@enumeratio/hopf

Combinatorial Hopf algebra extensions for compute-engine: NSym and QSym on compositions, with product, coproduct and antipode. Every other algebra in this section has a product and nothing else; a Hopf algebra also has a coproduct, and the two must satisfy $\Delta(x \cdot y) = \Delta(x) \cdot \Delta(y)$ — the identity the whole package is tested against. See the guide.

Declaring

import { ComputeEngine } from "@cortex-js/compute-engine";
import { declareHopf } from "@enumeratio/hopf";

const ce = new ComputeEngine();
declareHopf(ce);

The two algebras

Both are indexed by compositions (ordered lists of positive parts) and dual to each other:

	basis	product	coproduct
NSym	$H_\alpha$	concatenation	deconcatenation, extended multiplicatively
QSym	$M_\alpha$	quasi-shuffle	deconcatenation

Heads

• NSym — NSymAlgebra, NSymH (complete basis), NSymR (ribbon basis)
• QSym — QSymAlgebra, QSymM (monomial basis), QSymF (fundamental basis)
• Structure — Coproduct, Antipode, Counit, HopfDegree, HopfTensor
• Basis conversion — InCompleteBasis, InRibbonBasis, InMonomialBasis, InFundamentalBasis, ConjugateComposition

src/hopf.ts also exports the underlying Composition/Element/Tensor model (nsymProduct, qsymCoproduct, quasiShuffle, antipode, …) for use without going through compute-engine.

Next

Both bases at degree $n$ have dimension $2^{n-1}$, one basis element per composition of $n$ — the guide checks the compatibility identity directly against that count.