A characterization of the Delannoy category by Adams operations

TL;DR

This paper characterizes the Delannoy category as a uniquely determinate object among semisimple pre-Tannakian categories with trivial Adams operations, via a generator fixed by $\\psi^2$ and simple exterior powers, and it shows this category is determined by its Grothendieck semiring. It develops a robust framework linking Adams operations, Schur functors, and growth conditions to o-Frobenius and ordered étale algebras, culminating in universal properties that realize $\\mathcal{D}$ as a template for other categories through functors $\\mathcal{D} \\to \\mathcal{C}$. The work also ties the Delannoy category to oligomorphic tensor categories, providing a Kazhdan–Wenzl–style recognition phenomenon and showing that, under mild hypotheses, a category with trivial Adams operations arises from a pro-oligomorphic group with a compatible measure. A central technical achievement is promoting character-theoretic data to algebraic structures (o-Frobenius, Kriz, ordered étale) to produce universal constructions, and a detailed Case IIb analysis outlines potential new pre-Tannakian phenomena yet leaves this case open. Overall, the results illuminate how Adams-operator constraints control categorical structure and enable precise classification via Grothendieck data and universal properties.

Abstract

In recent joint work with Harman, we studied a pre-Tannakian category called the Delannoy category, and showed that it had numerous special properties. One of these is that the Adams operations on its Grothendieck group are trivial. In this paper, we prove three theorems inspired by this. Theorem A states that the Delannoy category is the unique semi-simple pre-Tannakian category having a generator that is fixed by the second Adams operation and whose exterior powers are simple. Theorem B states the Delannoy category is uniquely determined by its Grothendieck semi-ring (among semi-simple pre-Tannakian categories). This is reminiscent of Kazhdan--Wenzl's recognition theorem for quantum $\mathfrak{gl}_n$, and many subsequent results. Finally, Theorem C establishes some properties of pre-Tannakian categories where the second Adams operation fixes a generator.

Figures (1)

• Figure 1: The character table of $\mathbb{H}$. Here $\omega$ is a primitive 3rd root of unity, and $z=\zeta+\zeta^2+\zeta^4$ and $z'=\zeta^3+\zeta^5+\zeta^6$, where $\zeta$ is a primitive 7th root of unity.

Theorems & Definitions (103)

• Theorem A
• Theorem B
• Remark 1.1
• Theorem C
• Remark 1.4
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3