Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized diagram categories and monoids, and their representations

Matthias Fresacher, Willow Stewart, Daniel Tubbenhauer

TL;DR

This work extends classical diagram algebras by formulating a unified framework of generalized diagram categories via cobordism quotients and twistings. It develops a sandwich-cellular approach to their representation theory, proves a general reduction theorem showing tight twistings largely preserve simple-module indexing and dimensions, and establishes Schur–Weyl dualities that connect diagrammatic endomorphisms with group or supergroup representations. The results yield uniform dimension formulas, typical-highest-weight phenomena, and asymptotic growth descriptions across semisimple and nonsemisimple regimes, revealing that the core representation theory is largely insensitive to many parameter choices and twistings. Collectively, the paper maps a broad landscape of diagram monoids and algebras, linking topology-inspired constructions to concrete algebraic and combinatorial structures with implications for centralizer algebras and categorical dualities.

Abstract

Classical diagram categories and monoids, including the Temperley--Lieb, Brauer, and partition cases, arise as special instances of the category of two dimensional cobordisms and admit additional twists that produce a large new family of diagram categories and monoids. In this paper we introduce this family and develop a unified approach to their representation theory.

Generalized diagram categories and monoids, and their representations

TL;DR

This work extends classical diagram algebras by formulating a unified framework of generalized diagram categories via cobordism quotients and twistings. It develops a sandwich-cellular approach to their representation theory, proves a general reduction theorem showing tight twistings largely preserve simple-module indexing and dimensions, and establishes Schur–Weyl dualities that connect diagrammatic endomorphisms with group or supergroup representations. The results yield uniform dimension formulas, typical-highest-weight phenomena, and asymptotic growth descriptions across semisimple and nonsemisimple regimes, revealing that the core representation theory is largely insensitive to many parameter choices and twistings. Collectively, the paper maps a broad landscape of diagram monoids and algebras, linking topology-inspired constructions to concrete algebraic and combinatorial structures with implications for centralizer algebras and categorical dualities.

Abstract

Classical diagram categories and monoids, including the Temperley--Lieb, Brauer, and partition cases, arise as special instances of the category of two dimensional cobordisms and admit additional twists that produce a large new family of diagram categories and monoids. In this paper we introduce this family and develop a unified approach to their representation theory.

Paper Structure

This paper contains 34 sections, 54 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Generalized diagram categories
  3. What we prove
  4. Generalized diagram monoids: cobordisms
  5. Cobordism categories
  6. Some first generalized diagram monoids
  7. Cellularity
  8. Generalized diagram monoids: twists
  9. Representation theoretic motivation for what follows
  10. Equivalences
  11. Twistings
  12. Green's relations
  13. Main theorem
  14. Representation theoretic interpretation
  15. Diagrammatic Schur--Weyl dualities
  16. ...and 19 more sections

Key Result

Lemma 1

For $p\in\{0,1\}[x]$ (a binary polynomial) and $q=1+(-1)^rx^r$ for odd $r\in\mathbb{Z}_{\geq 0}$, $\mathbf{Cob}_{{p/q}}$ can be viewed as a settheoretic category (i.e. without fixing an underlying field).

Theorems & Definitions (143)

  • Example 1
  • Lemma 1
  • proof
  • Definition 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 2
  • proof
  • Remark 4
  • ...and 133 more