# Ticket #14111: trac_14111-qsym_tutorial-sb.patch

File trac_14111-qsym_tutorial-sb.patch, 15.1 KB (added by billey, 9 years ago)
• ## doc/en/reference/combinat/ncsf_qsym.rst

# HG changeset patch
# User Sara Billey <billey@math.washington.edu>
# Date 1362103878 28800
# Node ID e08dcba42369938033c098524510eb3687006b98
# Parent  ec1fb07db6e23f9fbd4c34f0d48198d08ec76473
#14111: Tutorial for quasisymmetric functions (QSYM)

diff --git a/doc/en/reference/combinat/ncsf_qsym.rst b/doc/en/reference/combinat/ncsf_qsym.rst
 a .. toctree:: :maxdepth: 2 ../sage/combinat/ncsf_qsym/tutorial ../sage/combinat/ncsf_qsym/combinatorics ../sage/combinat/ncsf_qsym/generic_basis_code ../sage/combinat/ncsf_qsym/ncsf
• ## sage/combinat/ncsf_qsym/__init__.py

diff --git a/sage/combinat/ncsf_qsym/__init__.py b/sage/combinat/ncsf_qsym/__init__.py
 a import tutorial
• ## new file sage/combinat/ncsf_qsym/tutorial.py

diff --git a/sage/combinat/ncsf_qsym/tutorial.py b/sage/combinat/ncsf_qsym/tutorial.py
new file mode 100755
 - 1# -*- coding: utf-8 -*- r""" ===================================== Tutorial for Quasisymmetric Functions ===================================== In this document we briefly explain the quasisymmetric function bases and related functionality in Sage.   We assume the reader is familar with the package :class:SymmetricFunctions Quasisymmetric functions, denoted QSym, form a subring of the power series ring in countably many variables.  QSym contains the symmetric functions.  These functions first arose in the theory of $P$-partitions.  The initial ideas in this field are attributed to MacMahon, Knuth, Kreweras, Glâffrwd Thomas, Stanley.  In 1984, Gessel formalized the study of quasisymmetric functions and introduced the basis of fundamental quasisymmetric functions.  In 1995, Gelfand, Krob, Lascoux, Leclerc, Retakh, and Thibon showed that the ring of quasisymmetric functions is Hopf dual to the noncommutative symmetric functions.  Many results have built on these. One advantage of working in QSym is that many interesting families of symmetric functions have explicit expansions in fundamental quasisymmetric functions such as Schur functions [Ges]_, Macdonald polynomials [Haiman-Haglund-Loehr], and plethysm of Schur functions [Loehr-Warrington]. For more background see :wikipedia:Quasisymmetric_function. To begin, initialize the ring.  Below we chose to use the rational numbers \QQ.  Other options include the integers \ZZ and \CC:: sage: QSym = QuasiSymmetricFunctions(QQ) sage: QSym Quasisymmetric functions over the Rational Field sage: QSym = QuasiSymmetricFunctions(CC); QSym Quasisymmetric functions over the Complex Field with 53 bits of precision sage: QSym = QuasiSymmetricFunctions(ZZ); QSym Quasisymmetric functions over the Integer Ring All bases of QSym are indexed by compositions e.g. [3,1,1,4].  The convention is to use capitol letters for bases of QSym and lowercase letters for bases of Sym.  Next set up names for the  known bases by running :meth:QuasiSymmetricFunctions.inject_shorthands. As with symmetric functions, you do not need to run this commmand and you could assign these bases other names.  :: sage: QSym = QuasiSymmetricFunctions(QQ) sage: QSym.inject_shorthands() Injecting M as shorthand for Quasisymmetric functions over the Rational Field in the Monomial basis Injecting F as shorthand for Quasisymmetric functions over the Rational Field in the Fundamental basis Injecting dI as shorthand for Quasisymmetric functions over the Rational Field in the dualImmaculate basis Now one can start constructing quasisymmetric functions.  Note, it is best to use variables other than M,F.  :: sage: x = M[2,1]+M[1,2] sage: x M[1, 2] + M[2, 1] sage: y=3*M[1,2]+M[3]^2; y 3*M[1, 2] + 2*M[3, 3] + M[6] sage: F[3,1,3]+7*F[2,1] 7*F[2, 1] + F[3, 1, 3] sage: 3*F[2,1,2]+F[3]^2 F[1, 2, 2, 1] + F[1, 2, 3] + 2*F[1, 3, 2] + F[1, 4, 1] + F[1, 5] + 3*F[2, 1, 2] + 2*F[2, 2, 2] + 2*F[2, 3, 1] + 2*F[2, 4] + F[3, 2, 1] + 3*F[3, 3] + 2*F[4, 2] + F[5, 1] + F[6] To convert from one basis to another is easy:: sage: z=M[1,2,1] sage: z M[1, 2, 1] sage: F(z) -F[1, 1, 1, 1] + F[1, 2, 1] sage: M(F(z)) M[1, 2, 1] To expand in variables, one can specify a finite size alphabet $x_1,x_2,\ldots, x_m$. :: sage: y=M[1,2,1] sage: y.expand(4) x0*x1^2*x2 + x0*x1^2*x3 + x0*x2^2*x3 + x1*x2^2*x3 The usual methods on free modules are available such as coefficients, degrees, and the support. :: sage: z=3*M[1,2]+M[3]^2; z 3*M[1, 2] + 2*M[3, 3] + M[6] sage: z.coefficient([1,2]) 3 sage: z.degree() 6 sage: z.coefficients() [3, 2, 1] sage: z.monomials() [M[1, 2], M[3, 3], M[6]] sage: z.monomial_coefficients() {[3, 3]: 2, [1, 2]: 3, [6]: 1} As with the symmetric functions package, the quasisymmetric function 1 has several instantiations.  However, the most obvious way to write 1 leads to an error:: sage: M[[]] M[] sage: M.one() M[] sage: M(1) M[] sage: M[[]] == 1 True sage: M[] Traceback (most recent call last): ... SyntaxError: invalid syntax Working with symmetric functions -------------------------------- The quasisymmetric functions are a ring which contains the symmetric functions as a subring.  The Monomial quasisymmetric functions are related to the monomial symmetric functions by m_\lambda = \sum_{sort(c) = \lambda} M_c~:: sage: SymmetricFunctions(QQ).inject_shorthands() doctest:1075: RuntimeWarning: redefining global value e doctest:1075: RuntimeWarning: redefining global value m sage: m[2,1] m[2, 1] sage: M(m[2,1]) M[1, 2] + M[2, 1] sage: M(s[2,1]) 2*M[1, 1, 1] + M[1, 2] + M[2, 1] There are methods to test if an expression in the quasisymmetric functions is a symmetric function and, if it is, send it to an expression in the symmetric functions. :: sage: f = M[1,1,2] + M[1,2,1] sage: f.is_symmetric() False sage: g = M[3,1] + M[1,3] sage: g.is_symmetric() True sage: g.to_symmetric_function() m[3, 1] The expansion of the Schur function in terms of the Fundamental quasisymmetric functions is due to [Ges]_.  There is one term in the expansion for each standard tableau of shape equal to the partition indexing the Schur function. :: sage: f = F[3,2] + F[2,2,1] + F[2,3] + F[1,3,1] + F[1,2,2] sage: f.is_symmetric() True sage: f.to_symmetric_function() 5*m[1, 1, 1, 1, 1] + 3*m[2, 1, 1, 1] + 2*m[2, 2, 1] + m[3, 1, 1] + m[3, 2] sage: s(f.to_symmetric_function()) s[3, 2] It is also possible to convert any symmetric function to the quasisymmetric function expansion in any known basis.  The converse is not true :: sage: M( m[3,1,1] ) M[1, 1, 3] + M[1, 3, 1] + M[3, 1, 1] sage: F( s[2,2,1] ) F[1, 1, 2, 1] + F[1, 2, 1, 1] + F[1, 2, 2] + F[2, 1, 2] + F[2, 2, 1] sage: s(M[2,1]) Traceback (most recent call last): ... TypeError: do not know how to make x (= M[2, 1]) an element of self It is possible to experiment with the quasisymmetric function expansion of other bases, but it is important that the base ring be the same for both algebras. :: sage: R = QQ['t'] sage: Qp = SymmetricFunctions(R).hall_littlewood().Qp() sage: QSymt = QuasiSymmetricFunctions(R) sage: Ft = QSymt.F() sage: Ft( Qp[2,2] ) F[1, 2, 1] + t*F[1, 3] + (t+1)*F[2, 2] + t*F[3, 1] + t^2*F[4] :: sage: K = QQ['q','t'].fraction_field() sage: Ht = SymmetricFunctions(K).macdonald().Ht() sage: Fqt = QuasiSymmetricFunctions(Ht.base_ring()).F() sage: Fqt(Ht[2,1]) q*t*F[1, 1, 1] + (q+t)*F[1, 2] + (q+t)*F[2, 1] + F[3] The following will raise an error because the base ring of F is not equal to the base ring of Ht :: sage: F(Ht[2,1]) Traceback (most recent call last): ... TypeError: do not know how to make x (= McdHt[2, 1]) an element of self (=Quasisymmetric functions over the Rational Field in the Fundamental basis) QSym is a Hopf algebra ---------------------- The product on this space is commutative and is inherited from the product by the realization within the polynomial ring:: sage: M[3]*M[1,1] == M[1,1]*M[3] True sage: M[3]*M[1,1] M[1, 1, 3] + M[1, 3, 1] + M[1, 4] + M[3, 1, 1] + M[4, 1] sage: F[3]*F[1,1] F[1, 1, 3] + F[1, 2, 2] + F[1, 3, 1] + F[1, 4] + F[2, 1, 2] + F[2, 2, 1] + F[2, 3] + F[3, 1, 1] + F[3, 2] + F[4, 1] sage: M[3]*F[2] M[1, 1, 3] + M[1, 3, 1] + M[1, 4] + M[2, 3] + M[3, 1, 1] + M[3, 2] + M[4, 1] + M[5] sage: F[2]*M[3] F[1, 1, 1, 2] - F[1, 2, 2] + F[2, 1, 1, 1] - F[2, 1, 2] - F[2, 2, 1] + F[5] There is a coproduct on this ring as well, which in the Monomial basis acts by cutting the composition into a left half and a right half.  The co-product is non-co-commutative :: sage: M[1,3,1].coproduct() M[] # M[1, 3, 1] + M[1] # M[3, 1] + M[1, 3] # M[1] + M[1, 3, 1] # M[] sage: F[1,3,1].coproduct() F[] # F[1, 3, 1] + F[1] # F[3, 1] + F[1, 1] # F[2, 1] + F[1, 2] # F[1, 1] + F[1, 3] # F[1] + F[1, 3, 1] # F[] .. rubric:: The duality pairing with non-commutative symmetric functions These two operations endow the quasisymmetric functions QSym with the structure of a Hopf algebra. It is the dual Hopf algebra of the non-commutative symmetric functions NCSF. Under this duality, the Monomial basis of QSym is dual to the Complete basis of NCSF, and the Fundamental basis of QSym is dual to the Ribbon basis of NCSF (see [MR]_) :: sage: S = M.dual(); S Non-Commutative Symmetric Functions over the Rational Field in the Complete basis sage: M[1,3,1].duality_pairing( S[1,3,1] ) 1 sage: M.duality_pairing_matrix( S, degree=4 ) [1 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0] [0 0 1 0 0 0 0 0] [0 0 0 1 0 0 0 0] [0 0 0 0 1 0 0 0] [0 0 0 0 0 1 0 0] [0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 1] sage: F.duality_pairing_matrix( S, degree=4 ) [1 0 0 0 0 0 0 0] [1 1 0 0 0 0 0 0] [1 0 1 0 0 0 0 0] [1 1 1 1 0 0 0 0] [1 0 0 0 1 0 0 0] [1 1 0 0 1 1 0 0] [1 0 1 0 1 0 1 0] [1 1 1 1 1 1 1 1] sage: NCSF = M.realization_of().dual() sage: R = NCSF.Ribbon() sage: F.duality_pairing_matrix( R, degree=4 ) [1 0 0 0 0 0 0 0] [0 1 0 0 0 0 0 0] [0 0 1 0 0 0 0 0] [0 0 0 1 0 0 0 0] [0 0 0 0 1 0 0 0] [0 0 0 0 0 1 0 0] [0 0 0 0 0 0 1 0] [0 0 0 0 0 0 0 1] sage: M.duality_pairing_matrix( R, degree=4 ) [ 1  0  0  0  0  0  0  0] [-1  1  0  0  0  0  0  0] [-1  0  1  0  0  0  0  0] [ 1 -1 -1  1  0  0  0  0] [-1  0  0  0  1  0  0  0] [ 1 -1  0  0 -1  1  0  0] [ 1  0 -1  0 -1  0  1  0] [-1  1  1 -1  1 -1 -1  1] Let H and G be elements of QSym and h an element of NCSF. Then if we represent the duality pairing with the mathematical notation [ \cdot, \cdot ], [H G, h] = [H \otimes G, \Delta(h)]~. For example, the coefficient of M[2,1,4,1] in M[1,3]*M[2,1,1] may be computed with the duality pairing:: sage: I, J = Composition([1,3]), Composition([2,1,1]) sage: (M[I]*M[J]).duality_pairing(S[2,1,4,1]) 1 And the coefficient of S[1,3] # S[2,1,1] in S[2,1,4,1].coproduct() is equal to this result :: sage: S[2,1,4,1].coproduct() S[] # S[2, 1, 4, 1] + ... + S[1, 3] # S[2, 1, 1] + ... + S[4, 1] # S[2, 1] The duality pairing on the tensor space is another way of getting this coefficient, but currently the method duality_pairing is not defined on the tensor squared space. However, we can extend this functionality by applying a linear morphism to the terms in the coproduct, as follows :: sage: X = S[2,1,4,1].coproduct() sage: def linear_morphism(x, y): ...     return x.duality_pairing(M[1,3]) * y.duality_pairing(M[2,1,1]) sage: X.apply_multilinear_morphism(linear_morphism, codomain=ZZ) 1 Similarly, if H is an element of QSym and g and h are elements of NCSF, then [ H, g h ] = [ \Delta(H), g \otimes h ]~. For example, the coefficient of R[2,3,1] in R[2,1]*R[2,1] is computed with the duality pairing by the following command :: sage: (R[2,1]*R[2,1]).duality_pairing(F[2,3,1]) 1 sage: R[2,1]*R[2,1] R[2, 1, 2, 1] + R[2, 3, 1] This coefficient should then be equal to the coefficient of F[2,1] # F[2,1] in F[2,3,1].coproduct() :: sage: F[2,3,1].coproduct() F[] # F[2, 3, 1] + ... + F[2, 1] # F[2, 1]  + ... + F[2, 3, 1] # F[] This can also be computed by the duality pairing on the tensor space, as above :: sage: X = F[2,3,1].coproduct() sage: def linear_morphism(x, y): ...     return x.duality_pairing(R[2,1]) * y.duality_pairing(R[2,1]) sage: X.apply_multilinear_morphism(linear_morphism, codomain=ZZ) 1 .. rubric:: The operation adjoint to multiplication by a non-commutative symmetric function Let g \in NCSF and consider the linear endomorphism of NCSF defined by left (respectively, right) multiplication by g. Since there is a duality between QSym and NCSF, this linear transformation induces an operator g^\perp on QSym satisfying [ g^\perp(H), h ] = [ H, gh ]~. for any non-commutative symmetric function h. This is implemented by the method :meth:~sage.combinat.ncsf_qsym.generic_basis_code.BasesOfQSymOrNCSF.ElementMethods.skew_by. Explicitly, if H is a quasisymmetric function and g a non-commutative symmetric function, then H.skew_by(g) and H.skew_by(g, side='right') are expressions that satisfy for any non-commutative symmetric function h :: H.skew_by(g).duality_pairing(h) == H.duality_pairing(g*h) H.skew_by(g, side='right').duality_pairing(h) == H.duality_pairing(h*g) For example, M[J].skew_by(S[I]) is 0 unless the composition J begins with I and M(J).skew_by(S(I), side='right') is 0 unless the composition J ends with I :: sage: M[3,2,2].skew_by(S[3]) M[2, 2] sage: M[3,2,2].skew_by(S[2]) 0 sage: M[3,2,2].coproduct().apply_multilinear_morphism( lambda x,y: x.duality_pairing(S[3])*y ) M[2, 2] sage: M[3,2,2].skew_by(S[3], side='right') 0 sage: M[3,2,2].skew_by(S[2], side='right') M[3, 2] .. rubric:: The antipode The antipode sends the Fundamental basis element indexed by the composition I to -1 to the size of I times the Fundamental basis element indexed by the conjugate composition to I :: sage: F[3,2,2].antipode() -F[1, 2, 2, 1, 1] sage: Composition([3,2,2]).conjugate() [1, 2, 2, 1, 1] sage: M[3,2,2].antipode() -M[2, 2, 3] - M[2, 5] - M[4, 3] - M[7] We demonstrate here the defining relation of the antipode:: sage: X = F[3,2,2].coproduct() sage: X.apply_multilinear_morphism(lambda x,y: x*y.antipode()) 0 sage: X.apply_multilinear_morphism(lambda x,y: x.antipode()*y) 0 """