Insights on the homogeneous $3$-local representations of the twin groups
Mohamad N. Nasser
TL;DR
This work addresses the problem of classifying homogeneous $3$-local representations of the twin-related groups $T_n$, $VT_n$, and $WT_n$ for all $n\ge 4$, and examines their irreducibility and faithfulness. The authors provide explicit classifications: $T_n$ yields eleven $3$-local families $\tau_j$, $VT_n$ yields fourteen $\delta_j$ and $WT_n$ yields five $\gamma_j$, all realized via block matrices in $GL_{n+1}(\mathbb{C})$. A key finding is that every homogeneous $3$-local representation is reducible to degree $n$, with invariant subspaces that yield lower-dimensional factors; irreducibility criteria are sharpened in the $n=4$ case for the first two families ($\tau_1$ and $\tau_2$). Moreover, in the VT_n and WT_n settings, all $3$-local representations are unfaithful, with unfaithfulness determined by parameter values in several VT_n families and universally for WT_n. Collectively, these results illuminate the linear representation theory of these non-classical braid-type groups and constrain the existence of faithful finite-dimensional representations for them.
Abstract
We provide a complete classification of the homogeneous $3$-local representations of the twin group $T_n$, the virtual twin group $VT_n$, and the welded twin group $WT_n$, for all $n\geq 4$. Beyond this classification, we examine the main characteristics of these representations, particularly their irreducibility and faithfulness. More deeply, we show that all such representations are reducible, and most of them are unfaithful. Also, we find necessary and sufficient conditions of the first two types of the classified representations of $T_n$ to be irreducible in the case $n=4$. The obtained results provide insights into the algebraic structure of these three groups.