Affine diagram categories, algebras and monoids
Annular versions of the classical diagram algebras, treated uniformly as categories, algebras, monoids and representation-theoretic objects.
Data
- Title: Affine diagram categories, algebras and monoids
- Authors: David He and Daniel Tubbenhauer
- Status: preprint. Last update: Fri, 19 Dec 2025 07:05:59 UTC
- Code / errata: GitHub
- arXiv: https://arxiv.org/abs/2512.05510
Abstract
We introduce and study several affine (=annular in this paper) versions of the classical diagram algebras such as Temperley–Lieb, partition, Brauer, Motzkin, rook Brauer, rook, and planar rook algebras. We give generators and relation presentations for them and their associated categories, study their representation theory, and the asymptotic behavior of tensor products of their representations in the monoid case. Under a mild hypothesis, we also prove a previous conjecture concerning the asymptotic growth of the number of indecomposable summands in tensor powers of representations for finite monoids.
What is the point?
Classical diagram algebras are usually drawn in a rectangle. This paper asks what happens when one moves the same world to an annulus: diagrams may now wrap around, and this changes both the combinatorics and the representation theory. The payoff is a uniform framework containing many familiar diagram algebras at once.
diagrams live on an annulus.
composition gives categories, algebras and monoids.
tensor powers reveal asymptotic representation theory.
A picture
The abstract is spot on. Here is one of the affine diagrammatic examples.
