The second Delannoy category

TL;DR

The paper constructs the abelian second Delannoy category $ D$, a pre-Tannakian category arising from the non-quasi-regular Delannoy setting, and embeds the additive category $ A$ into it via a fully faithful tensor functor. It shows that $ A$ has exactly two local abelian envelopes, through $ C$ and $ D$, without admitting a single abelian envelope, and develops a derived-category framework that yields a semi-orthogonal decomposition of $D( B)$ into components equivalent to $D( ext{Vec})$, $D( C)$, and $D( D)$. The work provides a Ringel dual perspective by relating the pre-sheaf category $ B$ to $ D$ via tilting modules and establishes a universal property for $ D$ in terms of unbounded 2-Delannic algebras, enabling abelian versions of other oligomorphic tensor categories. The results are independent of the coefficient field in many respects and lay groundwork for systematic construction of abelian envelopes for other oligomorphic tensor categories. Overall, the paper advances the understanding of local envelopes, highest-weight structures, and derived tensor structures in oligomorphic contexts, offering a concrete and universal framework for non-quasi-regular Delannoy-type categories.

Abstract

In recent work, Harman and Snowden constructed a symmetric tensor category associated to an oligomorphic group equipped with a measure. The oligomorphic group $\mathbb{G}$ of order preserving automorphisms of the real line admits exactly four measures. The category $\mathcal{C}$ associated to the first measure is called the (first) Delannoy category; it is semi-simple and pre-Tannakian, with numerous special properties. In this paper, we study the (non-abelian) category $\mathcal{A}$ associated to the second measure, which we call the second Delannoy category. We construct a new pre-Tannakian category $\mathcal{D}$ together with a fully faithful tensor functor $Ψ\colon \mathcal{A} \to \mathcal{D}$. The category $\mathcal{D}$ is the correct ``abelian version'' of the second Delannoy category. Like $\mathcal{C}$, it has remarkable properties: for instance, it is non-semi-simple, but behaves uniformly in the coefficient field (e.g., it has the same Grothendieck ring and $\mathrm{Ext}^1$ quiver over any field). Additionally, we completely solve the problem of understanding how $\mathcal{A}$ relates to general pre-Tannakian categories. We show that $\mathcal{A}$ admits exactly two local abelian envelopes: the functor $Ψ$, and a previously constructed functor $Φ\colon \mathcal{A} \to \mathcal{C}$. This is the first case where the local envelopes of a category have been completely determined, outside of cases where there is at most one envelope. This work opens the door to constructing abelian versions of other oligomorphic tensor categories that do not admit a unique envelope.

Figures (2)

• Figure 1: The four main categories and their relationships: $\mathcal{A}$ is the second Delannoy category, $\mathcal{B}$ is the pre-sheaf category of $\mathcal{A}$, $\mathcal{C}$ is the first Delannoy category, and $\mathcal{D}$ is the new tensor category constructed in this paper, which we call the abelian second Delannoy category. The categories $\mathcal{C}$ and $\mathcal{D}$ are pre-Tannakian, while $\mathcal{A}$ and $\mathcal{B}$ are not.
• Figure 2: Two $(4,4)$-Delannoy paths. The left one is quasi-diagonal, while the right one is not.

Theorems & Definitions (159)

• Theorem A
• Remark 2.1
• Remark 2.2
• Theorem B
• Remark 2.3
• Remark 2.4
• Theorem C
• Theorem D
• Remark 2.5
• Definition 3.1