A 2-local representation of the twin group reduces to an (n-1)-dimensional one, and this reduced representation is irreducible exactly when a avoids 1, -1 and roots of an explicit polynomial.
To this end, we first present several propositions, lemmas, and theorems, which will lead us to the statement of our main result
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On the classification and irreducibility of $2$-local representations of the twin group $T_n$
A 2-local representation of the twin group reduces to an (n-1)-dimensional one, and this reduced representation is irreducible exactly when a avoids 1, -1 and roots of an explicit polynomial.