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Kevin Piterman: Posets associated to vector spaces with non-degenerate forms

Time: Wed 2023-04-05 10.15 - 11.15

Location: KTH 3721

Participating: Kevin Piterman (Marburg University)

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Abstract: Given a finite vector space with a non-degenerate form, the frame complex is the simplicial complex whose simplices are the sets of pairwise orthogonal non-degenerate 1-dimensional subspaces. We will show how to explicitly compute the eigenvalues of the 1-skeleton of the frame complex in the Hermitian case, and then conclude vanishing results on the homology of these complexes by applying Garland's method. It is of particular interest to us to prove that the "top-1" degree homology group of the frame complex is nonzero. However, this is still an open problem. We will also study the Cohen-Macaulay property and show that this fails if the dimension is big enough with respect to the size of the ground field. Finally, we will address similar questions on the poset of orthogonal decompositions and the poset of non-degenerate subspaces. This is a joint work with Volkmar Welker.