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On Higher Representation Theory via Categories of type Charge-Conserving--with--Glue

Paul P Martin, Sarah Almateari, Eric C Rowell

TL;DR

The paper develops higher representation theory via charge-conserving targets augmented by glue, introducing the CCwg category ${\mathsf{Match}}_g^N$ to capture non-semisimple phenomena beyond standard CC and ACC frameworks. It proves glue yields a nilpotent radical (HoG-style result) and shows braid representations into ${\mathsf{Match}}_g^N$ restrict, up to radical factors, to charge-conserving content, enabling a structured analysis of representations in rank $N=2$ and $N=3$. By applying the glue calculus, the authors extend CC classifications to rank-$3$ braid representations, and provide detailed case studies in rank $2$ and unipotent braid representations to illustrate how glue shapes indecomposable blocks and radical growth. The framework bridges ordinary radical theory with higher representation theory in a controlled way, offering a pathway to partial classifications and deeper understanding of representation-theoretic structures arising in braid-related contexts with potential implications for topological quantum computation. The work highlights that glue can meaningfully enlarge representation spaces while preserving tractable, translatable CC content, suggesting practical routes to broader classifications in higher-rank settings.

Abstract

In this paper we introduce a strict monoidal subcategory of the category of matrices, suitable to address a higher representation theoretic analogue of radicals (non-semisimplicity) in ordinary representation theory. We show the extent to which this analogue has analogous representation theoretic properties. To illustrate, we apply to two key problems in the study of braid representations (strict monoidal functors from the braid category $\mathsf{B}$ to the matrix category): the classification problem; and the problem of analysing the ordinary braid group representations that braid representations generate in towers.

On Higher Representation Theory via Categories of type Charge-Conserving--with--Glue

TL;DR

The paper develops higher representation theory via charge-conserving targets augmented by glue, introducing the CCwg category to capture non-semisimple phenomena beyond standard CC and ACC frameworks. It proves glue yields a nilpotent radical (HoG-style result) and shows braid representations into restrict, up to radical factors, to charge-conserving content, enabling a structured analysis of representations in rank and . By applying the glue calculus, the authors extend CC classifications to rank- braid representations, and provide detailed case studies in rank and unipotent braid representations to illustrate how glue shapes indecomposable blocks and radical growth. The framework bridges ordinary radical theory with higher representation theory in a controlled way, offering a pathway to partial classifications and deeper understanding of representation-theoretic structures arising in braid-related contexts with potential implications for topological quantum computation. The work highlights that glue can meaningfully enlarge representation spaces while preserving tractable, translatable CC content, suggesting practical routes to broader classifications in higher-rank settings.

Abstract

In this paper we introduce a strict monoidal subcategory of the category of matrices, suitable to address a higher representation theoretic analogue of radicals (non-semisimplicity) in ordinary representation theory. We show the extent to which this analogue has analogous representation theoretic properties. To illustrate, we apply to two key problems in the study of braid representations (strict monoidal functors from the braid category to the matrix category): the classification problem; and the problem of analysing the ordinary braid group representations that braid representations generate in towers.
Paper Structure (24 sections, 3 theorems, 67 equations, 1 figure, 1 table)

This paper contains 24 sections, 3 theorems, 67 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Notation
  3. Charge-conservation with glue
  4. Introduction, motivation and preliminaries
  5. The main construction: the categories ${\mathsf{Match}}_g^N$
  6. Towards a glue calculus for representation theory
  7. Applications I: braid representations
  8. Preamble: Braid representations in rank $N=2$
  9. The parity property
  10. Braid representations in rank $N=3$
  11. The topography of a glue matrix
  12. Recalling the rank-3 CC classification
  13. Braid representations extending the CC type aa0
  14. Braid representations extending the CC type fff, unipotent case
  15. Some $a//$ cases
  16. ...and 9 more sections

Key Result

Lemma 4.2

Let $N \in {\mathbb{N}}$. If a YBO $R\in Aut({\mathbb{C}}^N\otimes {\mathbb{C}}^N) \subset {\mathsf{Mat}}^N(2,2)$ is invariant under the skew transpose $R_{ij,kl} \mapsto R_{\overline{kl},\overline{ij}}$ then for each $n \in {\mathbb{N}}$ the representation $\rho_R$ given by the functor acting on $B

Figures (1)

  • Figure 1: Alcove geometric framework for unipotent fff. This is the ${\mathbb{Z}}^3$ nearest neighbour lattice projected along the (1,1,1) line; with affine reflection walls at separation 2 (in blue). The origin and the start of the dominant region (in Lie terminology) is marked with a yellow triangle.

Theorems & Definitions (6)

  • Lemma 4.2
  • proof
  • Lemma 4.10
  • proof
  • Corollary 4.10.1
  • proof