Schur--Weyl duality for twin groups

2021/05/26 by Doty, Stephen, Giaquinto, Anthony
#05E10 #16D20 #20C30 #20G05 #20G42 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2105.12875

Abstract

The twin group TW_n on n strands is the group generated by t₁, …, t_n-1 with defining relations t_i²=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.

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