# Ticket #14111: trac_14111-qsym_tutorial-review-ts.patch

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• ## sage/combinat/ncsf_qsym/__init__.py

# HG changeset patch
# User Travis Scrimshaw <tscrim@ucdavis.edu>
# Date 1362169656 28800
diff --git a/sage/combinat/ncsf_qsym/__init__.py b/sage/combinat/ncsf_qsym/__init__.py
diff --git a/sage/combinat/ncsf_qsym/tutorial.py b/sage/combinat/ncsf_qsym/tutorial.py
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 a 1# -*- coding: utf-8 -*- # -*- coding: utf-8 -*- r""" ===================================== Tutorial for Quasisymmetric Functions ===================================== Introduction to 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 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 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 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, basis of fundamental quasisymmetric functions [Ges]_. 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. functions [NCSF]_.  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]. 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 [HHL05]_, and plethysm of Schur functions [LW12]_. For more background see :wikipedia:Quasisymmetric_function. numbers \QQ. Other options include th 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. :: 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 the symmetric functions Sym.  Next set up names for the known bases by running 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() you could assign these bases other names 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. Now one can start constructing quasisymmetric functions.  Note, it is best to use variables other than M,F.  :: .. NOTE:: sage: x = M[2,1]+M[1,2] It is best to use variables other than M and F. :: sage: x = M[2,1] + M[1,2] sage: x M[1, 2] + M[2, 1] sage: y=3*M[1,2]+M^2; y sage: y = 3*M[1,2] + M^2; y 3*M[1, 2] + 2*M[3, 3] + M sage: F[3,1,3]+7*F[2,1] sage: F[3,1,3] + 7*F[2,1] 7*F[2, 1] + F[3, 1, 3] sage: 3*F[2,1,2]+F^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 sage: 3*F[2,1,2] + F^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 To convert from one basis to another is easy:: sage: z=M[1,2,1] sage: z = M[1,2,1] sage: z M[1, 2, 1] To convert from one basis to another is sage: M(F(z)) M[1, 2, 1] To expand in variables, one can specify a finite size alphabet $x_1,x_2,\ldots, x_m$. :: 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 = 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. :: The usual methods on free modules are available such as coefficients, degrees, and the support:: sage: z=3*M[1,2]+M^2; z 3*M[1, 2] + 2*M[3, 3] + M The usual methods on free modules are av sage: z.monomial_coefficients() {[3, 3]: 2, [1, 2]: 3, : 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:: 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 (this is due to the semantics of python):: sage: M[[]] sage: M[[]] M[] sage: M.one() M[] 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~:: \sum_{sort(c) = \lambda} M_c:: sage: SymmetricFunctions(QQ).inject_shorthands() doctest:1075: RuntimeWarning: redefining global value e related to the monomial symmetric functi 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. :: There are methods to test if an expression f in the quasisymmetric functions is a symmetric function:: 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() sage: f = M[3,1] + M[1,3] sage: f.is_symmetric() True sage: g.to_symmetric_function() If f is symmetric, there are methods to convert f to an expression in the symmetric functions:: sage: f.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. 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] tableau of shape equal to the partition 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 :: 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] bases, but it is important that the base q*t*F[1, 1, 1] + (q+t)*F[1, 2] + (q+t)*F[2, 1] + F The following will raise an error because the base ring of F is not equal to the base ring of Ht :: 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 QSym is a Hopf algebra ---------------------- The product on this space is commutative and is inherited from the product by the realization within the po F[1, 1, 1, 2] - F[1, 2, 2] + F[2, 1, 1, 1] - F[2, 1, 2] - F[2, 2, 1] + F 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 :: 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 # M[3, 1] + M[1, 3] # M + M[1, 3, 1] # M[] sage: F[1,3,1].coproduct() F[] # F[1, 3, 1] + F # F[3, 1] + F[1, 1] # F[2, 1] + F[1, 2] # F[1, 1] + F[1, 3] # F + F[1, 3, 1] # F[] .. rubric:: The duality pairing with non-commutative symmetric functions .. 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]_) :: These two operations endow 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 Fundamental basis of QSym is dual to t 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 ], \cdot ], we have: [H G, h] = [H \otimes G, \Delta(h)]~. .. MATH:: [H \cdot 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:: computed with the duality pairing:: 1 And the coefficient of S[1,3] # S[2,1,1] in S[2,1,4,1].coproduct() is equal to this result :: 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 :: coefficient, but currently the method :meth:~sage.combinat.ncsf_qsym.generic_basis_code.BasesOfQSymOrNCSF.ParentMethods.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): applying a linear morphism to the terms 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 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 ]~. .. MATH:: 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 :: [ H, g \cdot 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 the duality pairing by the following com 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() :: 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 :: as above:: sage: X = F[2,3,1].coproduct() sage: def linear_morphism(x, y): as above sage: X.apply_multilinear_morphism(linear_morphism, codomain=ZZ) 1 .. rubric:: The operation adjoint to multiplication by a non-commutative symmetric function .. 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 ]~. .. MATH:: [ g^\perp(H), h ] = [ H, g \cdot h ]. 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. 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 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 :: 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) M[2, 2] the composition J ends with I 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 :: basis element indexed by the conjugate composition to I:: sage: F[3,2,2].antipode() -F[1, 2, 2, 1, 1] We demonstrate here the defining relatio sage: X.apply_multilinear_morphism(lambda x,y: x.antipode()*y) 0 REFERENCES: .. [HHL05] *A combinatorial formula for Macdonald polynomials*. Haiman, Haglund, and Loehr. J. Amer. Math. Soc. 18 (2005), no. 3, 735-761. .. [LW12] *Quasisymmetric expansions of Schur-function plethysms*. Loehr and Warrington. Proc. Amer. Math. Soc. 140 (2012), no. 4, 1159-1171. .. [KT97] *Noncommutative symmetric functions IV: Quantum linear groups and Hecke algebras at* q = 0. Krob and Thibon. Journal of Algebraic Combinatorics 6 (1997), 339-376. """