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Classification of Homogeneous Local Representations of the Singular Braid Monoid

Taher I. Mayassi, Mohamad N. Nasser

TL;DR

The paper addresses the classification of homogeneous $k$-local representations of the braid group $B_n$ and their extensions to the singular braid monoid $SM_n$, extending Mikhalchishina’s results to two new frontiers. It provides a complete classification of non-trivial homogeneous $2$-local representations of $SM_n$ for all $n\ge 2$ and, for $n\ge 3$, all $\Phi$-type extensions of homogeneous $2$-local $B_n$ representations to $SM_n$; it also classifies non-trivial homogeneous $3$-local representations of $B_n$ for $n\ge 4$ and their homogeneous $3$-local extensions to $SM_n$, with explicit matrix parametrizations. The results identify three base $2$-local families $\rho_1,\rho_2,\rho_3$ and eight $3$-local families $\nu_j$, each accompanied by corresponding $SM_n$-extensions, and they underscore when such extensions yield representations of the full singular braid group $SB_n$ (invertibility of certain blocks). Computational tools (e.g., solving systems via Mathematica) accompany the analytic classification, and the work clarifies the relationship between $k$-local extensions and $\Phi$-type extensions, while raising open questions about reducibility of the $F$-representation’s $3$-local extension. This advances the understanding of local representations on braid-related structures and informs potential connections to known representations like Burau and $F$.

Abstract

For a natural number $n$, denote by $B_n$ the braid group on $n$ strings and by $SM_n$ the singular braid monoid on $n$ strings. $SM_n$ is one of the most important extensions of $B_n$. In [13], Y. Mikhalchishina classified all homogeneous $2$-local representations of $B_n$ for all $n \geq 3$. In this article, we extend the result of Mikhalchishina in two ways. First, we classify all homogeneous $3$-local representations of $B_n$ for all $n \geq 4$. Second, we classify all homogeneous $2$-local representations of $SM_n$ for all $n\geq 2$ and all homogeneous $3$-local representations of $SM_n$ for all $n\geq 4$.

Classification of Homogeneous Local Representations of the Singular Braid Monoid

TL;DR

The paper addresses the classification of homogeneous -local representations of the braid group and their extensions to the singular braid monoid , extending Mikhalchishina’s results to two new frontiers. It provides a complete classification of non-trivial homogeneous -local representations of for all and, for , all -type extensions of homogeneous -local representations to ; it also classifies non-trivial homogeneous -local representations of for and their homogeneous -local extensions to , with explicit matrix parametrizations. The results identify three base -local families and eight -local families , each accompanied by corresponding -extensions, and they underscore when such extensions yield representations of the full singular braid group (invertibility of certain blocks). Computational tools (e.g., solving systems via Mathematica) accompany the analytic classification, and the work clarifies the relationship between -local extensions and -type extensions, while raising open questions about reducibility of the -representation’s -local extension. This advances the understanding of local representations on braid-related structures and informs potential connections to known representations like Burau and .

Abstract

For a natural number , denote by the braid group on strings and by the singular braid monoid on strings. is one of the most important extensions of . In [13], Y. Mikhalchishina classified all homogeneous -local representations of for all . In this article, we extend the result of Mikhalchishina in two ways. First, we classify all homogeneous -local representations of for all . Second, we classify all homogeneous -local representations of for all and all homogeneous -local representations of for all .
Paper Structure (5 sections, 11 theorems, 158 equations)

This paper contains 5 sections, 11 theorems, 158 equations.

Table of Contents

  1. Introduction
  2. Generalities
  3. Homogeneous $2$-Local Extensions of Homogeneous $2$-Local Representations of $B_n$ to $SM_n$
  4. Homogeneous $3$-Local Representations of $B_n$ and its Homogeneous $3$-Local Extensions to $SM_n$
  5. Conclusions

Key Result

Proposition 3

13 Let $\rho: B_n \rightarrow G_n$ be a representation of the braid group $B_n$ to a group $G_n$ and let $\mathbb{K}$ be a field with $a,b,c \in \mathbb{K}$. Then, the map $\Phi_{a,b,c}:SM_n\rightarrow \mathbb{K}[G_n]$ which acts on the generators of $SM_n$ by the rules defines a representation of $SM_n$ to $\mathbb{K}[G_n]$. Here $e$ is a neutral element of $G_n$.

Theorems & Definitions (21)

  • Definition 1
  • Definition 2
  • Proposition 3
  • Definition 4
  • Definition 5
  • Theorem 6
  • Theorem 7
  • Definition 8
  • Theorem 9
  • Theorem 10
  • ...and 11 more