FI–sets with relations

Eric Ramos, David Speyer, Graham White

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Let FI denote the category whose objects are the sets [n] = {1, . . ., n}, and whose morphisms are injections. We study functors from the category FI into the category of finite sets. We write Sn for the symmetric group on [n]. Our first main result is that, if the functor [n] 7→ Xn is “finitely generated” there is a finite sequence of integers mi and a finite sequence of subgroups Hi of Smi such that, for n sufficiently large, Xn = Fi Sn/(Hi ×Sn−mi) as a set with Sn action. Our second main result is that, if [n] 7→ Xn and [n] 7→ Yn are two such finitely generated functors and Rn ⊂ Xn × Yn is an FI–invariant family of relations, then the (0, 1) matrices encoding the relation Rn, when written in an appropriate basis, vary polynomially with n. In particular, if Rn is an FI–invariant family of relations from Xn to itself, then the eigenvalues of this matrix are algebraic functions of n. As an application of this theorem we provide a proof of a result about eigenvalues of adjacency matrices claimed by the first and last author. This result recovers, for instance, that the adjacency matrices of the Kneser graphs have eigenvalues which are algebraic functions of n, while also expanding this result to a larger family of graphs.

Original languageEnglish
Pages (from-to)1079-1098
Number of pages20
JournalAlgebraic Combinatorics
Volume3
Issue number5
DOIs
StatePublished - 2020

Keywords

  • FI-modules
  • Kneser graphs
  • Representation Stability

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