Let me share an announcement of a zoom lecture by Sheila Sundaram. The lecture (tomorrow, Sunday) is about the paper
Koszulity, supersolvability, and Stirling representations,
by Ayah Almousa, Victor Reiner, and Sheila Sundaram.
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Bar-Ilan Combinatorics Seminar
The next meeting will be, IYH, on Sunday 11 Iyar (May 19th), 18:05 pm
(18:05 refers to Israel time.)
*Note change to unusual time
on zoom
Speaker: Sheila Sundaram
Title: Stirling numbers, Koszul duality and cohomology of configuration spaces
Abstract: (Joint work with Ayah Almousa and Vic Reiner)
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.
You are welcome to attend
Combinatorics and more 