TY - JOUR
T1 - Families of nested graphs with compatible symmetric-group actions
AU - Ramos, Eric
AU - White, Graham
N1 - Publisher Copyright:
© 2019, Springer Nature Switzerland AG.
PY - 2019/12/1
Y1 - 2019/12/1
N2 - 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.
AB - 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.
KW - FI-modules
KW - Graph theory
KW - Kneser graphs
KW - Representation stability
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U2 - 10.1007/s00029-019-0520-9
DO - 10.1007/s00029-019-0520-9
M3 - Article
AN - SCOPUS:85074868666
SN - 1022-1824
VL - 25
JO - Selecta Mathematica, New Series
JF - Selecta Mathematica, New Series
IS - 5
M1 - 70
ER -