On representations of the triplet group and some of its extensions

Abstract

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 $V{L}_n$ and $W{L}_n$, 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 $\Theta :L_n⟶{\mathrm{GL}}_{n-1}(\mathbb{C})$ over the complex field $\mathbb{C}$ and constructing a new representation $\mu :L_n⟶{\mathrm{GL}}_n(\mathbb{Z}[t^{\pm 1}])$, where $t$ is an indeterminate. For the representation $\mu$, 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 $\mu$. Finally, we examine extensions of $L_n$ representations to $V{L}_n$ and $W{L}_n$, 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 $V{L}_n$ and $W{L}_n$.