Bases for \(NCSym\).

AUTHORS:

  • Travis Scrimshaw (08-04-2013): Initial version
sage.combinat.ncsym.bases.MultiplicativeNCSymBases

Category of multiplicative bases of symmetric functions in non-commuting variables.

A multiplicative basis is one for which \(\mathbf{b}_A \mathbf{b}_B = \mathbf{b}_{A|B}\) where \(A|B\) is the pipe() operation on set partitions.

EXAMPLES:

sage: from sage.combinat.ncsym.bases import MultiplicativeNCSymBases
sage: NCSym = SymmetricFunctionsNonCommutingVariables(QQ)
sage: MultiplicativeNCSymBases(NCSym)
Category of multiplicative bases of symmetric functions in non-commuting variables over the Rational Field
sage.combinat.ncsym.bases.NCSymBases

Category of bases of symmetric functions in non-commuting variables.

EXAMPLES:

sage: from sage.combinat.ncsym.bases import NCSymBases
sage: NCSym = SymmetricFunctionsNonCommutingVariables(QQ)
sage: NCSymBases(NCSym)
Category of bases of symmetric functions in non-commuting variables over the Rational Field
sage.combinat.ncsym.bases.NCSymBasis_abstract

Abstract base class for a basis of \(NCSym\) or its dual.

sage.combinat.ncsym.bases.NCSymDualBases

Category of bases of dual symmetric functions in non-commuting variables.

EXAMPLES:

sage: from sage.combinat.ncsym.bases import NCSymDualBases
sage: DNCSym = SymmetricFunctionsNonCommutingVariables(QQ).dual()
sage: NCSymDualBases(DNCSym)
Category of bases of dual symmetric functions in non-commuting variables over the Rational Field
sage.combinat.ncsym.bases.NCSymOrNCSymDualBases

Base category for the category of bases of symmetric functions in non-commuting variables or its Hopf dual for the common code.