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Affine diagram categories, algebras and monoids

David He, Daniel Tubbenhauer

TL;DR

The paper develops a comprehensive framework for affine (annular) diagram algebras and their categories, presenting explicit generators and relations for a wide range of algebras (Temperley–Lieb, Brauer, partition, Motzkin, rook variants) and their periodic/reduced forms. It establishes representation-theoretic structure through sandwich cellularity, classifies simple modules (including via finite quotients) and analyzes top-set sizes, while also introducing a growth theory for tensor powers in monoid settings and linking asymptotics to the underlying unit groups (e.g., extended affine symmetric groups and wreath products). The results unify diagrammatic algebras on annuli with categorical perspectives, provide concrete presentations and counts, and demonstrate growth phenomena in both finite and infinite contexts, including explicit formulas and generating-function techniques. Overall, the work advances systematic understanding of the algebraic, categorical, and asymptotic aspects of affine diagram algebras and their representations.

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, planar partition, and planar rook algebras. We give generators and relation presentation for them and their associated categories, study their representation theory, and the asymptotic behavior of tensor products of their representations in the monoid case.

Affine diagram categories, algebras and monoids

TL;DR

The paper develops a comprehensive framework for affine (annular) diagram algebras and their categories, presenting explicit generators and relations for a wide range of algebras (Temperley–Lieb, Brauer, partition, Motzkin, rook variants) and their periodic/reduced forms. It establishes representation-theoretic structure through sandwich cellularity, classifies simple modules (including via finite quotients) and analyzes top-set sizes, while also introducing a growth theory for tensor powers in monoid settings and linking asymptotics to the underlying unit groups (e.g., extended affine symmetric groups and wreath products). The results unify diagrammatic algebras on annuli with categorical perspectives, provide concrete presentations and counts, and demonstrate growth phenomena in both finite and infinite contexts, including explicit formulas and generating-function techniques. Overall, the work advances systematic understanding of the algebraic, categorical, and asymptotic aspects of affine diagram algebras and their representations.

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, planar partition, and planar rook algebras. We give generators and relation presentation for them and their associated categories, study their representation theory, and the asymptotic behavior of tensor products of their representations in the monoid case.

Paper Structure

This paper contains 40 sections, 39 theorems, 39 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Diagram algebras
  3. Affine diagram algebras
  4. What we do
  5. Affine diagram algebras
  6. Definitions
  7. Algebras and monoids
  8. Categories
  9. Relations to affine diagram algebras in the literature
  10. Affine Temperley--Lieb
  11. Affine Brauer
  12. Others
  13. Presentations for algebras
  14. Temperley--Lieb
  15. Planar rook
  16. ...and 25 more sections

Key Result

Proposition 1

$\overline{pTL_m}(\beta)$ is the subalgebra of $\overline{aTL_m}(\beta)$ generated by all even diagrams, excluding even powers of $\tau$.

Figures (10)

  • Figure 1: An affine partition $5$-diagram with the fundamental rectangle shaded.
  • Figure 2: A table of diagram algebras.
  • Figure 3: The image of $e_0$ under the embedding $\overline{pTL_4}(\beta)\hookrightarrow \overline{aTL_4}(\beta)$.
  • Figure 4: The global twist $\tau$ and the individual twist $t_2$.
  • Figure 5: The affine diagram on the left corresponds to the colored diagram on the right. The labels which are $0$ are not shown.
  • ...and 5 more figures

Theorems & Definitions (107)

  • Remark 1
  • Definition 1
  • Definition 2
  • Remark 2
  • Remark 3
  • Remark 4
  • Definition 3
  • Remark 5
  • Definition 4
  • Definition 5
  • ...and 97 more