Topology of augmented Bergman complexes

Abstract: (based on arxiv:2108.13394; joint with REU students E. Bullock, A. Kelley, K. Ren, G. Shemy, D. Shen, B. Sun, A. Tao, Z. Zhang)

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 the order complex of the lattice of flats.

After recalling the relevance of the augmented Bergman complex in the B-H-M-P-W work, we show that it is shellable, via two different families of shelling orders. Both shellings determine the homotopy type, and comparing the two answers re-interprets a known convolution formula counting bases of the matroid. One of the shellings leads to a surprisingly simple description for how symmetries of the matroid act on the homology of the complex.

computational geometrydiscrete mathematicscommutative algebracombinatorics

Audience: researchers in the topic

Copenhagen-Jerusalem Combinatorics Seminar

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