Schur--Weyl duality for twin groups
Stephen Doty, Anthony Giaquinto

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 on strands is the group generated by with defining relations , if . We find a new instance of semisimple Schur--Weyl duality for tensor powers of a natural -dimensional reflection representation of , depending on a parameter . At the representation coincides with the natural permutation representation of the symmetric group, so the new Schur--Weyl duality may be regarded as a -analogue of the one motivating the definition of the partition algebra.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Algebraic structures and combinatorial models · Advanced Combinatorial Mathematics
