Partition algebras are finite-dimensional "diagram" algebras with a combinatorial basis given by set partitions, with multiplication defining 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. |