FI–sets with relations
Algebraic Combinatorics, Volume 3 (2020) no. 5, pp. 1079-1098.

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 𝔖 n for the symmetric group on [n]. Our first main result is that, if the functor [n]X n is “finitely generated” there is a finite sequence of integers m i and a finite sequence of subgroups H i of 𝔖 m i such that, for n sufficiently large, X n i 𝔖 n /(H i ×𝔖 n-m i ) as a set with 𝔖 n action. Our second main result is that, if [n]X n and [n]Y n are two such finitely generated functors and R n X n ×Y n is an FI–invariant family of relations, then the (0,1) matrices encoding the relation R n , when written in an appropriate basis, vary polynomially with n. In particular, if R n is an FI–invariant family of relations from X n 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.

Published online:
DOI: 10.5802/alco.128
Classification: 05E18, 18A25, 05C25, 05C75
Keywords: FI-modules, Representation Stability, Kneser graphs.
Ramos, Eric 1; Speyer, David 2; White, Graham 3

1 Department of Mathematics University of Oregon Fenton Hall, Eugene, OR 97401, USA
2 Department of Mathematics University of Michigan 530 Church St., Ann Arbor, MI 48109, USA
3 Department of Mathematics Indiana University - Bloomington Rawles Hall, Bloomington, IN 47405, USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Ramos, Eric; Speyer, David; White, Graham. FI–sets with relations. Algebraic Combinatorics, Volume 3 (2020) no. 5, pp. 1079-1098. doi : 10.5802/alco.128.

[1] Brouwer, Andries E.; Cohen, Arjeh M.; Neumaier, Arnold Distance-regular graphs, Springer-Verlag, 1989 | Zbl

[2] Church, Thomas; Ellenberg, Jordan S.; Farb, Benson FI-modules and stability for representations of symmetric groups, Duke Math. J., Volume 164 (2015) no. 9, pp. 1833-1910 | DOI | MR | Zbl

[3] Church, Thomas; Ellenberg, Jordan S.; Farb, Benson; Nagpal, Rohit FI-modules over Noetherian rings, Geom. Topol., Volume 18 (2014) no. 5, pp. 2951-2984 | DOI | MR | Zbl

[4] Church, Thomas; Farb, Benson Representation theory and homological stability, Adv. Math., Volume 245 (2013), pp. 250-314 | DOI | MR | Zbl

[5] Djament, Aurélien Des propriétés de finitude des foncteurs polynomiaux, Fund. Math., Volume 233 (2016) no. 3, pp. 197-256 | DOI | MR | Zbl

[6] Djament, Aurélien; Vespa, Christine Foncteurs faiblement polynomiaux, Int. Math. Res. Not. IMRN (2019) no. 2, pp. 321-391 | DOI | MR | Zbl

[7] Gadish, Nir Representation stability for families of linear subspace arrangements, Adv. Math., Volume 322 (2017), pp. 341-377 | DOI | MR | Zbl

[8] Godsil, Chris; Royle, Gordon F. Algebraic graph theory, Graduate Texts in Mathematics, 207, Springer Science & Business Media, 2001 | MR | Zbl

[9] Nagpal, Rohit FI-modules and the cohomology of modular representations of symmetric groups (2015) ( | MR

[10] Putman, Andrew Stability in the homology of congruence subgroups, Invent. Math., Volume 202 (2015) no. 3, pp. 987-1027 | DOI | MR | Zbl

[11] Ramos, Eric Homological invariants of FI-modules and FI G -modules, J. Algebra, Volume 502 (2018), pp. 163-195 | DOI | MR | Zbl

[12] Ramos, Eric An application of the theory of FI-algebras to graph configuration spaces, Math. Z., Volume 294 (2020) no. 1-2, pp. 1-15 | DOI | MR | Zbl

[13] Ramos, Eric; White, Graham Families of Markov chains with compatible symmetric-group actions (2018) (

[14] Ramos, Eric; White, Graham Families of nested graphs with compatible symmetric-group actions, Selecta Math. (N.S.), Volume 25 (2019) no. 5, Paper no. Paper No. 70, 42 pages | DOI | MR | Zbl

[15] Sam, Steven V.; Snowden, Andrew GL-equivariant modules over polynomial rings in infinitely many variables, Trans. Amer. Math. Soc., Volume 368 (2016) no. 2, pp. 1097-1158 | DOI | MR | Zbl

[16] Snowden, Andrew Syzygies of Segre embeddings and Δ-modules, Duke Math. J., Volume 162 (2013) no. 2, pp. 225-277 | DOI | MR | Zbl

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