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, together with their 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 $Θ: L_n \longrightarrow\mathrm{GL}_{n-1}(\mathbb{C})$ over the complex field $\mathbb{C}$ and constructing a new representation $μ: L_n \longrightarrow \mathrm{GL}_{n}(\mathbb{Z}[t^{\pm 1}])$, where $t$ is an indeterminate. For the representation $μ$, we completely study its faithfulness and irreducibility. We also classify all complex homogeneous $2$-local representations of $L_n$ for $n \ge 3$ and all complex 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 concerning further extensions of representations of $L_n$ to $VL_n$ and $WL_n$.