NSymR

The ribbon basis of NSym, and the fundamental basis QSymF of QSym. A composition of $n$ is a subset of $\{1,\ldots,n-1\}$, 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

Typed signatures

Type	Package	Applies
(list<integer>) -> number	hopf	always

Domain: Combinatorial Hopf algebras

Details

• $H_\alpha = \sum_{\beta \text{ coarsens } \alpha} R_\beta$, and the inverse carries the sign $(-1)^{\ell(\alpha)-\ell(\beta)}$ — Möbius inversion over the Boolean lattice
• $F_\alpha = \sum_{\beta \text{ refines } \alpha} M_\beta$ — the OPPOSITE direction, because NSym and QSym are dual with $H$ dual to $M$ and $R$ dual to $F$
• $\langle R_\alpha, F_\beta\rangle = \delta_{\alpha\beta}$, which holds only if both transition matrices are right; they are written down separately
• The ribbon product has two terms: $R_\alpha R_\beta = R_{\alpha\cdot\beta} + R_{\alpha\triangleright\beta}$, for concatenation and near-concatenation
• The ribbon antipode has ONE term: $S(R_\alpha) = (-1)^{|\alpha|}R_{\alpha^{*}}$ — against an alternating sum over all coarsenings in the $H$ basis
• Products, coproducts and antipodes answer in the basis they were asked in; the two algebras still refuse to mix

Examples

Basic #

$R_{(1,1)} = H_{(1,1)} - H_{(2)}$

live ↗

In$\mathrm{InCompleteBasis}(\mathrm{NSymR}(\bigl\lbrack1, 1\bigr\rbrack))$Out$\mathrm{NSymH}(\bigl\lbrack1, 1\bigr\rbrack)-\mathrm{NSymH}(\bigl\lbrack2\bigr\rbrack)$

the fundamental basis sums over refinements

live ↗

In$\mathrm{InMonomialBasis}(\mathrm{QSymF}(\bigl\lbrack2\bigr\rbrack))$Out$\mathrm{QSymM}(\bigl\lbrack2\bigr\rbrack)+\mathrm{QSymM}(\bigl\lbrack1, 1\bigr\rbrack)$

Properties #

one signed term

live ↗

In$\mathrm{Antipode}(\mathrm{NSymR}(\bigl\lbrack3\bigr\rbrack))$Out$-\mathrm{NSymR}(\bigl\lbrack1, 1, 1\bigr\rbrack)$

Possible issues #

self-conjugate — the convention is settled by the antipode, not by taste

live ↗

In$\mathrm{ConjugateComposition}(\bigl\lbrack2, 1\bigr\rbrack)$Out$\bigl\lbrack2, 1\bigr\rbrack$

About

Examples	4
Crosswalk	Wikipedia ×2Wikidata

See also: Coproduct, Antipode