Schur–Weyl Duality for Twin Groups

The twin group TWn on n strands is the generated by $t_{1}, \dots , t_{n-1}$ with defining relations ${t_{i}^{2}}=1$ titj = tjti if |i − j| > 1. We find a new instance of semisimple Schur–Weyl duality for tensor powers natural n-dimensional reflection representation TWn, depending parameter q. At q 1, coincides permutation symmetric group, so may be regarded as q-analogue one motivating definition partition algebra.