Irreducible representations of welded braid group

TL;DR

This paper classifies irreducible matrix representations of the welded braid group, revealing new representations and extending known ones, with implications for understanding the group's structure.

Contribution

It provides a complete classification of certain irreducible representations of the welded braid group and introduces new families of such representations.

Findings

No irreducible representations of dimension 2 to n-2 for n≥5.

Complete classification of extensions of braid group representations to welded braid groups.

Discovery of new one-parameter family of n-1 dimensional irreducible representations.

Abstract

In this paper we study irreducible matrix representations of the welded braid group $WB_n$, also known as the group of conjugating automorphisms of a free group $F_n.$ We prove that $WB_n$ has no irreducible representations of dimension $r,$ where $2\leqslant r\leqslant n-2$ for $n\geqslant 5.$ We give complete classification of all extensions of irreducible representations of the braid group $B_n$ to the welded braid group $WB_n$ of dimensions $n-1$ (for $n\geqslant 7$) and $n$ (for $n\geqslant 7,$ $n\neq 8$). Classification of all extensions of the irreducible $n-1-$ dimensional reduced Burau representation is given for $n\geqslant 5,$ $n\neq 6.$ A new one-parameter family of $n-1-$ dimensional irreducible representations of $WB_n$ is discovered. Classification of all extensions of the irreducible $n-$dimensional standard representation (also known as Tong-Yang-Ma representation) is given for all $n\geqslant 3.$ New families of the 3-dimensional irreducible representations of $WB_3$ are discovered.