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""" 

Symmetric functions, with their multiple realizations 

""" 

from __future__ import absolute_import 

#***************************************************************************** 

# Copyright (C) 2007 Mike Hansen <mhansen@gmail.com> 

# 2009-2012 Jason Bandlow <jbandlow@gmail.com> 

# 2012 Anne Schilling <anne at math.ucdavis.edu> 

# 2009-2012 Nicolas M. Thiery <nthiery at users.sf.net> 

# 2012 Mike Zabrocki <mike.zabrocki@gmail.com> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# 

# This code is distributed in the hope that it will be useful, 

# but WITHOUT ANY WARRANTY; without even the implied warranty of 

# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU 

# General Public License for more details. 

# 

# The full text of the GPL is available at: 

# 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from sage.structure.parent import Parent 

from sage.structure.unique_representation import UniqueRepresentation 

from sage.categories.graded_hopf_algebras import GradedHopfAlgebras 

from sage.categories.fields import Fields 

from sage.categories.rings import Rings 

from sage.combinat.partition import Partitions 

from sage.combinat.free_module import CombinatorialFreeModule 

from sage.rings.rational_field import QQ 

 

from . import schur 

from . import monomial 

from . import powersum 

from . import elementary 

from . import homogeneous 

from . import hall_littlewood 

from . import jack 

from . import macdonald 

from . import llt 

 

class SymmetricFunctions(UniqueRepresentation, Parent): 

r""" 

The abstract algebra of commutative symmetric functions 

 

.. rubric:: Symmetric Functions in Sage 

 

.. MODULEAUTHOR:: Jason Bandlow, Anne Schilling, Nicolas M. Thiery, Mike Zabrocki 

 

This document is an introduction to working with symmetric function 

theory in Sage. 

It is not intended to be an introduction to the theory 

of symmetric functions ([MAC]_ and [STA]_, Chapter 7, are two excellent 

references.) The reader is also expected to be familiar with Sage. 

 

.. rubric:: The algebra of symmetric functions 

 

The algebra of symmetric functions is the unique free commutative graded 

connected algebra over the given ring, with one generator in each degree. It 

can also be thought of as the inverse limit (in the category of graded 

algebras) of the algebra of symmetric polynomials in `n` variables as `n \rightarrow \infty`. 

Sage allows us to construct the algebra of symmetric functions over 

any ring. We will use a base ring of rational numbers in these first 

examples:: 

 

sage: Sym = SymmetricFunctions(QQ) 

sage: Sym 

Symmetric Functions over Rational Field 

 

Sage knows certain categorical information about this algebra:: 

 

sage: Sym.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 coalgebras over Rational Field with realizations 

 

Notice that ``Sym`` is an *abstract* algebra. This reflects the fact that 

there are multiple natural bases. To work with specific 

elements, we need a *realization* of this algebra. In practice, this 

means we need to specify a basis. 

 

.. rubric:: An example basis - power sums 

 

Here is an example of how one might use the power sum realization:: 

 

sage: p = Sym.powersum() 

sage: p 

Symmetric Functions over Rational Field in the powersum basis 

 

``p`` now represents the realization of the symmetric function algebra on 

the power sum basis. The basis itself is accessible through:: 

 

sage: p.basis() 

Lazy family (Term map from Partitions to Symmetric Functions over Rational Field in the powersum basis(i))_{i in Partitions} 

sage: p.basis().keys() 

Partitions 

 

This last line means that ``p.basis()`` is an association between the set 

of Partitions and the basis elements of the algebra ``p``. To construct a 

specific element one can therefore do:: 

 

sage: p.basis()[Partition([2,1,1])] 

p[2, 1, 1] 

 

As this is rather cumbersome, realizations of the symmetric function 

algebra allow for the following abuses of notation:: 

 

sage: p[Partition([2, 1, 1])] 

p[2, 1, 1] 

sage: p[[2, 1, 1]] 

p[2, 1, 1] 

sage: p[2, 1, 1] 

p[2, 1, 1] 

 

or even:: 

 

sage: p[(i for i in [2, 1, 1])] 

p[2, 1, 1] 

 

In the special case of the empty partition, due to a limitation in 

Python syntax, one cannot use:: 

 

sage: p[] # todo: not implemented 

 

Please use instead:: 

 

sage: p[[]] 

p[] 

 

.. note:: When elements are constructed using the ``p[something ]`` syntax , 

an error will be raised if the input cannot be interpreted as a partition. 

This is *not* the case when ``p.basis()`` is used:: 

 

sage: p['something'] 

Traceback (most recent call last): 

... 

ValueError: ['s', 'o', 'm', 'e', 't', 'h', 'i', 'n', 'g'] is not an element of Partitions 

sage: p.basis()['something'] 

p'something' 

 

Elements of ``p`` are linear combinations of such compositions:: 

 

sage: p.an_element() 

2*p[] + 2*p[1] + 3*p[2] 

 

.. rubric:: Algebra structure 

 

Algebraic combinations of basis elements can be entered in a natural way:: 

 

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

3*p[2, 1, 1] + 2*p[4, 1] 

 

Let us explore the other operations of ``p``. We can ask for 

the mathematical properties of ``p``:: 

 

sage: p.categories() 

[Category of graded bases of Symmetric Functions over Rational Field, 

Category of filtered bases of Symmetric Functions over Rational Field, 

Category of bases of Symmetric Functions over Rational Field, 

Category of graded hopf algebras with basis over Rational Field, 

...] 

 

To start with, ``p`` is a graded algebra, the grading being induced 

by the size of the partitions. Due to this, the one is the basis 

element indexed by the empty partition:: 

 

sage: p.one() 

p[] 

 

The ``p`` basis is multiplicative; that is, multiplication is induced by 

linearity from the (nonincreasingly sorted) concatenation of partitions:: 

 

sage: p[3,1] * p[2,1] 

p[3, 2, 1, 1] 

 

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

p[4, 2] + 2*p[4, 3, 2, 1] 

 

.. rubric:: The classical bases 

 

In addition to the power sum basis, other classical bases of the 

symmetric function algebra include the elementary, complete 

homogeneous, monomial, and Schur bases:: 

 

sage: e = Sym.elementary() 

sage: h = Sym.homogeneous() 

sage: m = Sym.monomial() 

sage: s = Sym.schur() 

 

These and others can be defined all at once with the single command:: 

 

sage: Sym.inject_shorthands() 

Defining e as shorthand for Symmetric Functions over Rational Field in the elementary basis 

Defining f as shorthand for Symmetric Functions over Rational Field in the forgotten basis 

Defining h as shorthand for Symmetric Functions over Rational Field in the homogeneous basis 

Defining m as shorthand for Symmetric Functions over Rational Field in the monomial basis 

Defining p as shorthand for Symmetric Functions over Rational Field in the powersum basis 

Defining s as shorthand for Symmetric Functions over Rational Field in the Schur basis 

 

We can then do conversions from one basis to another:: 

 

sage: s(p[2,1]) 

-s[1, 1, 1] + s[3] 

 

sage: m(p[3]) 

m[3] 

sage: m(p[3,2]) 

m[3, 2] + m[5] 

 

For computations which mix bases, Sage will return a result with respect 

to a single (not necessarily predictable) basis:: 

 

sage: p[2] * s[2] - m[4] 

1/2*p[2, 1, 1] + 1/2*p[2, 2] - p[4] 

 

sage: p( m[1] * ( e[3]*s[2] + 1 )) 

p[1] + 1/12*p[1, 1, 1, 1, 1, 1] - 1/6*p[2, 1, 1, 1, 1] - 1/4*p[2, 2, 1, 1] + 1/6*p[3, 1, 1, 1] + 1/6*p[3, 2, 1] 

 

 

The one for different bases such as the power sum and Schur function is the same:: 

 

sage: s.one() == p.one() 

True 

 

.. rubric:: Basic computations 

 

In this section, we explore some of the many methods that can be applied 

to an arbitrary symmetric function:: 

 

sage: f = s[2]^2; f 

s[2, 2] + s[3, 1] + s[4] 

 

For more methods than discussed here, create a symmetric function as 

above, and use ``f.<tab>``. 

 

.. _`Representation theory of the symmetric group`: 

 

.. rubric:: Representation theory of the symmetric group 

 

The Schur functions `s_\lambda` can also be interpreted as irreducible characters of the symmetric 

group `S_n`, where `n` is the size of the partition `\lambda`. Since the Schur functions of 

degree `n` form a basis of the symmetric functions of degree `n`, it 

follows that an arbitrary symmetric function (homogeneous of degree 

`n`) may be interpreted as a function on the symmetric group. In this 

interpretation the power sum symmetric function `p_\lambda` is the characteristic 

function of the conjugacy class with shape `\lambda`, multiplied by the order of 

the centralizer of an element. Hence the irreducible characters can be computed 

as follows:: 

 

sage: Sym = SymmetricFunctions(QQ) 

sage: s = Sym.schur() 

sage: p = Sym.power() 

sage: P = Partitions(5).list() 

sage: P = [P[i] for i in range(len(P)-1,-1,-1)] 

sage: M = matrix([[s[P[i]].scalar(p[P[j]]) for j in range(len(P))] for i in range(len(P))]) 

sage: M 

[ 1 -1 1 1 -1 -1 1] 

[ 4 -2 0 1 1 0 -1] 

[ 5 -1 1 -1 -1 1 0] 

[ 6 0 -2 0 0 0 1] 

[ 5 1 1 -1 1 -1 0] 

[ 4 2 0 1 -1 0 -1] 

[ 1 1 1 1 1 1 1] 

 

We can indeed check that this agrees with the character table of `S_5`:: 

 

sage: SymmetricGroup(5).character_table() == M 

True 

 

In this interpretation of symmetric functions as characters on the 

symmetric group, the multiplication and comultiplication are 

interpreted as induction (from `S_n\times S_m` to `S_{n+m}`) 

and restriction, respectively. The Schur functions can also be interpreted 

as characters of `GL_n`, see `Partitions and Schur functions`__. 

 

__ ../../../../../thematic_tutorials/lie/lie_basics.html#partitions-and-schur-polynomials 

 

.. rubric:: The omega involution 

 

The `\omega` involution is the linear extension of the map which sends 

`e_\lambda` to `h_{\lambda}`:: 

 

sage: h(f) 

h[2, 2] 

sage: e(f.omega()) 

e[2, 2] 

 

.. rubric:: The Hall scalar product 

 

The Hall scalar product on the algebra of symmetric functions makes the 

Schur functions into an orthonormal basis:: 

 

sage: f.scalar(f) 

3 

 

.. rubric:: Skewing 

 

*Skewing* is the adjoint operation to multiplication with respect to 

this scalar product:: 

 

sage: f.skew_by(s[1]) 

2*s[2, 1] + 2*s[3] 

 

In general, ``s[la].skew_by(s[mu])`` is the symmetric function typically 

denoted `s_{\lambda \setminus \mu}` or `s_{\lambda / \mu}`. 

 

.. rubric:: Expanding into variables 

 

We can expand a symmetric function into a symmetric polynomial in a 

specified number of variables:: 

 

sage: f.expand(2) 

x0^4 + 2*x0^3*x1 + 3*x0^2*x1^2 + 2*x0*x1^3 + x1^4 

 

See the documentation for ``expand`` for more examples. 

 

.. rubric:: The Kronecker product 

 

As in the section on the `Representation theory of 

the symmetric group`_, a symmetric function may be considered as a 

class function on the symmetric group where the elements 

`p_\mu/z_\mu` are the indicators of a permutation having 

cycle structure `\mu`. The Kronecker product of two 

symmetric functions corresponds to the pointwise product 

of these class functions. 

 

Since the Schur functions are the irreducible characters 

of the symmetric group under this identification, the Kronecker 

product of two Schur functions corresponds to the internal 

tensor product of two irreducible symmetric group representations. 

 

Under this identification, the Kronecker 

product of `p_\mu/z_\mu` and `p_\nu/z_\nu` is `p_\mu/z_\mu` 

if `\mu=\nu`, and the result is equal to `0` otherwise. 

 

``internal_product``, ``kronecker_product``, ``inner_tensor`` and 

``itensor`` are different names for the same function. 

 

:: 

 

sage: f.kronecker_product(f) 

s[1, 1, 1, 1] + 3*s[2, 1, 1] + 4*s[2, 2] + 5*s[3, 1] + 3*s[4] 

 

.. rubric:: Plethysm 

 

The *plethysm* of symmetric functions is the operation corresponding to 

composition of representations of the general linear group. See [STA]_ 

Chapter 7, Appendix 2 for details. 

:: 

 

sage: s[2].plethysm(s[2]) 

s[2, 2] + s[4] 

 

Plethysm can also be written as a composition of functions:: 

 

sage: s[2]( s[2] ) 

s[2, 2] + s[4] 

 

If the coefficient ring contains degree 1 elements, these are handled 

properly by plethysm:: 

 

sage: R.<t> = QQ[]; s = SymmetricFunctions(R).schur() 

sage: s[2]( (1-t)*s[1] ) 

(t^2-t)*s[1, 1] + (-t+1)*s[2] 

 

See the documentation for ``plethysm`` for more information. 

 

.. rubric:: Inner plethysm 

 

The operation of inner plethysm ``f.inner_plethysm(g)`` models the 

composition of the `S_n` representation represented by `g` with the 

`GL_m` representation whose character is `f`. See the documentation of 

``inner_plethysm``, [ST94]_ or [STA]_, exercise 7.74 solutions for more 

information:: 

 

sage: s = SymmetricFunctions(QQ).schur() 

sage: f = s[2]^2 

sage: f.inner_plethysm(s[2]) 

s[2] 

 

.. rubric:: Hopf algebra structure 

 

The ring of symmetric functions is further endowed with a coalgebra 

structure. The coproduct is an algebra morphism, and therefore 

determined by its values on the generators; the power sum generators 

are primitive:: 

 

sage: p[1].coproduct() 

p[] # p[1] + p[1] # p[] 

sage: p[2].coproduct() 

p[] # p[2] + p[2] # p[] 

 

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

 

sage: p[2, 1].coproduct() 

p[] # p[2, 1] + p[1] # p[2] + p[2] # p[1] + p[2, 1] # p[] 

 

This coproduct, along with the counit which sends every symmetric function 

to its `0`-th homogeneous component, makes the ring of symmetric functions 

into a graded connected bialgebra. It is known that every graded connected 

bialgebra has an antipode. For the ring of symmetric functions, the antipode 

can be characterized explicitly: The antipode is an anti-algebra morphism 

(thus an algebra morphism, since our algebra is commutative) which sends 

`p_{\lambda}` to `(-1)^{\mathrm{length}(\lambda)} p_{\lambda}` for every 

partition `\lambda`. Thus, in particular, it sends the generators on the 

``p`` basis to their opposites:: 

 

sage: p[3].antipode() 

-p[3] 

sage: p[3,2,1].antipode() 

-p[3, 2, 1] 

 

The graded connected bialgebra of symmetric functions over a `\QQ`-algebra 

has a rather simply-understood structure: It is (isomorphic to) the 

symmetric algebra of its space of primitives (which is spanned by the 

power-sum symmetric functions). 

 

Here are further examples:: 

 

sage: f = s[2]^2 

sage: f.antipode() 

s[1, 1, 1, 1] + s[2, 1, 1] + s[2, 2] 

sage: f.coproduct() 

s[] # s[2, 2] + s[] # s[3, 1] + s[] # s[4] + 2*s[1] # s[2, 1] + 2*s[1] # s[3] + s[1, 1] # s[1, 1] 

+ s[1, 1] # s[2] + s[2] # s[1, 1] + 3*s[2] # s[2] + 2*s[2, 1] # s[1] + s[2, 2] # s[] + 2*s[3] # s[1] 

+ s[3, 1] # s[] + s[4] # s[] 

sage: f.coproduct().apply_multilinear_morphism( lambda x,y: x*y.antipode() ) 

0 

 

.. rubric:: Transformations of symmetric functions 

 

There are many methods in Sage which make it easy to manipulate symmetric 

functions. For example, if we have some function which acts on partitions 

(say, conjugation), it is a simple matter to apply it to the support of a 

symmetric function. Here is an example:: 

 

sage: conj = lambda mu: mu.conjugate() 

sage: f = h[4] + 2*h[3,1] 

sage: f.map_support(conj) 

h[1, 1, 1, 1] + 2*h[2, 1, 1] 

 

We can also easily modify the coefficients:: 

 

sage: def foo(mu, coeff): return mu.conjugate(), -coeff 

sage: f.map_item(foo) 

-h[1, 1, 1, 1] - 2*h[2, 1, 1] 

 

See also ``map_coefficients``. 

 

There are also methods for building functions directly:: 

 

sage: s.sum_of_monomials(mu for mu in Partitions(3)) 

s[1, 1, 1] + s[2, 1] + s[3] 

sage: s.sum_of_monomials(Partitions(3)) 

s[1, 1, 1] + s[2, 1] + s[3] 

sage: s.sum_of_terms( (mu, mu[0]) for mu in Partitions(3)) 

s[1, 1, 1] + 2*s[2, 1] + 3*s[3] 

 

These are the preferred way to build elements within a program; 

the result will usually be faster than using :func:`sum`. It also 

guarantees that empty sums yields the zero of ``s`` (see also 

``s.sum``). 

 

Note also that it is a good idea to use:: 

 

sage: s.one() 

s[] 

sage: s.zero() 

0 

 

instead of ``s(1)`` and ``s(0)`` within programs where speed is important, 

in order to prevent unnecessary coercions. 

 

.. rubric:: Different base rings 

 

Depending on the base ring, the different realizations of the symmetric 

function algebra may not span the same space:: 

 

sage: SZ = SymmetricFunctions(ZZ) 

sage: p = SZ.power(); s = SZ.schur() 

sage: p(s[1,1,1]) 

Traceback (most recent call last): 

... 

TypeError: no conversion of this rational to integer 

 

Because of this, some functions may not behave as expected when working over 

the integers, even though they make mathematical sense:: 

 

sage: s[1,1,1].plethysm(s[1,1,1]) 

Traceback (most recent call last): 

... 

TypeError: no conversion of this rational to integer 

 

It is possible to work over different base rings simultaneously:: 

 

sage: s = SymmetricFunctions(QQ).schur() 

sage: p = SymmetricFunctions(QQ).power() 

sage: sz = SymmetricFunctions(ZZ).schur(); sz._prefix = 'sz' 

sage: pz = SymmetricFunctions(ZZ).power(); pz._prefix = 'pz' 

sage: p(sz[1,1,1]) 

1/6*p[1, 1, 1] - 1/2*p[2, 1] + 1/3*p[3] 

sage: sz( 1/6*p[1, 1, 1] - 1/2*p[2, 1] + 1/3*p[3] ) 

sz[1, 1, 1] 

 

As shown in this example, if you are working over multiple base rings 

simultaneously, it is a good idea to change the prefix in some cases, so that 

you can tell from the output which realization your result is in. 

 

Let us change the notation back for the remainder of this tutorial:: 

 

sage: sz._prefix = 's' 

sage: pz._prefix = 'p' 

 

One can also use the Sage standard renaming idiom to get shorter outputs:: 

 

sage: Sym = SymmetricFunctions(QQ) 

sage: Sym.rename("Sym") 

sage: Sym 

Sym 

sage: Sym.rename() 

 

And we name it back:: 

 

sage: Sym.rename("Symmetric Functions over Rational Field"); Sym 

Symmetric Functions over Rational Field 

 

.. rubric:: Other bases 

 

There are two additional basis of the symmetric functions which are not 

considered as classical bases: 

 

* forgotten basis 

* Witt basis 

 

The forgotten basis is the dual basis of the elementary symmetric 

functions basis with respect to the Hall scalar product. The Witt basis 

can be constructed by 

 

.. MATH:: 

 

\prod_{d=1}^{\infty} (1 - w_d t^d)^{-1} = \sum_{n=0}^{\infty} h_n t^n 

 

where `t` is a formal variable. 

 

There are further bases of the ring of symmetric functions, in general over 

fields with parameters such as `q` and `t`: 

 

* Hall-Littlewood bases 

* Jack bases 

* Macdonald bases 

* `k`-Schur functions 

 

We briefly demonstrate how to access these bases. For more information, see 

the documentation of the individual bases. 

 

The *Jack polynomials* can be obtained as:: 

 

sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) 

sage: Jack = Sym.jack() 

sage: P = Jack.P(); J = Jack.J(); Q = Jack.Q() 

sage: J(P[2,1]) 

(1/(t+2))*JackJ[2, 1] 

 

The parameter `t` can be specialized as follows:: 

 

sage: Sym = SymmetricFunctions(QQ) 

sage: Jack = Sym.jack(t = 1) 

sage: P = Jack.P(); J = Jack.J(); Q = Jack.Q() 

sage: J(P[2,1]) 

1/3*JackJ[2, 1] 

 

Similarly one can access the Hall-Littlewood and Macdonald polynomials, etc:: 

 

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t'])) 

sage: Mcd = Sym.macdonald() 

sage: P = Mcd.P(); J = Mcd.J(); Q = Mcd.Q() 

sage: J(P[2,1]) 

(1/(-q*t^4+2*q*t^3-q*t^2+t^2-2*t+1))*McdJ[2, 1] 

 

.. rubric:: `k`-Schur functions 

 

The `k`-Schur functions live in the `k`-bounded subspace of the ring of 

symmetric functions. It is possible to compute in the `k`-bounded subspace 

directly:: 

 

sage: Sym = SymmetricFunctions(QQ) 

sage: ks = Sym.kschur(3,1) 

sage: f = ks[2,1]*ks[2,1]; f 

ks3[2, 2, 1, 1] + ks3[2, 2, 2] + ks3[3, 1, 1, 1] 

 

or to lift to the ring of symmetric functions:: 

 

sage: f.lift() 

s[2, 2, 1, 1] + s[2, 2, 2] + s[3, 1, 1, 1] + 2*s[3, 2, 1] + s[3, 3] + s[4, 1, 1] + s[4, 2] 

 

However, it is not always possible to convert a symmetric function to the `k`-bounded subspace:: 

 

sage: s = Sym.schur() 

sage: ks(s[2,1,1]) 

Traceback (most recent call last): 

... 

ValueError: s[2, 1, 1] is not in the image 

 

The `k`-Schur functions are more generally defined with a parameter `t` and they are 

a basis of the subspace spanned by the Hall-Littlewood ``Qp`` symmetric functions 

indexed by partitions whose first part is less than or equal to `k`:: 

 

sage: Sym = SymmetricFunctions(QQ['t'].fraction_field()) 

sage: SymS3 = Sym.kBoundedSubspace(3) # default t='t' 

sage: ks = SymS3.kschur() 

sage: Qp = Sym.hall_littlewood().Qp() 

sage: ks(Qp[2,1,1,1]) 

ks3[2, 1, 1, 1] + (t^2+t)*ks3[2, 2, 1] + (t^3+t^2)*ks3[3, 1, 1] + t^4*ks3[3, 2] 

 

The subspace spanned by the `k`-Schur functions with a parameter `t` are not known 

to form a natural algebra. However it is known that the product of a `k`-Schur 

function and an `\ell`-Schur function is in the linear span of the `k+\ell`-Schur 

functions:: 

 

sage: ks(ks[2,1]*ks[1,1]) 

Traceback (most recent call last): 

... 

ValueError: s[2, 1, 1, 1] + s[2, 2, 1] + s[3, 1, 1] + s[3, 2] is not in the image 

sage: ks[2,1]*ks[1,1] 

s[2, 1, 1, 1] + s[2, 2, 1] + s[3, 1, 1] + s[3, 2] 

sage: ks6 = Sym.kBoundedSubspace(6).kschur() 

sage: ks6(ks[3,1,1]*ks[3]) 

ks6[3, 3, 1, 1] + ks6[4, 2, 1, 1] + (t+1)*ks6[4, 3, 1] + t*ks6[4, 4] 

+ ks6[5, 1, 1, 1] + ks6[5, 2, 1] + t*ks6[5, 3] + ks6[6, 1, 1] 

 

The `k`-split basis is a second basis of the ring spanned by the `k`-Schur 

functions with a parameter `t`. The `k`-split basis has the property that 

`Q'_\lambda[X;t]` expands positively in the `k`-split basis and the 

`k`-split basis conjecturally expands positively in the `k`-Schur functions. 

The definition can be found in [LLMSSZ]_ p. 81.:: 

 

sage: ksp3 = SymS3.ksplit() 

sage: ksp3(Qp[2,1,1,1]) 

ksp3[2, 1, 1, 1] + t^2*ksp3[2, 2, 1] + (t^3+t^2)*ksp3[3, 1, 1] + t^4*ksp3[3, 2] 

sage: [ks(ksp3(la)) for la in ksp3(Qp[2,1,1,1]).support()] 

[ks3[2, 2, 1], ks3[2, 1, 1, 1] + t*ks3[2, 2, 1], ks3[3, 2], ks3[3, 1, 1]] 

 

.. rubric:: dual `k`-Schur functions 

 

The dual space to the subspace spanned by the `k`-Schur functions is most naturally 

realized as a quotient of the ring of symmetric functions by an ideal. When `t=1` 

the ideal is generated by the monomial symmetric functions indexed by partitions 

whose first part is greater than `k`.:: 

 

sage: Sym = SymmetricFunctions(QQ) 

sage: SymQ3 = Sym.kBoundedQuotient(3,t=1) 

sage: km = SymQ3.kmonomial() 

sage: km[2,1]*km[2,1] 

4*m3[2, 2, 1, 1] + 6*m3[2, 2, 2] + 2*m3[3, 2, 1] + 2*m3[3, 3] 

sage: F = SymQ3.affineSchur() 

sage: F[2,1]*F[2,1] 

2*F3[1, 1, 1, 1, 1, 1] + 4*F3[2, 1, 1, 1, 1] + 4*F3[2, 2, 1, 1] + 4*F3[2, 2, 2] 

+ 2*F3[3, 1, 1, 1] + 4*F3[3, 2, 1] + 2*F3[3, 3] 

 

When `t` is not equal to `1`, the subspace spanned by the `k`-Schur functions is 

realized as a quotient of the ring of symmetric functions by the ideal generated by 

the Hall-Littlewood symmetric functions in the P basis indexed by partitions with 

first part greater than `k`.:: 

 

sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) 

sage: SymQ3 = Sym.kBoundedQuotient(3) 

sage: kHLP = SymQ3.kHallLittlewoodP() 

sage: kHLP[2,1]*kHLP[2,1] 

(t^2+2*t+1)*HLP3[2, 2, 1, 1] + (t^3+2*t^2+2*t+1)*HLP3[2, 2, 2] 

+ (-t^4-t^3+t+1)*HLP3[3, 1, 1, 1] + (-t^2+t+2)*HLP3[3, 2, 1] + (t+1)*HLP3[3, 3] 

sage: HLP = Sym.hall_littlewood().P() 

sage: kHLP(HLP[3,1]) 

HLP3[3, 1] 

sage: kHLP(HLP[4]) 

0 

 

In this space, the basis which is dual to the `k`-Schur functions conjecturally 

expands positively in the `k`-bounded Hall-Littlewood functions and has positive 

structure coefficients.:: 

 

sage: dks = SymQ3.dual_k_Schur() 

sage: kHLP(dks[2,2]) 

(t^4+t^2)*HLP3[1, 1, 1, 1] + t*HLP3[2, 1, 1] + HLP3[2, 2] 

sage: dks[2,1]*dks[1,1] 

(t^2+t)*dks3[1, 1, 1, 1, 1] + (t+1)*dks3[2, 1, 1, 1] + (t+1)*dks3[2, 2, 1] 

+ dks3[3, 1, 1] + dks3[3, 2] 

 

At `t=1` the `k`-bounded Hall-Littlewood basis is equal to the `k`-bounded monomial 

basis and the dual `k`-Schur elements are equal to the affine Schur basis. The 

`k`-bounded monomial basis and affine Schur functions are faster and should be used 

instead of the `k`-bounded Hall-Littlewood P basis and dual `k`-Schur functions when 

`t=1`.:: 

 

sage: SymQ3 = Sym.kBoundedQuotient(3,t=1) 

sage: dks = SymQ3.dual_k_Schur() 

sage: F = SymQ3.affineSchur() 

sage: F[3,1]==dks[3,1] 

True 

 

.. rubric:: Implementing new bases 

 

In order to implement a new symmetric function basis, Sage will need 

to know at a minimum how to change back and forth between at least one 

other basis (although they do not necessarily have to be the same basis). 

All of the standard functions associated with the basis will have a 

default implementation (although a more specific implementation may 

be more efficient). 

 

To present an idea of how this is done, we will create 

here the example of how to implement the basis `s_\mu[X(1-t)]`. 

 

To begin, we import the class 

:class:`sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic()`. Our 

new basis will inherit all of the default methods from this class:: 

 

sage: from sage.combinat.sf.sfa import SymmetricFunctionAlgebra_generic as SFA_generic 

 

Now the basis we are creating has a parameter `t` which is possible 

to specialize. In this example we will convert to and from the Schur 

basis. For this we implement methods ``_self_to_s`` and ``_s_to_self``. 

By registering these two functions as coercions, Sage then knows 

automatically how it possible to change between any two bases for 

which there is a path of changes of bases. :: 

 

sage: from sage.categories.morphism import SetMorphism 

sage: class SFA_st(SFA_generic): 

....: def __init__(self, Sym, t): 

....: SFA_generic.__init__(self, Sym, basis_name= 

....: "Schur functions with a plethystic substitution of X -> X(1-t)", 

....: prefix='st') 

....: self._s = Sym.s() 

....: self.t = Sym.base_ring()(t) 

....: cat = HopfAlgebras(Sym.base_ring()).WithBasis() 

....: self.register_coercion( 

....: SetMorphism(Hom(self._s, self, cat), self._s_to_self)) 

....: self._s.register_coercion( 

....: SetMorphism(Hom(self, self._s, cat), self._self_to_s)) 

....: def _s_to_self(self, f): 

....: # f is a Schur function and the output is in the st basis 

....: return self._from_dict(f.theta_qt(0,self.t)._monomial_coefficients) 

....: def _self_to_s(self, f): 

....: # f is in the st basis and the output is in the Schur basis 

....: return self._s.sum(cmu*self._s(mu).theta_qt(self.t,0) for mu,cmu in f) 

....: class Element(SFA_generic.Element): 

....: pass 

 

An instance of this basis is created by calling it with a symmetric 

function ring ``Sym`` and a parameter ``t`` which is in the base ring 

of ``Sym``. The ``Element`` class inherits all of the methods from 

:class:`sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element`. 

 

In the reference [MAC]_ on page 354, this basis is denoted 

`S_\lambda(x;t)` and the change of basis coefficients of the 

Macdonald ``J`` basis are the coefficients `K_{\lambda\mu}(q,t)`. 

Here is an example of its use:: 

 

sage: QQqt = QQ['q','t'].fraction_field() 

sage: (q,t) = QQqt.gens() 

sage: st = SFA_st(SymmetricFunctions(QQqt),t) 

sage: st 

Symmetric Functions over Fraction Field of Multivariate Polynomial 

Ring in q, t over Rational Field in the Schur functions with a 

plethystic substitution of X -> X(1-t) basis 

sage: st[2,1] * st[1] 

st[2, 1, 1] + st[2, 2] + st[3, 1] 

sage: st([2]).coproduct() 

st[] # st[2] + st[1] # st[1] + st[2] # st[] 

sage: J = st.symmetric_function_ring().macdonald().J() 

sage: st(J[2,1]) 

q*st[1, 1, 1] + (q*t+1)*st[2, 1] + t*st[3] 

 

.. rubric:: Acknowledgements 

 

The design is heavily inspired from the implementation of 

symmetric functions in MuPAD-Combinat (see [HT04]_ and [FD06]_). 

 

REFERENCES: 

 

.. [FD06] Francois Descouens, Making research on symmetric functions using MuPAD-Combinat. 

In Andres Iglesias and Nobuki Takayama, editors, 2nd International Congress on Mathematical Software (ICMS'06), 

volume 4151 of LNCS, pages 407-418, Castro Urdiales, Spain, September 2006. Springer-Verlag. 

:arXiv:`0806.1873` 

 

.. [HT04] Florent Hivert and Nicolas M. Thiery, 

MuPAD-Combinat, an open-source package for research in algebraic combinatorics. 

Sem. Lothar. Combin., 51 :Art. B51z, 70 pp. (electronic), 2004. 

http://mupad-combinat.sf.net/. 

 

.. [MAC] Ian Macdonald, Symmetric Functions and Orthogonal Polynomials, 

Second edition. With contributions by A. Zelevinsky. Oxford Mathematical Monographs. 

Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1995. x+475 pp. 

ISBN: 0-19-853489-2 

 

.. [STA] Richard Stanley, Enumerative combinatorics. Vol. 2. 

With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. 

Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999. xii+581 pp. 

ISBN: 0-521-56069-1; 0-521-78987-7 

 

.. [ST94] Scharf, Thomas, Thibon, Jean-Yves, 

A Hopf-algebra approach to inner plethysm. 

Adv. Math. 104 (1994), no. 1, 30-58. 

:doi:`10.1006/aima.1994.1019` 

 

.. rubric:: Further tests 

 

TESTS:: 

 

sage: Sym = SymmetricFunctions(QQ) 

sage: Sym 

Symmetric Functions over Rational Field 

sage: h = Sym.h(); e = Sym.e(); s = Sym.s(); m = Sym.m(); p = Sym.p() 

sage: ( ( h[2,1] * ( 1 + 3 * h[2,1]) ) + s[2]. antipode()) . coproduct() 

h[] # h[1, 1] - h[] # h[2] + h[] # h[2, 1] + 3*h[] # h[2, 2, 1, 1] + h[1] # h[1] + h[1] # h[1, 1] 

+ h[1] # h[2] + 6*h[1] # h[2, 1, 1, 1] + 6*h[1] # h[2, 2, 1] + h[1, 1] # h[] + h[1, 1] # h[1] 

+ 3*h[1, 1] # h[1, 1, 1, 1] + 12*h[1, 1] # h[2, 1, 1] + 3*h[1, 1] # h[2, 2] + 6*h[1, 1, 1] # h[1, 1, 1] 

+ 6*h[1, 1, 1] # h[2, 1] + 3*h[1, 1, 1, 1] # h[1, 1] - h[2] # h[] + h[2] # h[1] + 6*h[2] # h[2, 1, 1] 

+ h[2, 1] # h[] + 6*h[2, 1] # h[1, 1, 1] + 12*h[2, 1] # h[2, 1] + 12*h[2, 1, 1] # h[1, 1] 

+ 6*h[2, 1, 1] # h[2] + 6*h[2, 1, 1, 1] # h[1] + 3*h[2, 2] # h[1, 1] + 6*h[2, 2, 1] # h[1] + 3*h[2, 2, 1, 1] # h[] 

 

.. TODO:: 

 

- Introduce fields with degree 1 elements as in 

MuPAD-Combinat, to get proper plethysm. 

- Use UniqueRepresentation to get rid of all the manual cache 

handling for the bases 

- Devise a mechanism so that pickling bases of symmetric 

functions pickles the coercions which have a cache. 

""" 

 

def __init__(self, R): 

r""" 

Initialization of ``self``. 

 

INPUT: 

 

- ``R`` -- a ring 

 

EXAMPLES:: 

 

sage: Sym = SymmetricFunctions(QQ) 

 

TESTS:: 

 

sage: Sym1 = SymmetricFunctions(FiniteField(23)) 

sage: Sym2 = SymmetricFunctions(Integers(23)) 

sage: TestSuite(Sym).run() 

 

""" 

# change the line below to assert(R in Rings()) once MRO issues from #15536, #15475 are resolved 

assert(R in Fields() or R in Rings()) # side effect of this statement assures MRO exists for R 

self._base = R # Won't be needed when CategoryObject won't override anymore base_ring 

Parent.__init__(self, category = GradedHopfAlgebras(R).WithRealizations()) 

 

def a_realization(self): 

r""" 

Return a particular realization of ``self`` (the Schur basis). 

 

EXAMPLES:: 

 

sage: Sym = SymmetricFunctions(QQ) 

sage: Sym.a_realization() 

Symmetric Functions over Rational Field in the Schur basis 

""" 

return self.schur() 

 

def _repr_(self): # could be taken care of by the category 

r""" 

Representation of ``self`` 

 

TESTS:: 

 

sage: SymmetricFunctions(RR) # indirect doctest 

Symmetric Functions over Real Field with 53 bits of precision 

""" 

return "Symmetric Functions over %s"%self.base_ring() 

 

def schur(self): 

r""" 

The Schur basis of the Symmetric Functions 

 

EXAMPLES:: 

 

sage: SymmetricFunctions(QQ).schur() 

Symmetric Functions over Rational Field in the Schur basis 

""" 

return schur.SymmetricFunctionAlgebra_schur(self) 

s = schur 

Schur = schur # Currently needed by SymmetricFunctions.__init_extra__ 

# and sfa.GradedSymmetricFunctionsBases.corresponding_basis_over 

 

def powersum(self): 

r""" 

The power sum basis of the Symmetric Functions 

 

EXAMPLES:: 

 

sage: SymmetricFunctions(QQ).powersum() 

Symmetric Functions over Rational Field in the powersum basis 

""" 

return powersum.SymmetricFunctionAlgebra_power(self) 

p = powersum 

power = powersum # Todo: get rid of this one when it won't be needed anymore 

 

def complete(self): 

r""" 

The complete basis of the Symmetric Functions 

 

EXAMPLES:: 

 

sage: SymmetricFunctions(QQ).complete() 

Symmetric Functions over Rational Field in the homogeneous basis 

""" 

return homogeneous.SymmetricFunctionAlgebra_homogeneous(self) 

h = complete 

homogeneous = complete 

 

def elementary(self): 

r""" 

The elementary basis of the Symmetric Functions 

 

EXAMPLES:: 

 

sage: SymmetricFunctions(QQ).elementary() 

Symmetric Functions over Rational Field in the elementary basis 

""" 

return elementary.SymmetricFunctionAlgebra_elementary(self) 

e = elementary 

 

def monomial(self): 

r""" 

The monomial basis of the Symmetric Functions 

 

EXAMPLES:: 

 

sage: SymmetricFunctions(QQ).monomial() 

Symmetric Functions over Rational Field in the monomial basis 

""" 

return monomial.SymmetricFunctionAlgebra_monomial(self) 

m = monomial 

 

def witt(self, coerce_h=True, coerce_e=False, coerce_p=False): 

r""" 

The Witt basis of the symmetric functions. 

 

EXAMPLES:: 

 

sage: SymmetricFunctions(QQ).witt() 

Symmetric Functions over Rational Field in the Witt basis 

sage: SymmetricFunctions(QQ).witt(coerce_p=True) 

Symmetric Functions over Rational Field in the Witt basis 

sage: SymmetricFunctions(QQ).witt(coerce_h=False, coerce_e=True, coerce_p=True) 

Symmetric Functions over Rational Field in the Witt basis 

""" 

from . import witt 

return witt.SymmetricFunctionAlgebra_witt(self, coerce_h=coerce_h, coerce_e=coerce_e, coerce_p=coerce_p) 

w = witt 

# Currently needed by sfa.GradedSymmetricFunctionsBases.corresponding_basis_over 

Witt = witt 

 

def irreducible_symmetric_group_character(self): 

r""" 

The irreducible `S_n` character basis of the Symmetric Functions. 

 

This basis has the property that if the element indexed by the 

partition `\lambda` is evaluated at the roots of a permutation of 

cycle structure `\rho` then the value is the irreducible character 

`\chi^{(|\rho|-|\lambda|,\lambda)}(\rho)`. 

 

In terms of methods that are implemented in Sage, if ``n`` is 

a sufficiently large integer, then 

``st(lam).character_to_frobenius_image(n)`` is equal the Schur function 

indexed by ``[n-sum(lam)]+lam``. 

 

This basis is introduced in [OZ2015]_. 

 

.. SEEALSO:: 

 

:meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.character_to_frobenius_image`, 

:meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.eval_at_permutation_roots` 

 

EXAMPLES:: 

 

sage: SymmetricFunctions(QQ).irreducible_symmetric_group_character() 

Symmetric Functions over Rational Field in the irreducible symmetric group character basis 

sage: st = SymmetricFunctions(QQ).st() 

sage: s = SymmetricFunctions(QQ).s() 

sage: s(st([3,2]).character_to_frobenius_image(9)) 

s[4, 3, 2] 

sage: s(st([3,2]).character_to_frobenius_image(7)) 

0 

sage: s(st([3,2]).character_to_frobenius_image(6)) 

-s[2, 2, 2] 

sage: list(SymmetricGroup(5).character_table()[-2]) 

[4, 2, 0, 1, -1, 0, -1] 

sage: list(reversed([st([1]).eval_at_permutation_roots(rho) \ 

....: for rho in Partitions(5)])) 

[4, 2, 0, 1, -1, 0, -1] 

""" 

from .character import irreducible_character_basis 

return irreducible_character_basis(self, 'st') 

st = irreducible_symmetric_group_character 

 

def induced_trivial_character(self): 

r""" 

The induced trivial character basis of the Symmetric Functions. 

 

The trivial character of 

 

.. MATH:: 

 

S_{n-|\lambda|} \times S_{\lambda_1} \times S_{\lambda_2} \times 

\cdots \times S_{\lambda_\ell(\lambda)} 

 

induced to the group `S_{n}` is a symmetric function in the 

eigenvalues of a permutation matrix. This basis is that character. 

 

It has the property that if the element indexed by the 

partition `\lambda` is evaluated at the roots of a permutation of 

cycle structure `\rho` then the value is the coefficient 

`\left< h_{(n-|\lambda|,\lambda)}, p_\rho \right>`. 

 

In terms of methods that are implemented in Sage, if ``n`` is 

a sufficiently large integer, then 

``ht(lam).character_to_frobenius_image(n)`` is equal the complete 

function indexed by ``[n-sum(lam)]+lam``. 

 

This basis is introduced in [OZ2015]_. 

 

.. SEEALSO:: 

 

:meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.character_to_frobenius_image`, 

:meth:`~sage.combinat.sf.sfa.SymmetricFunctionAlgebra_generic_Element.eval_at_permutation_roots` 

 

EXAMPLES:: 

 

sage: SymmetricFunctions(QQ).induced_trivial_character() 

Symmetric Functions over Rational Field in the induced trivial character basis 

sage: ht = SymmetricFunctions(QQ).ht() 

sage: h = SymmetricFunctions(QQ).h() 

sage: h(ht([3,2]).character_to_frobenius_image(9)) 

h[4, 3, 2] 

sage: h(ht([3,2]).character_to_frobenius_image(7)) 

h[3, 2, 2] 

sage: h(ht([3,2]).character_to_frobenius_image(5)) 

h[3, 2] 

sage: h(ht([3,2]).character_to_frobenius_image(4)) 

0 

sage: p = SymmetricFunctions(QQ).p() 

sage: [h([4,1]).scalar(p(rho)) for rho in Partitions(5)] 

[0, 1, 0, 2, 1, 3, 5] 

sage: [ht([1]).eval_at_permutation_roots(rho) for rho in Partitions(5)] 

[0, 1, 0, 2, 1, 3, 5] 

""" 

from .character import character_basis 

return character_basis(self, self.h(), "induced trivial character", 'ht') 

ht = induced_trivial_character 

 

def forgotten(self): 

r""" 

The forgotten basis of the Symmetric Functions (or the basis dual to 

the elementary basis with respect to the Hall scalar product). 

 

EXAMPLES:: 

 

sage: SymmetricFunctions(QQ).forgotten() 

Symmetric Functions over Rational Field in the forgotten basis 

 

TESTS: 

 

Over the rationals:: 

 

sage: Sym = SymmetricFunctions(QQ) 

sage: e = Sym.e() 

sage: f = Sym.f() 

sage: h = Sym.h() 

sage: p = Sym.p() 

sage: s = Sym.s() 

sage: m = Sym.m() 

sage: e(f([2,1])) 

-2*e[1, 1, 1] + 5*e[2, 1] - 3*e[3] 

sage: f(e([2,1])) 

3*f[1, 1, 1] + 2*f[2, 1] + f[3] 

sage: h(f([2,1])) 

h[2, 1] - 3*h[3] 

sage: f(h([2,1])) 

3*f[1, 1, 1] + f[2, 1] 

sage: p(f([2,1])) 

-p[2, 1] - p[3] 

sage: f(p([2,1])) 

-f[2, 1] - f[3] 

sage: s(f([2,1])) 

s[2, 1] - 2*s[3] 

sage: f(s([2,1])) 

2*f[1, 1, 1] + f[2, 1] 

sage: m(f([2,1])) 

-m[2, 1] - 2*m[3] 

sage: f(m([2,1])) 

-f[2, 1] - 2*f[3] 

 

Over the integers:: 

 

sage: Sym = SymmetricFunctions(ZZ) 

sage: e = Sym.e() 

sage: f = Sym.f() 

sage: h = Sym.h() 

sage: p = Sym.p() 

sage: s = Sym.s() 

sage: m = Sym.m() 

sage: e(f([2,1])) 

-2*e[1, 1, 1] + 5*e[2, 1] - 3*e[3] 

sage: f(e([2,1])) 

3*f[1, 1, 1] + 2*f[2, 1] + f[3] 

sage: h(f([2,1])) 

h[2, 1] - 3*h[3] 

sage: f(h([2,1])) 

3*f[1, 1, 1] + f[2, 1] 

sage: f(p([2,1])) 

-f[2, 1] - f[3] 

sage: s(f([2,1])) 

s[2, 1] - 2*s[3] 

sage: f(s([2,1])) 

2*f[1, 1, 1] + f[2, 1] 

sage: m(f([2,1])) 

-m[2, 1] - 2*m[3] 

sage: f(m([2,1])) 

-f[2, 1] - 2*f[3] 

 

Conversion from the forgotten basis to the power-sum basis over the 

integers is not well-defined in general, even if the result happens 

to have integral coefficients:: 

 

sage: p(f([2,1])) 

Traceback (most recent call last): 

... 

TypeError: no conversion of this rational to integer 

 

Fun exercise: prove that `p(f_{\lambda})` and `p(m_{\lambda})` have 

integral coefficients whenever `\lambda` is a strict partition. 

""" 

return self.elementary().dual_basis() 

f = forgotten 

 

def symplectic(self): 

""" 

The symplectic basis of the symmetric functions. 

 

.. SEEALSO:: :class:`~sage.combinat.sf.symplectic.SymmetricFunctionAlgebra_symplectic` 

 

EXAMPLES:: 

 

sage: SymmetricFunctions(QQ).symplectic() 

Symmetric Functions over Rational Field in the symplectic basis 

""" 

from . import symplectic 

return symplectic.SymmetricFunctionAlgebra_symplectic(self) 

sp = symplectic 

 

def orthogonal(self): 

""" 

The orthogonal basis of the symmetric functions. 

 

.. SEEALSO:: :class:`~sage.combinat.sf.orthogonal.SymmetricFunctionAlgebra_orthogonal` 

 

EXAMPLES:: 

 

sage: SymmetricFunctions(QQ).orthogonal() 

Symmetric Functions over Rational Field in the orthogonal basis 

""" 

from . import orthogonal 

return orthogonal.SymmetricFunctionAlgebra_orthogonal(self) 

o = orthogonal 

 

def macdonald(self, q='q', t='t'): 

r""" 

Returns the entry point for the various Macdonald bases. 

 

INPUT: 

 

- ``q``, ``t`` -- parameters 

 

Macdonald symmetric functions including bases `P`, `Q`, `J`, `H`, `Ht`. 

This also contains the `S` basis which is dual to the Schur basis with 

respect to the `q,t` scalar product. 

 

The parameters `q` and `t` must be in the base_ring of parent. 

 

EXAMPLES:: 

 

sage: Sym = SymmetricFunctions(FractionField(QQ['q','t'])) 

sage: P = Sym.macdonald().P(); P 

Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald P basis 

sage: P[2] 

McdP[2] 

sage: Q = Sym.macdonald().Q(); Q 

Symmetric Functions over Fraction Field of Multivariate Polynomial Ring in q, t over Rational Field in the Macdonald Q basis 

sage: S = Sym.macdonald().S() 

sage: s = Sym.schur() 

sage: matrix([[S(la).scalar_qt(s(mu)) for la in Partitions(3)] for mu in Partitions(3)]) 

[1 0 0] 

[0 1 0] 

[0 0 1] 

sage: H = Sym.macdonald().H() 

sage: s(H[2,2]) 

q^2*s[1, 1, 1, 1] + (q^2*t+q*t+q)*s[2, 1, 1] + (q^2*t^2+1)*s[2, 2] + (q*t^2+q*t+t)*s[3, 1] + t^2*s[4] 

 

sage: Sym = SymmetricFunctions(QQ['z','q'].fraction_field()) 

sage: (z,q) = Sym.base_ring().gens() 

sage: Hzq = Sym.macdonald(q=z,t=q).H() 

sage: H1z = Sym.macdonald(q=1,t=z).H() 

sage: s = Sym.schur() 

sage: s(H1z([2,2])) 

s[1, 1, 1, 1] + (2*z+1)*s[2, 1, 1] + (z^2+1)*s[2, 2] + (z^2+2*z)*s[3, 1] + z^2*s[4] 

sage: s(Hzq[2,2]) 

z^2*s[1, 1, 1, 1] + (z^2*q+z*q+z)*s[2, 1, 1] + (z^2*q^2+1)*s[2, 2] + (z*q^2+z*q+q)*s[3, 1] + q^2*s[4] 

sage: s(H1z(Hzq[2,2])) 

z^2*s[1, 1, 1, 1] + (z^2*q+z*q+z)*s[2, 1, 1] + (z^2*q^2+1)*s[2, 2] + (z*q^2+z*q+q)*s[3, 1] + q^2*s[4] 

""" 

return macdonald.Macdonald(self, q=q, t=t) 

 

def hall_littlewood(self, t='t'): 

""" 

Returns the entry point for the various Hall-Littlewood bases. 

 

INPUT: 

 

- ``t`` -- parameter 

 

Hall-Littlewood symmetric functions including bases `P`, `Q`, `Qp`. 

The Hall-Littlewood `P` and `Q` functions at `t=-1` are the 

Schur-P and Schur-Q functions when indexed by strict partitions. 

 

The parameter `t` must be in the base ring of parent. 

 

EXAMPLES:: 

 

sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) 

sage: P = Sym.hall_littlewood().P(); P 

Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Hall-Littlewood P basis 

sage: P[2] 

HLP[2] 

sage: Q = Sym.hall_littlewood().Q(); Q 

Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Hall-Littlewood Q basis 

sage: Q[2] 

HLQ[2] 

sage: Qp = Sym.hall_littlewood().Qp(); Qp 

Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Hall-Littlewood Qp basis 

sage: Qp[2] 

HLQp[2] 

""" 

return hall_littlewood.HallLittlewood(self, t=t) 

 

def jack(self, t='t'): 

""" 

Returns the entry point for the various Jack bases. 

 

INPUT: 

 

- ``t`` -- parameter 

 

Jack symmetric functions including bases `P`, `Q`, `Qp`. 

 

The parameter `t` must be in the base ring of parent. 

 

EXAMPLES:: 

 

sage: Sym = SymmetricFunctions(FractionField(QQ['t'])) 

sage: JP = Sym.jack().P(); JP 

Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack P basis 

sage: JQ = Sym.jack().Q(); JQ 

Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack Q basis 

sage: JJ = Sym.jack().J(); JJ 

Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack J basis 

sage: JQp = Sym.jack().Qp(); JQp 

Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the Jack Qp basis 

""" 

return jack.Jack( self, t=t ) 

 

def zonal(self): 

""" 

The zonal basis of the Symmetric Functions 

 

EXAMPLES:: 

 

sage: SymmetricFunctions(QQ).zonal() 

Symmetric Functions over Rational Field in the zonal basis 

""" 

return jack.SymmetricFunctionAlgebra_zonal( self ) 

 

def llt(self, k, t='t'): 

""" 

The LLT symmetric functions. 

 

INPUT: 

 

- ``k`` -- a positive integer indicating the level 

- ``t`` -- a parameter (default: `t`) 

 

LLT polynomials in `hspin` and `hcospin` bases. 

 

EXAMPLES:: 

 

sage: llt3 = SymmetricFunctions(QQ['t'].fraction_field()).llt(3); llt3 

level 3 LLT polynomials over Fraction Field of Univariate Polynomial Ring in t over Rational Field 

sage: llt3.hspin() 

Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the level 3 LLT spin basis 

sage: llt3.hcospin() 

Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the level 3 LLT cospin basis 

sage: llt3.hcospin() 

Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field in the level 3 LLT cospin basis 

""" 

return llt.LLT_class( self, k, t=t ) 

 

def from_polynomial(self, f): 

""" 

Converts a symmetric polynomial ``f`` to a symmetric function. 

 

INPUT: 

 

- ``f`` -- a symmetric polynomial 

 

This function converts a symmetric polynomial `f` in a polynomial ring in finitely 

many variables to a symmetric function in the monomial 

basis of the ring of symmetric functions over the same base ring. 

 

EXAMPLES:: 

 

sage: P = PolynomialRing(QQ, 'x', 3) 

sage: x = P.gens() 

sage: f = x[0] + x[1] + x[2] 

sage: S = SymmetricFunctions(QQ) 

sage: S.from_polynomial(f) 

m[1] 

 

sage: f = x[0] + 2*x[1] + x[2] 

sage: S.from_polynomial(f) 

Traceback (most recent call last): 

... 

ValueError: x0 + 2*x1 + x2 is not a symmetric polynomial 

""" 

return self.m().from_polynomial(f) 

 

def register_isomorphism(self, morphism, only_conversion=False): 

""" 

Register an isomorphism between two bases of ``self``, as a canonical coercion 

(unless the optional keyword ``only_conversion`` is set to ``True``, 

in which case the isomorphism is registered as conversion only). 

 

EXAMPLES: 

 

We override the canonical coercion from the Schur basis to the 

powersum basis by a (stupid!) map `s_\lambda\mapsto 2p_\lambda`. 

:: 

 

sage: Sym = SymmetricFunctions(QQ['zorglub']) # make sure we are not going to screw up later tests 

sage: s = Sym.s(); p = Sym.p().dual_basis() 

sage: phi = s.module_morphism(diagonal = lambda t: 2, codomain = p) 

sage: phi(s[2, 1]) 

2*d_p[2, 1] 

sage: Sym.register_isomorphism(phi) 

sage: p(s[2,1]) 

2*d_p[2, 1] 

 

The map is supposed to implement the canonical isomorphism 

between the two bases. Otherwise, the results will be 

mathematically wrong, as above. Use with care! 

""" 

if only_conversion: 

morphism.codomain().register_conversion(morphism) 

else: 

morphism.codomain().register_coercion(morphism) 

 

_shorthands = ['e', 'f', 'h', 'm', 'p', 's'] 

_shorthands_all = sorted(_shorthands + ['ht', 'o', 'sp', 'st', 'w']) 

 

def __init_extra__(self): 

""" 

Sets up the coercions between the different bases 

 

EXAMPLES:: 

 

sage: Sym = SymmetricFunctions(QQ) # indirect doctest 

sage: s = Sym.s(); p = Sym.p() 

sage: f = s.coerce_map_from(p); f 

Generic morphism: 

From: Symmetric Functions over Rational Field in the powersum basis 

To: Symmetric Functions over Rational Field in the Schur basis 

sage: p.an_element() 

2*p[] + 2*p[1] + 3*p[2] 

sage: f(p.an_element()) 

2*s[] + 2*s[1] - 3*s[1, 1] + 3*s[2] 

sage: f(p.an_element()) == p.an_element() 

True 

 

""" 

#powersum = self.powersum () 

#complete = self.complete () 

#elementary = self.elementary() 

#schur = self.schur () 

#monomial = self.monomial () 

 

iso = self.register_isomorphism 

 

from sage.combinat.sf.classical import conversion_functions 

 

for (basis1_name, basis2_name) in conversion_functions: 

basis1 = getattr(self, basis1_name)() 

basis2 = getattr(self, basis2_name)() 

on_basis = SymmetricaConversionOnBasis(t = conversion_functions[basis1_name,basis2_name], domain = basis1, codomain = basis2) 

from sage.rings.rational_field import RationalField 

if basis2_name != "powersum" or self._base.has_coerce_map_from(RationalField()): 

iso(basis1._module_morphism(on_basis, codomain = basis2)) 

else: 

# Don't register conversions to powersums as coercions, 

# unless the base ring is a `\QQ`-algebra 

# (otherwise the coercion graph loses commutativity). 

iso(basis1._module_morphism(on_basis, codomain = basis2), only_conversion = True) 

 

# Todo: fill in with other conversion functions on the classical bases 

 

def kBoundedSubspace(self, k, t='t'): 

r""" 

Return the `k`-bounded subspace of the ring of symmetric functions. 

 

INPUT: 

 

- ``k`` - a positive integer 

- ``t`` a formal parameter; `t=1` yields a subring 

 

The subspace of the ring of symmetric functions spanned by 

`\{ s_{\lambda}[X/(1-t)] \}_{\lambda_1\le k} = \{ s_{\lambda}^{(k)}[X,t]\}_{\lambda_1 \le k}` 

over the base ring `\mathbb{Q}[t]`. When `t=1`, this space is in fact a subalgebra of 

the ring of symmetric functions generated by the complete homogeneous symmetric functions 

`h_i` for `1\le i \le k`. 

 

.. SEEALSO:: :meth:`sage.combinat.sf.new_kschur.KBoundedSubspace` 

 

EXAMPLES:: 

 

sage: Sym = SymmetricFunctions(QQ) 

sage: KB = Sym.kBoundedSubspace(3,1); KB 

3-bounded Symmetric Functions over Rational Field with t=1 

 

sage: Sym = SymmetricFunctions(QQ['t']) 

sage: Sym.kBoundedSubspace(3) 

3-bounded Symmetric Functions over Univariate Polynomial Ring in t over Rational Field 

 

sage: Sym = SymmetricFunctions(QQ['z']) 

sage: z = Sym.base_ring().gens()[0] 

sage: Sym.kBoundedSubspace(3,t=z) 

3-bounded Symmetric Functions over Univariate Polynomial Ring in z over Rational Field with t=z 

""" 

from sage.combinat.sf.new_kschur import KBoundedSubspace 

return KBoundedSubspace(self, k, t=t) 

 

def kschur(self, k, t ='t'): 

r""" 

Returns the `k`-Schur functions. 

 

EXAMPLES:: 

 

sage: Sym = SymmetricFunctions(QQ) 

sage: ks = Sym.kschur(3,1) 

sage: ks[2]*ks[2] 

ks3[2, 2] + ks3[3, 1] 

sage: ks[2,1,1].lift() 

s[2, 1, 1] + s[3, 1] 

 

sage: Sym = SymmetricFunctions(QQ['t']) 

sage: ks = Sym.kschur(3) 

sage: ks[2,2,1].lift() 

s[2, 2, 1] + t*s[3, 2] 

""" 

return self.kBoundedSubspace(k, t=t).kschur() 

 

def ksplit(self, k, t ='t'): 

r""" 

Returns the `k`-split basis of the `k`-bounded subspace. 

 

EXAMPLES:: 

 

sage: Sym = SymmetricFunctions(QQ) 

sage: ksp = Sym.ksplit(3,1) 

sage: ksp[2]*ksp[2] 

ksp3[2, 2] + ksp3[3, 1] 

sage: ksp[2,1,1].lift() 

s[2, 1, 1] + s[2, 2] + s[3, 1] 

 

sage: Sym = SymmetricFunctions(QQ['t']) 

sage: ksp = Sym.ksplit(3) 

sage: ksp[2,1,1].lift() 

s[2, 1, 1] + t*s[2, 2] + t*s[3, 1] 

""" 

return self.kBoundedSubspace(k, t=t).ksplit() 

 

def khomogeneous(self, k): 

r""" 

Returns the homogeneous symmetric functions in the `k`-bounded subspace. 

 

EXAMPLES:: 

 

sage: Sym = SymmetricFunctions(QQ) 

sage: kh = Sym.khomogeneous(4) 

sage: kh[3]*kh[4] 

h4[4, 3] 

sage: kh[4].lift() 

h[4] 

""" 

return self.kBoundedSubspace(k, t=1).khomogeneous() 

 

def kBoundedQuotient(self, k, t='t'): 

r""" 

Returns the `k`-bounded quotient space of the ring of symmetric functions. 

 

INPUT: 

 

- ``k`` - a positive integer 

 

The quotient of the ring of symmetric functions ... 

 

.. SEEALSO:: :meth:`sage.combinat.sf.k_dual.KBoundedQuotient` 

 

EXAMPLES:: 

 

sage: Sym = SymmetricFunctions(QQ) 

sage: KQ = Sym.kBoundedQuotient(3); KQ 

Traceback (most recent call last): 

... 

TypeError: unable to convert 't' to a rational 

sage: KQ = Sym.kBoundedQuotient(3,t=1); KQ 

3-Bounded Quotient of Symmetric Functions over Rational Field with t=1 

sage: Sym = SymmetricFunctions(QQ['t'].fraction_field()) 

sage: KQ = Sym.kBoundedQuotient(3); KQ 

3-Bounded Quotient of Symmetric Functions over Fraction Field of Univariate Polynomial Ring in t over Rational Field 

""" 

from sage.combinat.sf.k_dual import KBoundedQuotient 

return KBoundedQuotient(self, k, t) 

 

class SymmetricaConversionOnBasis: 

def __init__(self, t, domain, codomain): 

""" 

Initialization of ``self``. 

 

INPUT: 

 

- ``t`` -- a function taking a monomial in CombinatorialFreeModule(QQ, Partitions()), 

and returning a (partition, coefficient) list. 

 

- ``domain``, ``codomain`` -- parents 

 

Construct a function mapping a partition to an element of ``codomain``. 

 

This is a temporary quick hack to wrap around the existing 

symmetrica conversions, without changing their specs. 

 

EXAMPLES:: 

 

sage: Sym = SymmetricFunctions(QQ['x']) 

sage: p = Sym.p(); s = Sym.s() 

sage: def t(x) : [(p,c)] = x; return [ (p,2*c), (p.conjugate(), c) ] 

sage: f = sage.combinat.sf.sf.SymmetricaConversionOnBasis(t, p, s) 

sage: f(Partition([3,1])) 

s[2, 1, 1] + 2*s[3, 1] 

""" 

self._domain = domain 

self.fake_sym = CombinatorialFreeModule(QQ, Partitions()) 

self._codomain = codomain 

self._t = t 

 

def __call__(self, partition): 

""" 

sage: Sym = SymmetricFunctions(QQ['x']) 

sage: p = Sym.p(); s = Sym.s() 

sage: p[1] + s[1] # indirect doctest 

2*p[1] 

""" 

# TODO: use self._codomain.sum_of_monomials, when the later 

# will have an optional optimization for the case when there 

# is no repetition in the support 

return self._codomain._from_dict(dict(self._t(self.fake_sym.monomial(partition))), coerce = True)