On The Irreducibility Of The Extension Of The Lawrence-Bigelow-Krammer Representation To The Virtual Braid Group v B 3

We consider the representation [Formula: see text] which extends the Lawrence-Bigelow-Krammer representation from the braid group [Formula: see text] to the virtual braid group [Formula: see text] on 3 strands. After specializing parameters [Formula: see text], [Formula: see text] to non-zero complex numbers, we study the irreducibility of [Formula: see text]. We prove that [Formula: see text] is irreducible for all non-zero complex numbers [Formula: see text], [Formula: see text] except for the case [Formula: see text].