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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
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 """
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 """
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 """ 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 """ 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 """ 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 """ 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 """ 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 """ 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] """ 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] """ 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. """ 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 """ 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 """ 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] """
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] """
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 """
def zonal(self): """ The zonal basis of the Symmetric Functions
EXAMPLES::
sage: SymmetricFunctions(QQ).zonal() Symmetric Functions over Rational Field in the zonal basis """
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 """
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 """
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! """ else:
_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 ()
else: # Don't register conversions to powersums as coercions, # unless the base ring is a `\QQ`-algebra # (otherwise the coercion graph loses commutativity).
# 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 """
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] """
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] """
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] """
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 """
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] """
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
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