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Supersolvable hyperplane arrangements and matroids are known to give rise to certain Koszul algebras, namely their Orlik–Solomon algebras and graded Varchenko–Gel’fand algebras. We explore how this interacts with group actions, particularly for the braid arrangement and the action of the symmetric group, where the Hilbert functions of the algebras and their Koszul duals are given by Stirling numbers of the first and second kinds, respectively. The corresponding symmetric group representations exhibit branching rules that interpret Stirling number recurrences, which are shown to apply to all supersolvable arrangements. They also enjoy representation stability properties that follow from Koszul duality.
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Given a representation of a finite group G over some commutative base ring k, the cofixed space is the largest quotient of the representation on which the group acts trivially. If G acts by k-algebra automorphisms, then the cofixed space is a module over the ring of G-invariants. When the order of G is not invertible in the base ring, little is known about this module structure. We study the cofixed space in the case that G is the symmetric group on n letters acting on a polynomial ring by permuting its variables. When k has characteristic 0, the cofixed space is isomorphic to an ideal of the ring of symmetric polynomials in n variables. Localizing k at a prime integer p while letting n vary reveals striking behavior in these ideals. As n grows, the ideals stay stable in a sense, then jump in complexity each time n reaches a multiple of p.more » « less
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The augmented Bergman complex of a matroid is a simplicial complex introduced recently in work of Braden, Huh, Matherne, Proudfoot and Wang. It may be viewed as a hybrid of two well-studied pure shellable simplicial complexes associated to matroids: the independent set complex and Bergman complex. It is shown here that the augmented Bergman complex is also shellable, via two different families of shelling orders. Furthermore, comparing the description of its homotopy type induced from the two shellings re-interprets a known convolution formula counting bases of the matroid. The representation of the automorphism group of the matroid on the homology of the augmented Bergman complex turns out to have a surprisingly simple description. This last fact is generalized to closures beyond those coming from a matroid.more » « less
