Stirling numbers, Koszul duality and cohomology of configuration spaces

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The unsigned Stirling numbers c(n,k) of the first kind give the Hilbert function for two algebras
associated to the hyperplane arrangement in type A, the Orlik-Solomon algebra and the graded Varchenko-Gelfand
algebra. Both algebras carry symmetric group actions with a rich structure, and
have been well studied by topologists, algebraists and combinatorialists: the first coincides with the Whitney homology
of the partition lattice, and the second with a well-known decomposition (Thrall's decomposition,
giving the higher Lie characters) of the universal enveloping algebra of the free Lie algebra.
In each case the graded representations have dimension c(n,k).

Both these algebras are examples of Koszul algebras, for which the Priddy resolution defines a group-equivariant dual Koszul algebra.
Now the Hilbert function is given by the Stirling numbers S(n,k) of the second kind, and hence the Koszul duality relation defines representations 

of the symmetric group whose dimensions are the numbers S(n,k).
Investigating this observation leads to the realisation that this situation generalises to all supersolvable matroids.

The Koszul duality recurrence is shown to have interesting general consequences.

For the resulting group representations, it implies the existence of branching rules which, in the case of the braid
arrangement, specialise by dimension to the classical enumerative recurrences satisfied by the Stirling numbers of both kinds.
It also implies representation stability in the sense of Church and Farb.

The associated Koszul dual representations appear to have other properties that are more mysterious;
for example, in the case of the braid arrangement, the Stirling representations of the Koszul dual
are sometimes tantalisingly close to being permutation modules.

We also show a connection with the homology of the subword order poset, from previous work of the speaker, where Stirling numbers
arise again in the representations.