Geodesic
Topology of augmented Bergman complexes
Elisabeth Bullock1 ; Aidan Kelley2 ; Victor Reiner3 ; Kevin Ren1 ; Gahl Shemy4 ; Dawei Shen2 ; Brian Sun5 ; Zhichun Joy Zhang6
1MIT
2Washington University in St. Louis
3University of Minnesota
4University of California - Santa Barbara
5Yale University
6Swarthmore College
The electronic journal of combinatorics, Tome 29 (2022) no. 1
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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.
DOI : 10.37236/10739
Classification : 05E45, 05B35
Mots-clés : shellable simplicial complexes, independent set complex, Bergman complex

Elisabeth Bullock  1   ; Aidan Kelley  2   ; Victor Reiner  3   ; Kevin Ren  1   ; Gahl Shemy  4   ; Dawei Shen  2   ; Brian Sun  5   ; Zhichun Joy Zhang  6

1 MIT
2 Washington University in St. Louis
3 University of Minnesota
4 University of California - Santa Barbara
5 Yale University
6 Swarthmore College 
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     title = {Topology of augmented {Bergman} complexes},
     journal = {The electronic journal of combinatorics},
     year = {2022},
     volume = {29},
     number = {1},
     doi = {10.37236/10739},
     zbl = {1486.05328},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/10739/}
}
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Elisabeth Bullock; Aidan Kelley; Victor Reiner; Kevin Ren; Gahl Shemy; Dawei  Shen; Brian Sun; Zhichun Joy Zhang. Topology of augmented Bergman complexes. The electronic journal of combinatorics, Tome 29 (2022) no. 1. doi: 10.37236/10739

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