Equivariant Orlik-Solomon algebra of a matroid

Our research group is member of the DFG Special Priority Programme SPP 2458 “Combinatorial Synergies”.

Our project has two intertwined goals: Compute the graded parts of Orlik-Solomon algebras of geometric lattices together with its algebra structure and generalise the Lehrer-Solomon conjecture, possibly altering its statement, to other classes of arrangements and nonrepresentable matroids.

For the first goal, a major strategy will be the exploitation of equivariant structures under (subgroups of) the automorphism group G of the underlying lattice of the Orlik-Solomon algebra.

For the second goal, reasonable classes of arrangements that allow both for an inductive process and have nontrivial automorphism groups are the class of inductively free arrangements and the larger class of free arrangements. Among these classes are the class of Coxeter arrangements and the larger classes of ideal subarrangements of Weyl arrangements and of crystallographic arrangements, all of which include arrangements with relatively large automorphism groups.

The need to combine both strategies naturally leads us to the use of the category-theoretic language, both as a theoretical framework and as an algorithmic machinery.

The results will be made accessible via an online database.