On representations of the triplet group and some of its extensions

In this paper, we study the representations of the triplet group $L_n$, where $n$ is a positive integer, and its extensions to the virtual and welded triplet groups $VL_n$ and $WL_n$, respectively. We first introduce $L_n$, its extensions, and its pure subgroup. We then investigate several representations, proving the irreducibility of the classical Tits representation $Θ: L_n \to \mathrm{GL}_{n-1}(\mathbb{C})$ over the complex field $\mathbb{C}$ and constructing a new representation $μ: L_n \to \mathrm{Aut}(\mathbb{F}_n)$, where $\mathbb{F}_n$ is the free group of rank $n$. For the representation $μ$, we determine its matrix form, faithfulness, and irreducibility. We also classify all complex homogeneous $2$-local representations of $L_n$ for $n \ge 3$ and all non-homogeneous $2$-local representations of $L_3$, establishing connections with the complex specialization of the representation $μ$. Finally, we examine extensions of $L_n$ representations to $VL_n$ and $WL_n$, proving their existence, classifying non-trivial complex homogeneous $2$-local representations, and analyzing their faithfulness and irreducibility. The paper concludes with an open question regarding further extension of representation of $L_n$ to $VL_n$ and $WL_n$.

💡 Research Summary

The paper investigates the representation theory of the triplet group Lₙ (for any positive integer n) and its two natural extensions, the virtual triplet group VLₙ and the welded triplet group WLₙ. After recalling the definition of Lₙ as a Coxeter‑type group generated by ℓ₁,…,ℓ_{n‑1} with relations ℓ_i²=1 and ℓ_iℓ_{i+1}ℓ_i=ℓ_{i+1}ℓ_iℓ_{i+1}, the authors introduce the pure triplet subgroup PLₙ and note that for n≥4 the pure subgroup is a non‑abelian free group of finite rank.

The first major result concerns the classical Tits representation Θ: Lₙ→GL_{n‑1}(ℂ). The authors give an explicit block‑matrix description of Θ(ℓ_i) and prove its irreducibility over ℂ. The proof hinges on two lemmas: any invariant subspace containing a standard basis vector must be the whole space, and a certain symmetric matrix A_{n‑1} (with diagonal entries –2 and off‑diagonal entries 1 or 2) is invertible. Assuming a non‑trivial invariant subspace leads to a homogeneous linear system A_{n‑1}u=0, which forces u=0, a contradiction. Hence Θ is completely irreducible.

Next the paper constructs a new representation μ: Lₙ→Aut(Fₙ), where Fₙ is the free group on n generators x₁,…,xₙ. For a fixed integer k and an indeterminate t, μ(ℓ_i) sends x_i↦t^{k}x_{i+1}, x_{i+1}↦t^{‑k}x_i, and fixes all other generators. Direct verification shows that μ respects both defining relations of Lₙ, so μ is a genuine group homomorphism. The authors translate μ into an (n×n) integer matrix form, demonstrating that each ℓ_i acts by a matrix Λ_i that reproduces the same action on the abelianisation of Fₙ. They then prove that μ is faithful (its kernel is trivial) and irreducible as a representation into Aut(Fₙ), because no non‑trivial proper free‑subgroup of Fₙ is invariant under all Λ_i.

The notion of a “2‑local” complex representation is introduced: each generator ℓ_i acts linearly on a 2‑dimensional complex vector space, while distinct generators commute. For n≥3 the authors completely classify all homogeneous 2‑local representations, showing that they fall into two families—those induced from the Tits representation and those induced from the complex specialization of μ. For the exceptional case n=3 they also list all non‑homogeneous 2‑local representations, again linking them to specializations of μ.

The paper then turns to extensions of these representations to the virtual and welded triplet groups. VLₙ is obtained by adjoining generators ρ_i (with ρ_i²=1, braid‑type relations among the ρ_i, and mixed relations ρ_iℓ_{i+1}ℓ_i=ℓ_{i+1}ρ_iρ_{i+1}). WLₙ is the quotient of VLₙ by the additional welded relation ρ_iℓ_{i+1}ℓ_i=ℓ_{i+1}ℓ_iρ_{i+1}. Proposition 23 shows that both Θ and μ extend naturally to VLₙ and WLₙ, preserving the defining relations. Theorem 24 classifies all non‑trivial homogeneous 2‑local complex representations of VLₙ and WLₙ for n≥3, proving that again only the Θ‑derived and μ‑derived families occur. Theorem 25 establishes the faithfulness of the extended μ representations, while Theorem 26 proves the irreducibility of both extended families (with the caveat that the extended Θ may lose faithfulness in certain cases).

Finally, the authors pose an open question: to what extent can representations of Lₙ be further generalized to VLₙ and WLₙ, especially beyond the 2‑local or linear setting? They suggest investigating higher‑dimensional, possibly non‑linear, or quantum‑type representations, which could illuminate deeper connections between triplet groups, virtual knot theory, and the topology of configuration spaces. The paper thus provides a thorough analysis of linear and non‑linear representations of Lₙ, their extensions, and a roadmap for future research.