The Delannoy category

Nate Harman; Andrew Snowden; Noah Snyder

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arXiv · 2211.15392

The Delannoy category

Nate Harman·Andrew Snowden·Noah Snyder

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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.

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Nate Harman, Andrew Snowden, Noah Snyder. 2022-11-28. The Delannoy category. https://arxiv.org/abs/2211.15392

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