Algebra Seminar: Doty -- Integral Schur-Weyl duality for partition algebras

Steve Doty (Loyola University Chicago) 

Friday 2 March, 12-1pm, Place: Carslaw 375 

Title: Integral Schur-Weyl duality for partition algebras.  

Abstract: Partition algebras are finite-dimensional "diagram" algebras with a
combinatorial basis given by set partitions, with multiplication defined by stacking
diagrams.  They were (independently) discovered in the 1990s by Vaughan Jones and Paul
P.  Martin in connection with the Potts model in physics.  Partition algebras can be
regarded as generic centralizers of the natural action of the Weyl group W of GL(V) on
the tensor algebra of V, where V is a finite-dimensional complex vector space.  By
construction, partition algebras satisfy a Schur-Weyl duality with the group algebra of
W, at least over a field of characteristic zero, in which case the group algebra is
semisimple.  I will try to explain why Schur-Weyl duality still holds, even when the
underlying field is replaced by an arbitrary ring (of any characteristic).  In
particular, it holds over the integers.  This vastly extends a result of Peter M.
Gibson (1980) on generalised doubly-stochastic matrices.  The result is joint work with
Chris Bowman and Stuart Martin.