Families of nested graphs with compatible symmetric-group actions

• Eric Ramos ^[1] ; Graham White ^[2]
  1. [1] University of Oregon, USA
  2. [2] Indiana University, USA
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 25, Nº. 5, 2019
• Idioma: inglés
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• Resumen

  • For fixed positive integers n and k, the Kneser graph KGn,k has vertices labeled by k-element subsets of {1,2,…,n} and edges between disjoint sets. Keeping k fixed and allowing n to grow, one obtains a family of nested graphs, each of which is acted on by a symmetric group in a way which is compatible with these inclusions and the inclusions of each symmetric group into the next. In this paper, we provide a framework for studying families of this kind using the FI-module theory of Church et al. (Duke Math J 164(9):1833–1910, 2015), and show that this theory has a variety of asymptotic consequences for such families of graphs. These consequences span a range of topics including enumeration, concerning counting occurrences of subgraphs, topology, concerning Hom-complexes and configuration spaces of the graphs, and algebra, concerning the changing behaviors in the graph spectra.