On the classification and irreducibility of $2$-local representations of the twin group $T_n$

Abstract

We investigate the homogeneous $2$-local representations of the twin group $T_n$ for all integers $n\geqslant 2$. A complete classification is obtained, yielding three distinct families of representations. We show that each of these families is reducible by explicitly constructing one-dimensional invariant subspaces, with particular emphasis on the first family, namely $\xi_1: T_n \rightarrow \text{GL}_n(\mathbb{C})$. Passing to the corresponding quotients, we construct a reduced representation of $\xi_1$, namely $\tilde{\xi}_1: T_n \rightarrow \text{GL}_{n-1}(\mathbb{C})$. The core of the paper is that we establish, through a precise criterion, a necessary and sufficient condition for the irreducibility of the representation $\tilde{\xi}_1$.