The Delannoy category

TL;DR

The paper provides a comprehensive analysis of the Delannoy category, a pre-Tannakian category associated with order-preserving self-maps of the real line, highlighting its classification, combinatorial structure, and remarkable properties.

Contribution

It classifies simple objects, determines tensor product rules, and introduces a Delannoy path model, revealing the category's semi-simplicity and unique features across all characteristics.

Findings

The Delannoy category is semi-simple in all characteristics.

All simple objects have categorical dimension ±1.

The Grothendieck group has trivial Adams operations.

Abstract

Let $G$ be the group of all order-preserving self-maps of the real line. In previous work, the first two authors constructed a pre-Tannakian category $\underline{\mathrm{Rep}}(G)$ associated to $G$. The present paper is a detailed study of this category, which we name the Delannoy category. We classify the simple objects, determine branching rules to open subgroups, and give a combinatorial rule for tensor products. The Delannoy category has some remarkable features: it is semi-simple in all characteristics; all simples have categorical dimension $\pm 1$; and the Adams operations on its Grothendieck group are trivial. We also give a combinatorial model for $\underline{\mathrm{Rep}}(G)$ based on Delannoy paths.