Schur-Weyl duality for twin groups (Q6594716)

Let \(\mathrm{TW}_{n}=\big \langle t_{1}, \ldots t_{n} \; \big | \; t_{i}^{2}, [t_{i},t_{j}] \mbox{ if } | i-j | >0 \big \rangle\) be twin group on \(n\) strands.\N\NThe authors find a new instance of semisimple Schur-Weyl duality for tensor powers of a natural \(n\)-dimensional reflection representation of \(\mathrm{TW}_{n}\), depending on a parameter \(q\). At \(q=1\), the representation coincides with the natural permutation representation of the symmetric group, so the new Schur-Weyl duality may be regarded as a \(q\)-analogue of the one motivating the definition of the partition algebra.