obtain the remaining immaculate functions (sensu lato), use the immaculate_function() method. Calling the immaculate basis with a list which is not a composition will currently return garbage!

EXAMPLES:

sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: I = NCSF.I()
sage: I([1,3,2])*I([1])
I[1, 3, 2, 1] + I[1, 3, 3] + I[1, 4, 2] + I[2, 3, 2]
sage: I([1])*I([1,3,2])
I[1, 1, 3, 2] - I[2, 2, 1, 2] - I[2, 2, 2, 1] - I[2, 2, 3] - I[3, 2, 2]
sage: I([1,3])*I([1,1])
I[1, 3, 1, 1] + I[1, 4, 1] + I[2, 3, 1] + I[2, 4]
sage: I([3,1])*I([2,1])
I[3, 1, 2, 1] + I[3, 2, 1, 1] + I[3, 2, 2] + I[3, 3, 1] + I[4, 1, 1, 1] + I[4, 1, 2] + 2*I[4, 2, 1] + I[4, 3] + I[5, 1, 1] + I[5, 2]
sage: R = NCSF.ribbon()
sage: I(R[1,3,1])
I[1, 3, 1] + I[2, 2, 1] + I[2, 3] + I[3, 1, 1] + I[3, 2]
sage: R(I(R([2,1,3])))
R[2, 1, 3]

class Element¶

Bases: sage.modules.with_basis.indexed_element.IndexedFreeModuleElement

An element in the Immaculate basis.

bernstein_creation_operator(n)¶

Return the image of self under the \(n\)-th Bernstein creation operator.

Let \(n\) be an integer. The \(n\)-th Bernstein creation operator \(\mathbb{B}_n\) is defined as the endomorphism of the space \(NSym\) of noncommutative symmetric functions given by

\[\mathbb{B}_n I_{(\alpha_1, \alpha_2, \ldots, \alpha_m)} = I_{(n, \alpha_1, \alpha_2, \ldots, \alpha_m)},\]

where \(I_{(\alpha_1, \alpha_2, \ldots, \alpha_m)}\) is the immaculate function associated to the \(m\)-tuple \((\alpha_1, \alpha_2, \ldots, \alpha_m) \in \ZZ^m\).

This has been introduced in [BBSSZ2012], section 3.1, in analogy to the Bernstein creation operators on the symmetric functions.

For more information on the \(n\)-th Bernstein creation operator, see bernstein_creation_operator().

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: I = NSym.I()
sage: b = I[1,3,2,1]
sage: b.bernstein_creation_operator(3)
I[3, 1, 3, 2, 1]
sage: b.bernstein_creation_operator(5)
I[5, 1, 3, 2, 1]
sage: elt = b + 3*I[4,1,2]
sage: elt.bernstein_creation_operator(1)
I[1, 1, 3, 2, 1] + 3*I[1, 4, 1, 2]

We check that this agrees with the definition on the Complete basis:

sage: S = NSym.S()
sage: S(elt).bernstein_creation_operator(1) == S(elt.bernstein_creation_operator(1))
True

Check on non-positive values of \(n\):

sage: I[2,2,2].bernstein_creation_operator(-1)
I[1, 1, 1, 2] + I[1, 1, 2, 1] + I[1, 2, 1, 1] - I[1, 2, 2]
sage: I[2,3,2].bernstein_creation_operator(0)
-I[1, 1, 3, 2] - I[1, 2, 2, 2] - I[1, 2, 3, 1] + I[2, 3, 2]

dual()¶

Return the dual basis to the Immaculate basis of NCSF.

The basis returned is the dualImmaculate basis of QSym.

OUTPUT:

The dualImmaculate basis of the quasi-symmetric functions.

EXAMPLES:

sage: I=NonCommutativeSymmetricFunctions(QQ).Immaculate()
sage: I.dual()
Quasisymmetric functions over the Rational Field in the dualImmaculate
basis

L¶: alias of NonCommutativeSymmetricFunctions.Elementary

class Monomial(NCSF)¶

Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass

The monomial basis defined in Tevlin’s paper [Tev2007].

The monomial basis is well-defined only when the base ring is a \(\QQ\)-algebra. It is the basis denoted by \((M^I)_I\) in [Tev2007].

class MultiplicativeBases(parent_with_realization)¶

Bases: sage.categories.realizations.Category_realization_of_parent

Category of multiplicative bases of non-commutative symmetric functions.

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: N.MultiplicativeBases()
Category of multiplicative bases of Non-Commutative Symmetric Functions over the Rational Field

The complete basis is a multiplicative basis, but the ribbon basis is not:

sage: N.Complete() in N.MultiplicativeBases()
True
sage: N.Ribbon() in N.MultiplicativeBases()
False

class ParentMethods¶

Bases: object

algebra_generators()¶

Return the algebra generators of a given multiplicative basis of non-commutative symmetric functions.

OUTPUT:

The family of generators of the multiplicative basis self.

EXAMPLES:

sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi()
sage: f = Psi.algebra_generators()
sage: f
Lazy family (<lambda>(i))_{i in Positive integers}
sage: f[1], f[2], f[3]
(Psi[1], Psi[2], Psi[3])

algebra_morphism(on_generators, **keywords)¶

Given a map defined on the generators of the multiplicative basis self, return the algebra morphism that extends this map to the whole algebra of non-commutative symmetric functions.

INPUT:

on_generators – a function defined on the index set of the generators (that is, on the positive integers)
anti – a boolean; defaults to False
category – a category; defaults to None

OUTPUT:

The algebra morphism of self which is defined by on_generators in the basis self. When anti is set to True, an algebra anti-morphism is computed instead of an algebra morphism.

EXAMPLES:

sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: Psi = NCSF.Psi()
sage: double = lambda i: Psi[i,i]
sage: f = Psi.algebra_morphism(double, codomain = Psi)
sage: f
Generic endomorphism of Non-Commutative Symmetric Functions over the Rational Field in the Psi basis
sage: f(2*Psi[[]] + 3 * Psi[1,3,2] + Psi[2,4] )
2*Psi[] + 3*Psi[1, 1, 3, 3, 2, 2] + Psi[2, 2, 4, 4]
sage: f.category()
Category of endsets of unital magmas and right modules over Rational Field and left modules over Rational Field

When extra properties about the morphism are known, one can specify the category of which it is a morphism:

sage: negate = lambda i: -Psi[i]
sage: f = Psi.algebra_morphism(negate, codomain = Psi, category = GradedHopfAlgebrasWithBasis(QQ))
sage: f
Generic endomorphism of Non-Commutative Symmetric Functions over the Rational Field in the Psi basis
sage: f(2*Psi[[]] + 3 * Psi[1,3,2] + Psi[2,4] )
2*Psi[] - 3*Psi[1, 3, 2] + Psi[2, 4]
sage: f.category()
Category of endsets of hopf algebras over Rational Field and graded modules over Rational Field

If anti is true, this returns an anti-algebra morphism:

sage: f = Psi.algebra_morphism(double, codomain = Psi, anti=True)
sage: f
Generic endomorphism of Non-Commutative Symmetric Functions over the Rational Field in the Psi basis
sage: f(2*Psi[[]] + 3 * Psi[1,3,2] + Psi[2,4] )
2*Psi[] + 3*Psi[2, 2, 3, 3, 1, 1] + Psi[4, 4, 2, 2]
sage: f.category()
Category of endsets of modules with basis over Rational Field

antipode()¶

Return the antipode morphism on the basis self.

The antipode of \(NSym\) is closely related to the omega involution; see omega_involution() for the latter.

OUTPUT:

The antipode module map from non-commutative symmetric functions on basis self.

EXAMPLES:

sage: S=NonCommutativeSymmetricFunctions(QQ).S()
sage: S.antipode
Generic endomorphism of Non-Commutative Symmetric Functions over the Rational Field in the Complete basis

coproduct()¶

Return the coproduct morphism in the basis self.

OUTPUT:

The coproduct module map from non-commutative symmetric functions on basis self.

EXAMPLES:

sage: S=NonCommutativeSymmetricFunctions(QQ).S()
sage: S.coproduct
Generic morphism:
  From: Non-Commutative Symmetric Functions over the Rational Field in the Complete basis
  To:   Non-Commutative Symmetric Functions over the Rational Field in the Complete basis # Non-Commutative Symmetric Functions over the Rational Field in the Complete basis

product_on_basis(composition1, composition2)¶

Return the product of two basis elements from the multiplicative basis. Multiplication is just concatenation on compositions.

INPUT:

composition1, composition2 – integer compositions

OUTPUT:

The product of the two non-commutative symmetric functions indexed by composition1 and composition2 in the multiplicative basis self. This will be again a non-commutative symmetric function.

EXAMPLES:

sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi()
sage: Psi[3,1,2] * Psi[4,2] == Psi[3,1,2,4,2]
True
sage: S = NonCommutativeSymmetricFunctions(QQ).S()
sage: S.product_on_basis(Composition([2,1]), Composition([1,2]))
S[2, 1, 1, 2]

to_symmetric_function()¶

Morphism to the algebra of symmetric functions.

This is constructed by extending the algebra morphism by the image of the generators.

OUTPUT:

The module morphism from the basis self to the symmetric functions which corresponds to taking a commutative image.

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: S = N.complete()
sage: S.to_symmetric_function(S[1,3])
h[3, 1]
sage: Phi = N.Phi()
sage: Phi.to_symmetric_function(Phi[1,3])
h[1, 1, 1, 1] - 3*h[2, 1, 1] + 3*h[3, 1]
sage: Psi = N.Psi()
sage: Psi.to_symmetric_function(Psi[1,3])
h[1, 1, 1, 1] - 3*h[2, 1, 1] + 3*h[3, 1]

to_symmetric_function_on_generators(i)¶

Morphism of the generators to symmetric functions.

This is constructed by coercion to the complete basis and applying the morphism.

OUTPUT:

The module morphism from the basis self to the symmetric functions which corresponds to taking a commutative image.

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: Phi = N.Phi()
sage: Phi.to_symmetric_function_on_generators(3)
h[1, 1, 1] - 3*h[2, 1] + 3*h[3]
sage: Phi.to_symmetric_function_on_generators(0)
h[]
sage: Psi = N.Psi()
sage: Psi.to_symmetric_function_on_generators(3)
h[1, 1, 1] - 3*h[2, 1] + 3*h[3]
sage: L = N.elementary()
sage: L.to_symmetric_function_on_generators(3)
h[1, 1, 1] - 2*h[2, 1] + h[3]

super_categories()¶

Return the super categories of the category of multiplicative bases of the non-commutative symmetric functions.

OUTPUT:

list

class MultiplicativeBasesOnGroupLikeElements(parent_with_realization)¶

Bases: sage.categories.realizations.Category_realization_of_parent

Category of multiplicative bases on grouplike elements of non-commutative symmetric functions.

Here, a “multiplicative basis on grouplike elements” means a multiplicative basis whose generators \((f_1, f_2, f_3, \ldots )\) satisfy

\[\Delta(f_i) = \sum_{j=0}^{i} f_j \otimes f_{i-j}\]

with \(f_0 = 1\). (In other words, the generators are to form a divided power sequence in the sense of a coalgebra.) This does not mean that the generators are grouplike, but means that the element \(1 + f_1 + f_2 + f_3 + \cdots\) in the completion of the ring of non-commutative symmetric functions with respect to the grading is grouplike.

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: N.MultiplicativeBasesOnGroupLikeElements()
Category of multiplicative bases on group like elements of Non-Commutative Symmetric Functions over the Rational Field

The complete basis is a multiplicative basis, but the ribbon basis is not:

sage: N.Complete() in N.MultiplicativeBasesOnGroupLikeElements()
True
sage: N.Ribbon() in N.MultiplicativeBasesOnGroupLikeElements()
False

class ParentMethods¶

Bases: object

antipode_on_basis(composition)¶

Return the application of the antipode to a basis element.

INPUT:

composition – a composition

OUTPUT:

The image of the basis element indexed by composition under the antipode map.

EXAMPLES:

sage: S = NonCommutativeSymmetricFunctions(QQ).complete()
sage: S.antipode_on_basis(Composition([2,1]))
-S[1, 1, 1] + S[1, 2]
sage: S[1].antipode() # indirect doctest
-S[1]
sage: S[2].antipode() # indirect doctest
S[1, 1] - S[2]
sage: S[3].antipode() # indirect doctest
-S[1, 1, 1] + S[1, 2] + S[2, 1] - S[3]
sage: S[2,3].coproduct().apply_multilinear_morphism(lambda be,ga: S(be)*S(ga).antipode())
0
sage: S[2,3].coproduct().apply_multilinear_morphism(lambda be,ga: S(be).antipode()*S(ga))
0

coproduct_on_generators(i)¶

Return the image of the \(i^{th}\) generator of the algebra under the coproduct.

INPUT:

i – a positive integer

OUTPUT:

The result of applying the coproduct to the \(i^{th}\) generator of self.

EXAMPLES:

sage: S = NonCommutativeSymmetricFunctions(QQ).complete()
sage: S.coproduct_on_generators(3)
S[] # S[3] + S[1] # S[2] + S[2] # S[1] + S[3] # S[]

super_categories()¶

Return the super categories of the category of multiplicative bases of group-like elements of the non-commutative symmetric functions.

OUTPUT:

list

class MultiplicativeBasesOnPrimitiveElements(parent_with_realization)¶

Bases: sage.categories.realizations.Category_realization_of_parent

Category of multiplicative bases of the non-commutative symmetric functions whose generators are primitive elements.

An element \(x\) of a bialgebra is primitive if \(\Delta(x) = x \otimes 1 + 1 \otimes x\), where \(\Delta\) is the coproduct of the bialgebra.

Given a multiplicative basis and knowing the coproducts and antipodes of its generators, one can compute the coproduct and the antipode of any element, since they are respectively algebra morphisms and anti-morphisms. See antipode_on_generators() and coproduct_on_generators().

Todo

this could be generalized to any free algebra.

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: N.MultiplicativeBasesOnPrimitiveElements()
Category of multiplicative bases on primitive elements of Non-Commutative Symmetric Functions over the Rational Field

The Phi and Psi bases are multiplicative bases whose generators are primitive elements, but the complete and ribbon bases are not:

sage: N.Phi() in N.MultiplicativeBasesOnPrimitiveElements()
True
sage: N.Psi() in N.MultiplicativeBasesOnPrimitiveElements()
True
sage: N.Complete() in N.MultiplicativeBasesOnPrimitiveElements()
False
sage: N.Ribbon() in N.MultiplicativeBasesOnPrimitiveElements()
False

class ParentMethods¶

Bases: object

antipode_on_generators(i)¶

Return the image of a generator of a primitive basis of the non-commutative symmetric functions under the antipode map.

INPUT:

i – a positive integer

OUTPUT:

The image of the \(i\)-th generator of the multiplicative basis self under the antipode of the algebra of non-commutative symmetric functions.

EXAMPLES:

sage: Psi=NonCommutativeSymmetricFunctions(QQ).Psi()
sage: Psi.antipode_on_generators(2)
-Psi[2]

coproduct_on_generators(i)¶

Return the image of the \(i^{th}\) generator of the multiplicative basis self under the coproduct.

INPUT:

i – a positive integer

OUTPUT:

The result of applying the coproduct to the \(i^{th}\) generator of self.

EXAMPLES:

sage: Psi = NonCommutativeSymmetricFunctions(QQ).Psi()
sage: Psi.coproduct_on_generators(3)
Psi[] # Psi[3] + Psi[3] # Psi[]

super_categories()¶

Return the super categories of the category of multiplicative bases of primitive elements of the non-commutative symmetric functions.

OUTPUT:

list

class Phi(NCSF)¶

Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass

The Hopf algebra of non-commutative symmetric functions in the Phi basis.

The Phi basis is defined in Definition 3.4 of [NCSF1], where it is denoted by \((\Phi^I)_I\). It is a multiplicative basis, and is connected to the elementary generators \(\Lambda_i\) of the ring of non-commutative symmetric functions by the following relation: Define a non-commutative symmetric function \(\Phi_n\) for every positive integer \(n\) by the power series identity

\[\sum_{k\geq 1} t^k \frac{1}{k} \Phi_k = -\log \left( \sum_{k \geq 0} (-t)^k \Lambda_k \right),\]

with \(\Lambda_0\) denoting \(1\). For every composition \((i_1, i_2, \ldots, i_k)\), we have \(\Phi^{(i_1, i_2, \ldots, i_k)} = \Phi_{i_1} \Phi_{i_2} \cdots \Phi_{i_k}\).

The \(\Phi\)-basis is well-defined only when the base ring is a \(\QQ\)-algebra. The elements of the \(\Phi\)-basis are known as the “power-sum non-commutative symmetric functions of the second kind”.

The generators \(\Phi_n\) are related to the (first) Eulerian idempotents in the descent algebras of the symmetric groups (see [NCSF1], 5.4 for details).

EXAMPLES:

sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: Phi = NCSF.Phi(); Phi
Non-Commutative Symmetric Functions over the Rational Field in the Phi basis
sage: Phi.an_element()
2*Phi[] + 2*Phi[1] + 3*Phi[1, 1]

class Element¶

Bases: sage.modules.with_basis.indexed_element.IndexedFreeModuleElement

psi_involution()¶

Return the image of the noncommutative symmetric function self under the involution \(\psi\).

\[\psi(R_I) = R_{I^c}, \quad \psi(S^I) = \Lambda^I, \quad \psi(\Lambda^I) = S^I, \quad \psi(\Phi^I) = (-1)^{|I| - \ell(I)} \Phi^I\]

The involution \(\psi\) of \(NCSF\) is adjoint to the involution \(\psi\) of \(QSym\) by the standard adjunction between \(NCSF\) and \(QSym\).

The involution \(\psi\) has been denoted by \(\psi\) in [LMvW13], section 3.6.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: Phi = NSym.Phi()
sage: Phi[3,2].psi_involution()
-Phi[3, 2]
sage: Phi[2,2].psi_involution()
Phi[2, 2]
sage: Phi[[]].psi_involution()
Phi[]
sage: (Phi[2,1] - 2*Phi[2]).psi_involution()
2*Phi[2] - Phi[2, 1]
sage: Phi(0).psi_involution()
0

The implementation at hand is tailored to the Phi basis. It is equivalent to the generic implementation via the ribbon basis:

sage: R = NSym.R()
sage: all( R(Phi[I].psi_involution()) == R(Phi[I]).psi_involution()
....:      for I in Compositions(4) )
True

star_involution()¶

Return the image of the noncommutative symmetric function self under the star involution.

\[(S^I)^{\ast} = S^{I^r}, \quad (\Lambda^I)^{\ast} = \Lambda^{I^r}, \quad R_I^{\ast} = R_{I^r}, \quad (\Phi^I)^{\ast} = \Phi^{I^r},\]

The star involution on \(NCSF\) is adjoint to the star involution on \(QSym\) by the standard adjunction between \(NCSF\) and \(QSym\).

The star involution has been denoted by \(\rho\) in [LMvW13], section 3.6. See [NCSF2], section 2.3 for the properties of this map.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(QQ)
sage: Phi = NSym.Phi()
sage: Phi[3,1,1,4].star_involution()
Phi[4, 1, 1, 3]
sage: Phi[4,2,1].star_involution()
Phi[1, 2, 4]
sage: (Phi[1,4] - Phi[2,3] + 2*Phi[5,4] - 3*Phi[3] + 4*Phi[[]]).star_involution()
4*Phi[] - 3*Phi[3] - Phi[3, 2] + Phi[4, 1] + 2*Phi[4, 5]
sage: (Phi[3,3] + 3*Phi[1]).star_involution()
3*Phi[1] + Phi[3, 3]
sage: Phi([2,1]).star_involution()
Phi[1, 2]

The implementation at hand is tailored to the Phi basis. It is equivalent to the generic implementation via the complete basis:

sage: S = NSym.S()
sage: all( S(Phi[I].star_involution()) == S(Phi[I]).star_involution()
....:      for I in Compositions(4) )
True

verschiebung(n)¶

Return the image of the noncommutative symmetric function self under the \(n\)-th Verschiebung operator.

\[\mathbf{V}_n(S_r) = S_{r/n}, \quad \mathbf{V}_n(\Lambda_r) = (-1)^{r - r/n} \Lambda_{r/n}, \quad \mathbf{V}_n(\Psi_r) = n \Psi_{r/n}, \quad \mathbf{V}_n(\Phi_r) = n \Phi_{r/n}\]

\[\mathbf{V}_n(S_r) = \mathbf{V}_n(\Lambda_r) = \mathbf{V}_n(\Psi_r) = \mathbf{V}_n(\Phi_r) = 0.\]

The \(n\)-th Verschiebung operator is also called the \(n\)-th Verschiebung endomorphism.

It is a lift of the \(n\)-th Verschiebung operator on the ring of symmetric functions (verschiebung()) to the ring of noncommutative symmetric functions.

The action of the \(n\)-th Verschiebung operator can also be described on the ribbon Schur functions. Namely, every composition \(I\) of size \(n \ell\) satisfies

\[\mathbf{V}_n ( R_I ) = (-1)^{\ell(I) - \ell(J)} \cdot R_{J / n},\]

INPUT:

n – a positive integer

OUTPUT:

The result of applying the \(n\)-th Verschiebung operator (on the ring of noncommutative symmetric functions) to self.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(ZZ)
sage: Phi = NSym.Phi()
sage: Phi([4,2]).verschiebung(2)
4*Phi[2, 1]
sage: Phi([2,4]).verschiebung(2)
4*Phi[1, 2]
sage: Phi([6]).verschiebung(2)
2*Phi[3]
sage: Phi([2,1]).verschiebung(3)
0
sage: Phi([3]).verschiebung(2)
0
sage: Phi([]).verschiebung(2)
Phi[]
sage: Phi([5, 1]).verschiebung(3)
0
sage: Phi([5, 1]).verschiebung(6)
0
sage: Phi([5, 1]).verschiebung(2)
0
sage: Phi([1, 2, 3, 1]).verschiebung(7)
0
sage: Phi([7]).verschiebung(7)
7*Phi[1]
sage: Phi([1, 2, 3, 1]).verschiebung(5)
0
sage: (Phi[1] - Phi[2] + 2*Phi[3]).verschiebung(1)
Phi[1] - Phi[2] + 2*Phi[3]

class Psi(NCSF)¶

Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass

The Hopf algebra of non-commutative symmetric functions in the Psi basis.

The Psi basis is defined in Definition 3.4 of [NCSF1], where it is denoted by \((\Psi^I)_I\). It is a multiplicative basis, and is connected to the elementary generators \(\Lambda_i\) of the ring of non-commutative symmetric functions by the following relation: Define a non-commutative symmetric function \(\Psi_n\) for every positive integer \(n\) by the power series identity

\[\frac{d}{dt} \sigma(t) = \sigma(t) \cdot \left( \sum_{k \geq 1} t^{k-1} \Psi_k \right),\]

where

\[\sigma(t) = \left( \sum_{k \geq 0} (-t)^k \Lambda_k \right)^{-1}\]

and where \(\Lambda_0\) denotes \(1\). For every composition \((i_1, i_2, \ldots, i_k)\), we have \(\Psi^{(i_1, i_2, \ldots, i_k)} = \Psi_{i_1} \Psi_{i_2} \cdots \Psi_{i_k}\).

The \(\Psi\)-basis is a basis only when the base ring is a \(\QQ\)-algebra (although the \(\Psi^I\) can be defined over any base ring). The elements of the \(\Psi\)-basis are known as the “power-sum non-commutative symmetric functions of the first kind”. The generators \(\Psi_n\) correspond to the Dynkin (quasi-)idempotents in the descent algebras of the symmetric groups (see [NCSF1], 5.2 for details).

Another (equivalent) definition of \(\Psi_n\) is

\[\Psi_n = \sum_{i=0}^{n-1} (-1)^i R_{1^i, n-i},\]

where \(R\) denotes the ribbon basis of \(NCSF\), and where \(1^i\) stands for \(i\) repetitions of the integer \(1\).

EXAMPLES:

sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: Psi = NCSF.Psi(); Psi
Non-Commutative Symmetric Functions over the Rational Field in the Psi basis
sage: Psi.an_element()
2*Psi[] + 2*Psi[1] + 3*Psi[1, 1]

Checking the equivalent definition of \(\Psi_n\):

sage: def test_psi(n):
....:     NCSF = NonCommutativeSymmetricFunctions(ZZ)
....:     R = NCSF.R()
....:     Psi = NCSF.Psi()
....:     a = R.sum([(-1) ** i * R[[1]*i + [n-i]]
....:                for i in range(n)])
....:     return a == R(Psi[n])
sage: test_psi(2)
True
sage: test_psi(3)
True
sage: test_psi(4)
True

class Element¶

Bases: sage.modules.with_basis.indexed_element.IndexedFreeModuleElement

verschiebung(n)¶

Return the image of the noncommutative symmetric function self under the \(n\)-th Verschiebung operator.

\[\mathbf{V}_n(S_r) = S_{r/n}, \quad \mathbf{V}_n(\Lambda_r) = (-1)^{r - r/n} \Lambda_{r/n}, \quad \mathbf{V}_n(\Psi_r) = n \Psi_{r/n}, \quad \mathbf{V}_n(\Phi_r) = n \Phi_{r/n}\]

\[\mathbf{V}_n(S_r) = \mathbf{V}_n(\Lambda_r) = \mathbf{V}_n(\Psi_r) = \mathbf{V}_n(\Phi_r) = 0.\]

The \(n\)-th Verschiebung operator is also called the \(n\)-th Verschiebung endomorphism.

It is a lift of the \(n\)-th Verschiebung operator on the ring of symmetric functions (verschiebung()) to the ring of noncommutative symmetric functions.

The action of the \(n\)-th Verschiebung operator can also be described on the ribbon Schur functions. Namely, every composition \(I\) of size \(n \ell\) satisfies

\[\mathbf{V}_n ( R_I ) = (-1)^{\ell(I) - \ell(J)} \cdot R_{J / n},\]

INPUT:

n – a positive integer

OUTPUT:

The result of applying the \(n\)-th Verschiebung operator (on the ring of noncommutative symmetric functions) to self.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(ZZ)
sage: Psi = NSym.Psi()
sage: Psi([4,2]).verschiebung(2)
4*Psi[2, 1]
sage: Psi([2,4]).verschiebung(2)
4*Psi[1, 2]
sage: Psi([6]).verschiebung(2)
2*Psi[3]
sage: Psi([2,1]).verschiebung(3)
0
sage: Psi([3]).verschiebung(2)
0
sage: Psi([]).verschiebung(2)
Psi[]
sage: Psi([5, 1]).verschiebung(3)
0
sage: Psi([5, 1]).verschiebung(6)
0
sage: Psi([5, 1]).verschiebung(2)
0
sage: Psi([1, 2, 3, 1]).verschiebung(7)
0
sage: Psi([7]).verschiebung(7)
7*Psi[1]
sage: Psi([1, 2, 3, 1]).verschiebung(5)
0
sage: (Psi[1] - Psi[2] + 2*Psi[3]).verschiebung(1)
Psi[1] - Psi[2] + 2*Psi[3]

internal_product_on_basis_by_bracketing(I, J)¶

The internal product of two elements of the Psi basis.

See internal_product() for a thorough documentation of this operation.

This is an implementation using [NCSF2] Lemma 3.10. It is fast when the length of \(I\) is small, but can get very slow otherwise. Therefore it is not being used by default for internally multiplying Psi functions.

INPUT:

I, J – compositions

OUTPUT:

The internal product of the elements of the Psi basis of \(NSym\) indexed by I and J, expressed in the Psi basis.

AUTHORS:

Travis Scrimshaw, 29 Mar 2014

EXAMPLES:

sage: N = NonCommutativeSymmetricFunctions(QQ)
sage: Psi = N.Psi()
sage: Psi.internal_product_on_basis_by_bracketing([2,2],[1,2,1])
0
sage: Psi.internal_product_on_basis_by_bracketing([1,2,1],[2,1,1])
4*Psi[1, 2, 1]
sage: Psi.internal_product_on_basis_by_bracketing([2,1,1],[1,2,1])
4*Psi[2, 1, 1]
sage: Psi.internal_product_on_basis_by_bracketing([1,2,1], [1,1,1,1])
0
sage: Psi.internal_product_on_basis_by_bracketing([3,1], [1,2,1])
-Psi[1, 2, 1] + Psi[2, 1, 1]
sage: Psi.internal_product_on_basis_by_bracketing([1,2,1], [3,1])
0
sage: Psi.internal_product_on_basis_by_bracketing([2,2],[1,2])
0
sage: Psi.internal_product_on_basis_by_bracketing([4], [1,2,1])
-Psi[1, 1, 2] + 2*Psi[1, 2, 1] - Psi[2, 1, 1]

R¶: alias of NonCommutativeSymmetricFunctions.Ribbon

class Ribbon(NCSF)¶

Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass

The Hopf algebra of non-commutative symmetric functions in the Ribbon basis.

The Ribbon basis is defined in Definition 3.12 of [NCSF1], where it is denoted by \((R_I)_I\). It is connected to the complete basis of the ring of non-commutative symmetric functions by the following relation: For every composition \(I\), we have

\[R_I = \sum_J (-1)^{\ell(I) - \ell(J)} S^J,\]

where the sum is over all compositions \(J\) which are coarser than \(I\) and \(\ell(I)\) denotes the length of \(I\). (See the proof of Proposition 4.13 in [NCSF1].)

The elements of the Ribbon basis are commonly referred to as the ribbon Schur functions.

EXAMPLES:

sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: R = NCSF.Ribbon(); R
Non-Commutative Symmetric Functions over the Rational Field in the Ribbon basis
sage: R.an_element()
2*R[] + 2*R[1] + 3*R[1, 1]

The following are aliases for this basis:

sage: NCSF.ribbon()
Non-Commutative Symmetric Functions over the Rational Field in the Ribbon basis
sage: NCSF.R()
Non-Commutative Symmetric Functions over the Rational Field in the Ribbon basis

class Element¶

Bases: sage.modules.with_basis.indexed_element.IndexedFreeModuleElement

star_involution()¶

Return the image of the noncommutative symmetric function self under the star involution.

\[(S^I)^{\ast} = S^{I^r}, \quad (\Lambda^I)^{\ast} = \Lambda^{I^r}, \quad R_I^{\ast} = R_{I^r}, \quad (\Phi^I)^{\ast} = \Phi^{I^r},\]

The star involution on \(NCSF\) is adjoint to the star involution on \(QSym\) by the standard adjunction between \(NCSF\) and \(QSym\).

The star involution has been denoted by \(\rho\) in [LMvW13], section 3.6. See [NCSF2], section 2.3 for the properties of this map.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(ZZ)
sage: R = NSym.R()
sage: R[3,1,4,2].star_involution()
R[2, 4, 1, 3]
sage: R[4,1,2].star_involution()
R[2, 1, 4]
sage: (R[1] - R[2] + 2*R[5,4] - 3*R[3] + 4*R[[]]).star_involution()
4*R[] + R[1] - R[2] - 3*R[3] + 2*R[4, 5]
sage: (R[3,3] - 21*R[1]).star_involution()
-21*R[1] + R[3, 3]
sage: R([14,1]).star_involution()
R[1, 14]

The implementation at hand is tailored to the ribbon basis. It is equivalent to the generic implementation via the complete basis:

sage: S = NSym.S()
sage: all( S(R[I].star_involution()) == S(R[I]).star_involution()
....:      for I in Compositions(4) )
True

verschiebung(n)¶

Return the image of the noncommutative symmetric function self under the \(n\)-th Verschiebung operator.

\[\mathbf{V}_n(S_r) = S_{r/n}, \quad \mathbf{V}_n(\Lambda_r) = (-1)^{r - r/n} \Lambda_{r/n}, \quad \mathbf{V}_n(\Psi_r) = n \Psi_{r/n}, \quad \mathbf{V}_n(\Phi_r) = n \Phi_{r/n}\]

\[\mathbf{V}_n(S_r) = \mathbf{V}_n(\Lambda_r) = \mathbf{V}_n(\Psi_r) = \mathbf{V}_n(\Phi_r) = 0.\]

The \(n\)-th Verschiebung operator is also called the \(n\)-th Verschiebung endomorphism.

It is a lift of the \(n\)-th Verschiebung operator on the ring of symmetric functions (verschiebung()) to the ring of noncommutative symmetric functions.

The action of the \(n\)-th Verschiebung operator can also be described on the ribbon Schur functions. Namely, every composition \(I\) of size \(n \ell\) satisfies

\[\mathbf{V}_n ( R_I ) = (-1)^{\ell(I) - \ell(J)} \cdot R_{J / n},\]

INPUT:

n – a positive integer

OUTPUT:

The result of applying the \(n\)-th Verschiebung operator (on the ring of noncommutative symmetric functions) to self.

EXAMPLES:

sage: NSym = NonCommutativeSymmetricFunctions(ZZ)
sage: R = NSym.R()
sage: R([4,2]).verschiebung(2)
R[2, 1]
sage: R([2,1]).verschiebung(3)
-R[1]
sage: R([3]).verschiebung(2)
0
sage: R([]).verschiebung(2)
R[]
sage: R([5, 1]).verschiebung(3)
-R[2]
sage: R([5, 1]).verschiebung(6)
-R[1]
sage: R([5, 1]).verschiebung(2)
-R[3]
sage: R([1, 2, 3, 1]).verschiebung(7)
-R[1]
sage: R([1, 2, 3, 1]).verschiebung(5)
0
sage: (R[1] - R[2] + 2*R[3]).verschiebung(1)
R[1] - R[2] + 2*R[3]

antipode_on_basis(composition)¶

Return the application of the antipode to a basis element of the ribbon basis self.

INPUT:

composition – a composition

OUTPUT:

The image of the basis element indexed by composition under the antipode map.

EXAMPLES:

sage: R = NonCommutativeSymmetricFunctions(QQ).ribbon()
sage: R.antipode_on_basis(Composition([2,1]))
-R[2, 1]
sage: R[3,1].antipode() # indirect doctest
R[2, 1, 1]
sage: R[[]].antipode() # indirect doctest
R[]

We check that the implementation of the antipode at hand does not contradict the generic one:

sage: S = NonCommutativeSymmetricFunctions(QQ).S()
sage: all( S(R[I].antipode()) == S(R[I]).antipode()
....:      for I in Compositions(4) )
True

dual()¶

Return the dual basis to the ribbon basis of the non-commutative symmetric functions. This is the Fundamental basis of the quasi-symmetric functions.

OUTPUT:

The fundamental basis of the quasi-symmetric functions.

EXAMPLES:

sage: R=NonCommutativeSymmetricFunctions(QQ).ribbon()
sage: R.dual()
Quasisymmetric functions over the Rational Field in the Fundamental basis

product_on_basis(I, J)¶

Return the product of two ribbon basis elements of the non-commutative symmetric functions.

INPUT:

I, J – compositions

OUTPUT:

The product of the ribbon functions indexed by I and J.

EXAMPLES:

sage: R = NonCommutativeSymmetricFunctions(QQ).ribbon()
sage: R[1,2,1] * R[3,1]
R[1, 2, 1, 3, 1] + R[1, 2, 4, 1]
sage: ( R[1,2] + R[3] ) * ( R[3,1] + R[1,2,1] )
R[1, 2, 1, 2, 1] + R[1, 2, 3, 1] + R[1, 3, 2, 1] + R[1, 5, 1] + R[3, 1, 2, 1] + R[3, 3, 1] + R[4, 2, 1] + R[6, 1]

to_symmetric_function_on_basis(I)¶

Return the commutative image of a ribbon basis element of the non-commutative symmetric functions.

INPUT:

I – a composition

OUTPUT:

The commutative image of the ribbon basis element indexed by I. This will be expressed as a symmetric function in the Schur basis.

EXAMPLES:

sage: R=NonCommutativeSymmetricFunctions(QQ).R()
sage: R.to_symmetric_function_on_basis(Composition([3,1,1]))
s[3, 1, 1]
sage: R.to_symmetric_function_on_basis(Composition([4,2,1]))
s[4, 2, 1] + s[5, 1, 1] + s[5, 2]
sage: R.to_symmetric_function_on_basis(Composition([]))
s[]

S¶: alias of NonCommutativeSymmetricFunctions.Complete

ZL¶: alias of NonCommutativeSymmetricFunctions.Zassenhaus_left

ZR¶: alias of NonCommutativeSymmetricFunctions.Zassenhaus_right

class Zassenhaus_left(NCSF)¶

Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass

The Hopf algebra of non-commutative symmetric functions in the left Zassenhaus basis.

This basis is the left-version of the basis defined in Section 2.5.1 of [HLNT09]. It is multiplicative, with \(Z_n\) defined as the element of \(NCSF_n\) satisfying the equation

\[\sigma_1 = \cdots exp(Z_n) \cdots exp(Z_2) exp(Z_1),\]

where

\[\sigma_1 = \sum_{n \geq 0} S_n .\]

It can be recursively computed by the formula

\[S_n = \sum_{\lambda \vdash n} \frac{1}{m_1(\lambda)! m_2(\lambda)! m_3(\lambda)! \cdots} Z_{\lambda_1} Z_{\lambda_2} Z_{\lambda_3} \cdots\]

for all \(n \geq 0\).

class Zassenhaus_right(NCSF)¶

Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass

The Hopf algebra of non-commutative symmetric functions in the right Zassenhaus basis.

This basis is defined in Section 2.5.1 of [HLNT09]. It is multiplicative, with \(Z_n\) defined as the element of \(NCSF_n\) satisfying the equation

\[\sigma_1 = exp(Z_1) exp(Z_2) exp(Z_3) \cdots exp(Z_n) \cdots\]

where

\[\sigma_1 = \sum_{n \geq 0} S_n .\]

It can be recursively computed by the formula

\[S_n = \sum_{\lambda \vdash n} \frac{1}{m_1(\lambda)! m_2(\lambda)! m_3(\lambda)! \cdots} \cdots Z_{\lambda_3} Z_{\lambda_2} Z_{\lambda_1}\]

for all \(n \geq 0\).

Note that there is a variant (called the “noncommutative power sum symmetric functions of the third kind”) in Definition 5.26 of [NCSF2] that satisfies:

\[\sigma_1 = exp(Z_1) exp(Z_2/2) exp(Z_3/3) \cdots exp(Z_n/n) \cdots.\]

a_realization()¶

Gives a realization of the algebra of non-commutative symmetric functions. This particular realization is the complete basis of non-commutative symmetric functions.

OUTPUT:

The realization of the non-commutative symmetric functions in the complete basis.

EXAMPLES:

sage: NonCommutativeSymmetricFunctions(ZZ).a_realization()
Non-Commutative Symmetric Functions over the Integer Ring in the Complete basis

complete¶: alias of NonCommutativeSymmetricFunctions.Complete

dQS¶: alias of NonCommutativeSymmetricFunctions.dualQuasisymmetric_Schur

dYQS¶: alias of NonCommutativeSymmetricFunctions.dualYoungQuasisymmetric_Schur

dual()¶

Return the dual to the non-commutative symmetric functions.

OUTPUT:

The dual of the non-commutative symmetric functions over a ring. This is the algebra of quasi-symmetric functions over the ring.

EXAMPLES:

sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: NCSF.dual()
Quasisymmetric functions over the Rational Field

class dualQuasisymmetric_Schur(NCSF)¶

Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass

The basis of NCSF dual to the Quasisymmetric-Schur basis of QSym.

The Quasisymmetric_Schur functions are defined in [QSCHUR] (see also Definition 5.1.1 of [LMvW13]). The dual basis in the algebra of non-commutative symmetric functions is defined by the following formula:

\[R_\alpha = \sum_{T} dQS_{shape(T)},\]

where the sum is over all standard composition tableaux with descent composition equal to \(\alpha\). The Quasisymmetric_Schur basis \(QS_\alpha\) has the property that

\[s_\lambda = \sum_{sort(\alpha) = \lambda} QS_\alpha.\]

As a consequence the commutative image of a dual Quasisymmetric-Schur element in the algebra of symmetric functions (the map defined in the method to_symmetric_function()) is equal to the Schur function indexed by the decreasing sort of the indexing composition.

EXAMPLES:

sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: dQS = NCSF.dQS()
sage: dQS([1,3,2])*dQS([1])
dQS[1, 2, 4] + dQS[1, 3, 2, 1] + dQS[1, 3, 3] + dQS[3, 2, 2]
sage: dQS([1])*dQS([1,3,2])
dQS[1, 1, 3, 2] + dQS[1, 3, 3] + dQS[1, 4, 2] + dQS[2, 3, 2]
sage: dQS([1,3])*dQS([1,1])
dQS[1, 3, 1, 1] + dQS[1, 4, 1] + dQS[3, 2, 1] + dQS[4, 2]
sage: dQS([3,1])*dQS([2,1])
dQS[1, 1, 4, 1] + dQS[1, 4, 2] + dQS[1, 5, 1] + dQS[2, 4, 1] + dQS[3, 1,
2, 1] + dQS[3, 2, 2] + dQS[3, 3, 1] + dQS[4, 3] + dQS[5, 2]
sage: dQS([1,1]).coproduct()
dQS[] # dQS[1, 1] + dQS[1] # dQS[1] + dQS[1, 1] # dQS[]
sage: dQS([3,3]).coproduct().monomial_coefficients()[(Composition([1,2]),Composition([1,2]))]
-1
sage: S = NCSF.complete()
sage: dQS(S[1,3,1])
dQS[1, 3, 1] + dQS[1, 4] + dQS[3, 2] + dQS[4, 1] + dQS[5]
sage: S(dQS[1,3,1])
S[1, 3, 1] - S[3, 2] - S[4, 1] + S[5]
sage: s = SymmetricFunctions(QQ).s()
sage: s(S(dQS([2,1,3])).to_symmetric_function())
s[3, 2, 1]

dual()¶

The dual basis to the dual Quasisymmetric-Schur basis of NCSF.

The basis returned is the Quasisymmetric_Schur basis of QSym.

OUTPUT:

the Quasisymmetric-Schur basis of the quasi-symmetric functions

EXAMPLES:

sage: dQS=NonCommutativeSymmetricFunctions(QQ).dualQuasisymmetric_Schur()
sage: dQS.dual()
Quasisymmetric functions over the Rational Field in the Quasisymmetric
Schur basis
sage: dQS.duality_pairing_matrix(dQS.dual(),3)
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]

to_symmetric_function_on_basis(I)¶

The commutative image of a dual quasi-symmetric Schur element

The commutative image of a basis element is obtained by sorting the indexing composition of the basis element.

INPUT:

I – a composition

OUTPUT:

The commutative image of the dual quasi-Schur basis element indexed by I. The result is the Schur symmetric function indexed by the partition obtained by sorting I.

EXAMPLES:

sage: dQS=NonCommutativeSymmetricFunctions(QQ).dQS()
sage: dQS.to_symmetric_function_on_basis([2,1,3])
s[3, 2, 1]
sage: dQS.to_symmetric_function_on_basis([])
s[]

class dualYoungQuasisymmetric_Schur(NCSF)¶

Bases: sage.combinat.free_module.CombinatorialFreeModule, sage.misc.bindable_class.BindableClass

The basis of NCSF dual to the Young Quasisymmetric-Schur basis of QSym.

The YoungQuasisymmetric_Schur functions are given in Definition 5.2.1 of [LMvW13]. The dual basis in the algebra of non-commutative symmetric functions are related by an involution reversing the indexing composition of the complete expansion of a quasi-Schur basis element. This basis has many of the same properties as the Quasisymmetric Schur basis and is related to that basis by an algebraic transformation.

EXAMPLES:

sage: NCSF = NonCommutativeSymmetricFunctions(QQ)
sage: dYQS = NCSF.dYQS()
sage: dYQS([1,3,2])*dYQS([1])
dYQS[1, 3, 2, 1] + dYQS[1, 3, 3] + dYQS[1, 4, 2] + dYQS[2, 3, 2]
sage: dYQS([1])*dYQS([1,3,2])
dYQS[1, 1, 3, 2] + dYQS[2, 3, 2] + dYQS[3, 3, 1] + dYQS[4, 1, 2]
sage: dYQS([1,3])*dYQS([1,1])
dYQS[1, 3, 1, 1] + dYQS[1, 4, 1] + dYQS[2, 3, 1] + dYQS[2, 4]
sage: dYQS([3,1])*dYQS([2,1])
dYQS[3, 1, 2, 1] + dYQS[3, 2, 2] + dYQS[3, 3, 1] + dYQS[4, 1, 1, 1]
 + dYQS[4, 1, 2] + dYQS[4, 2, 1] + dYQS[4, 3] + dYQS[5, 1, 1]
 + dYQS[5, 2]
sage: dYQS([1,1]).coproduct()
dYQS[] # dYQS[1, 1] + dYQS[1] # dYQS[1] + dYQS[1, 1] # dYQS[]
sage: dYQS([3,3]).coproduct().monomial_coefficients()[(Composition([1,2]),Composition([2,1]))]
1
sage: S = NCSF.complete()
sage: dYQS(S[1,3,1])
dYQS[1, 3, 1] + dYQS[1, 4] + dYQS[2, 3] + dYQS[4, 1] + dYQS[5]
sage: S(dYQS[1,3,1])
S[1, 3, 1] - S[1, 4] - S[2, 3] + S[5]
sage: s = SymmetricFunctions(QQ).s()
sage: s(S(dYQS([2,1,3])).to_symmetric_function())
s[3, 2, 1]

dual()¶

The dual basis to the dual Quasisymmetric-Schur basis of NCSF.

The basis returned is the Quasisymmetric_Schur basis of QSym.

OUTPUT:

the Young Quasisymmetric-Schur basis of quasi-symmetric functions

EXAMPLES:

sage: dYQS=NonCommutativeSymmetricFunctions(QQ).dualYoungQuasisymmetric_Schur()
sage: dYQS.dual()
Quasisymmetric functions over the Rational Field in the Young
Quasisymmetric Schur basis
sage: dYQS.duality_pairing_matrix(dYQS.dual(),3)
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]

to_symmetric_function_on_basis(I)¶

The commutative image of a dual Young quasi-symmetric Schur element.

The commutative image of a basis element is obtained by sorting the indexing composition of the basis element.

INPUT:

I – a composition

OUTPUT:

The commutative image of the dual Young quasi-Schur basis element indexed by I. The result is the Schur symmetric function indexed by the partition obtained by sorting I.

EXAMPLES:

sage: dYQS=NonCommutativeSymmetricFunctions(QQ).dYQS()
sage: dYQS.to_symmetric_function_on_basis([2,1,3])
s[3, 2, 1]
sage: dYQS.to_symmetric_function_on_basis([])
s[]

elementary¶: alias of NonCommutativeSymmetricFunctions.Elementary

monomial¶: alias of NonCommutativeSymmetricFunctions.Monomial

nM¶: alias of NonCommutativeSymmetricFunctions.Monomial

ribbon¶: alias of NonCommutativeSymmetricFunctions.Ribbon

Non-Commutative Symmetric Functions¶

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