Schur--Weyl duality for twin groups

TL;DR

This paper establishes a new semisimple Schur--Weyl duality for tensor powers of a reflection representation of twin groups, generalizing classical symmetric group duality with a parameter $q$.

Contribution

It introduces a $q$-analogue of Schur--Weyl duality for twin groups, expanding the understanding of their representation theory and connections to partition algebras.

Findings

A new semisimple Schur--Weyl duality for twin groups

Identification of a $q$-parameterized reflection representation

Connection to classical symmetric group duality at $q=1$

Abstract

The twin group $TW_n$ on $n$ strands is the group generated by $t_1, \dots, t_{n-1}$ with defining relations $t_i^2=1$, $t_it_j = t_jt_i$ if $|i-j|>1$. We find a new instance of semisimple Schur--Weyl duality for tensor powers of a natural $n$-dimensional reflection representation of $TW_n$, 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.