[Paper Review] Ordered forests and parking functions

Loïc Foissy|arXiv (Cornell University)|Jul 9, 2010

Advanced Topics in Algebra17 references6 citations

TL;DR

This paper establishes an isomorphism between the Hopf algebra of ordered forests and the Hopf algebra of parking functions using a rigidity theorem for Dup-Dend bialgebras—graded, connected bialgebras that unify duplicial and dendriform structures. The key result shows both algebras are isomorphic to the free duplicial algebra over a set of decorations, proving their structural equivalence via a symmetric Hopf pairing and morphism compatibility with the Malvenuto-Reutenauer algebra FQSym.

ABSTRACT

We prove that the Hopf algebra of parking functions and the Hopf algebra of ordered forests are isomorphic, using a rigidity theorem for a particular type of bialgebras.

Motivation & Objective

• To establish an algebraic isomorphism between the Hopf algebra of ordered forests and the Hopf algebra of parking functions.
• To study the Hopf algebra structure of ordered forests through a symmetric, degenerate pairing and its kernel.
• To prove that both algebras are isomorphic to the same free duplicial algebra, using a rigidity theorem for graded, connected Dup-Dend bialgebras.
• To show that the morphism Θ from ordered forests to FQSym is a morphism of Dup-Dend bialgebras and induces an isometry on quotients.
• To unify combinatorial Hopf algebra structures via the Dup-Dend bialgebra framework, extending known results on FQSym and PQSym.

Proposed method

• Introduce the notion of a Dup-Dend bialgebra as a structure combining duplicial algebra and dendriform coalgebra with compatibility conditions.
• Define a symmetric Hopf pairing on the Hopf algebra of ordered forests, showing it is degenerate with kernel equal to the kernel of the morphism Θ.
• Prove that the augmentation ideals of the Hopf algebras H_o, PQSym, and H_p^D are Dup-Dend bialgebras, with H_ho and FQSym as sub-bialgebras.
• Establish a rigidity theorem: any graded, connected Dup-Dend bialgebra is isomorphic to a free duplicial algebra, hence isomorphic to H_p^D.
• Use formal series manipulation to deduce that H_o and PQSym are both isomorphic to the same H_p^D, hence isomorphic to each other.
• Verify that the morphism Θ: H_o → FQSym is a morphism of Dup-Dend bialgebras, and that it induces an isometry on the quotient H_o / Ker(Θ) ≅ H_ho.

Experimental results

Research questions

• RQ1 Are the Hopf algebras of ordered forests and parking functions isomorphic as Hopf algebras?
• RQ2 Does the symmetric pairing on the Hopf algebra of ordered forests have a kernel that coincides with the kernel of the morphism Θ to FQSym?
• RQ3 Can the structure of the Hopf algebra of ordered forests be fully characterized via the Dup-Dend bialgebra framework?
• RQ4 Is there a rigidity theorem that forces any graded, connected Dup-Dend bialgebra to be isomorphic to a free duplicial algebra?
• RQ5 Do the morphisms between H_o, PQSym, and FQSym preserve the Dup-Dend bialgebra structure?

Key findings

• The Hopf algebra of ordered forests H_o and the Hopf algebra of parking functions PQSym are isomorphic as Hopf algebras.
• The symmetric pairing on H_o is degenerate, and its kernel is exactly the kernel of the morphism Θ: H_o → FQSym.
• The morphism Θ induces an isometry from H_o / Ker(Θ) (isomorphic to H_ho) onto FQSym, preserving the Hopf algebra structure.
• The augmentation ideals of H_o, PQSym, and H_p^D are all Dup-Dend bialgebras, and H_ho and FQSym are sub-bialgebras of H_o and PQSym respectively.
• A rigidity theorem proves that any graded, connected Dup-Dend bialgebra is isomorphic to a free duplicial algebra, hence isomorphic to H_p^D, which implies H_o ≅ PQSym.
• The morphism Θ: H_o → FQSym is a morphism of Dup-Dend bialgebras, and the sub-bialgebras H_ho and FQSym are isomorphic as Dup-Dend bialgebras.