Insights on the homogeneous $3$-local representations of the twin groups

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.

💡 Research Summary

The paper investigates homogeneous 3‑local linear representations of three closely related groups: the twin group Tₙ, its virtual extension VTₙ, and its welded extension WTₙ, for all n ≥ 4. A “k‑local” representation is one in which each generator is sent to a block‑diagonal matrix whose non‑trivial block has size k; “homogeneous” means that the same k × k block appears for every generator. The authors focus on the case k = 3, which yields representations of dimension n + 1.

First, the authors recall the Coxeter presentation of Tₙ (generators s₁,…,s_{n‑1} with s_i² = 1 and commuting relations for non‑adjacent indices) and the presentations of VTₙ and WTₙ, which add virtual generators ρ_i together with braid‑type and mixed relations. They then define homogeneous 3‑local representations formally and note that the well‑known Burau (2‑local) and F‑representations (3‑local) of the braid group are prototypes.

The core of the work is Theorem 12, which classifies every homogeneous 3‑local representation τ : Tₙ → GL_{n+1}(ℂ) up to equivalence. By writing the generic 3 × 3 block M =