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

We investigate the homogeneous $2$-local representations of the twin group $T_n$ for all integers $n⩾2$. 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 ${\xi }_1:T_n\to {\text{GL}}_n(\mathbb{C})$. Passing to the corresponding quotients, we construct a reduced representation of ${\xi }_1$, namely ${\tilde{\xi }}_1:T_n\to {\text{GL}}_{n-1}(\mathbb{C})$. The core of the paper is that we establish, through a precise criterion, a necessary and sufficient condition for the irreducibility of the representation ${\tilde{\xi }}_1$.