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Homotopy type of skeleta of the flag complex over a finite vector space and generalized Galois numbers

Abstract

Let \({\mathscr {F}}\) stand for the flag complex associated to the lattice of proper subspaces of a finite-dimensional vector space V. This paper aims at giving a (discrete) Morse theoretical proof of the fact that the k-th skeleton of \({\mathscr {F}}\) is homotopy equivalent to a wedge of spheres of dimension \(\min \{k,\dim ({\mathscr {F}})\}\). The tight control provided by Morse theoretic methods (through an explicit discrete gradient field) allows us to give a formula for the number of spheres appearing in each of these wedge summands. As an application, we derive an explicit formula for the number of flags on V of a given dimension, i.e., the number of simplices in \({\mathscr {F}}\) of the given dimension. Rather than depending on generalized Galois numbers, our formula for flags is given in terms of weighted inversion statistics of the symmetric group.

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Notes

  1. 1.

    We thank the anonymous referee of an earlier version of this paper for sharp observations that helped us clarify the relation of the discrete Morse theory methods with the shellability viewpoint. Actually, while addressing the referee’s comments, we came across the connection between (2) and (3).

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Correspondence to Jesús González.

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Jorge Aguilar-Guzmán and José Luis León-Medina were supported by Conacyt scholarships. Jesús González was supported by Conacyt research Grant CB-2013-01-221221.

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Aguilar-Guzmán, J., González, J. & León-Medina, J.L. Homotopy type of skeleta of the flag complex over a finite vector space and generalized Galois numbers. J Appl. and Comput. Topology 4, 181–198 (2020). https://doi.org/10.1007/s41468-020-00049-9

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Keywords

  • Flag complex
  • Discrete Morse theory
  • Acyclic pairing
  • Generalized Galois number

Mathematics Subject Classification

  • 06A07
  • 55P15
  • 57R70