[Paper Review] Some dendriform functors

Frédéric Chapoton|ArXiv.org|Sep 15, 2009

Algebraic structures and combinatorial models12 references3 citations

TL;DR

This paper proposes a categorification of the dendriform operad using module categories over Tamari lattices, constructing functors that categorify three key operations: the first composition map $\circ_1$, the associative product $*$, and a new product $\#$. The key contribution is proving that the $\#$ product preserves projective modules, enabling a well-defined functorial categorification of this operation on the level of Grothendieck groups.

ABSTRACT

We make a first step towards categorification of the dendriform operad, using categories of modules over the Tamari lattices. This means that we describe some functors that correspond to part of the operad structure.

Motivation & Objective

• To initiate a categorification of the dendriform operad $\operatorname{Dend}$ using representation theory of Tamari lattices.
• To define functors on module categories over Tamari posets that correspond to the operadic operations of $\operatorname{Dend}$.
• To show that the $\#$ product, introduced via Catalan alternative tableaux, preserves projective modules and thus induces a well-defined functor on the Grothendieck group.
• To demonstrate that the $\#$ product respects the Euler form and preserves positivity in the Grothendieck group.

Proposed method

• Categorification is achieved by interpreting the basis of planar binary trees as simple modules in the Grothendieck group of the Tamari poset category.
• Functors are constructed from module categories over $\mathbb{Y}_m \times \mathbb{Y}_n$ to $\mathbb{Y}_{m+n-1}$, corresponding to the $\circ_1$, $*$, and $\#$ operations.
• The $\#$ product is defined combinatorially via noncrossing configurations and shown to preserve projective modules using induction and decomposition lemmas.
• The Euler form is used to verify compatibility of the $\#$ functor with the Grothendieck group structure.
• The proof relies on structural lemmas (e.g., Lemma 10.3, Lemma 10.4) that decompose products using over/under products and noncrossing tree properties.
• The category of projective modules is shown to be closed under $\circ_1$ and $\#$, enabling the construction of the desired functors.

Experimental results

Research questions

• RQ1 Can the dendriform operad be partially categorified using module categories over Tamari lattices?
• RQ2 How can the $\#$ product, defined via Catalan alternative tableaux, be realized as a functor on module categories?
• RQ3 Does the $\#$ product preserve the subcategory of projective modules in the Tamari poset category?
• RQ4 Is the $\#$ product compatible with the Euler form on the Grothendieck group of Tamari posets?
• RQ5 Can the $\#$ product be shown to preserve positivity and multiplicity-free sums in the Grothendieck group?

Key findings

• The $\#$ product preserves projective modules, as proven by induction on degree and decomposition into over/under products.
• The $\#$ product induces a well-defined functor from the category of $\mathbb{Y}_m \times \mathbb{Y}_n$-modules to $\mathbb{Y}_{m+n-1}$-modules.
• The $\#$ product respects the Euler form, satisfying $E(x\#y) = E(x)\#E(y)$ on the Grothendieck group.
• The $\#$ product of two sums of planar binary trees without multiplicity results in a sum of trees without multiplicity, as shown in equation (40).
• The set of projective elements is generated by $\psfig{height=7.11317pt}$ and $\psfig{height=7.11317pt}$ under $\circ_1$ and $\#$, as established in Proposition 10.6.
• The $\#$ product is positive on positive elements, and the product of two positive elements in the Grothendieck group remains positive.