NSymR

The ribbon basis of NSym, and the fundamental basis `QSymF` of QSym. A composition of is a subset of , so the compositions form a Boolean lattice; these bases come from summing over it and inverting.

WikipediaQuasisymmetric function↗Quasisymmetric function↗WikidataQ7269535↗

NSymR(composition)a ribbon basis element of NSym

QSymF(composition)a fundamental basis element of QSym

InCompleteBasis(x) / InRibbonBasis(x)change of basis within NSym

InMonomialBasis(x) / InFundamentalBasis(x)change of basis within QSym

ConjugateComposition(composition)the transpose of the ribbon's skew shape

Domain: Combinatorial Hopf algebras

Details

• , and the inverse carries the sign — Möbius inversion over the Boolean lattice
• — the OPPOSITE direction, because NSym and QSym are dual with dual to and dual to
• , which holds only if both transition matrices are right; they are written down separately
• The ribbon product has two terms: , for concatenation and near-concatenation
• The ribbon antipode has ONE term: — against an alternating sum over all coarsenings in the basis
• Products, coproducts and antipodes answer in the basis they were asked in; the two algebras still refuse to mix

Examples

See also: Coproduct, Antipode