nLab FI-representation

Idea

In variation of how a symmetric group representation (or “$Sym(n)$-module”) may equivalently be thought of as a functor

• from the delooping groupoid $\mathbf{B}Sym(n)$

  or equivalently from the core groupoid of the full subcategory of $FinSet$ on the finite sets of cardinality $n$

• to Vect (or to $R$Mod for some ring $R$)

so an FI-representation or FI-modules is a functor to Vect (or $R$Mod) from the larger category $FinSet_{inj}$ of all finite sets with morphism all the injective maps between these.

So an FI-representation is, in particular, one $Sym(n)$-representation in the sense of ordinary representation theory, for all $n \in \mathbb{N}$, but in addition equipped with a system of intertwiners between these.

Properties

Relation to representation stability

One reason that FI-representation draw attention is that they exhibit the phenomenon called representation stability, and in fact the FI-representation theory thereby serves to “explain” the occurence of various stability phenomena seen the the study of moduli spaces, notably in the study of ordered configuration spaces of points.

References

Original articles on the notion of FI-modules:

• Thomas Church, Jordan S. Ellenberg, Benson Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 9 (2015) 1833-1910 [arXiv:1204.4533, doi:10.1215/00127094-3120274]

• Thomas Church, Jordan S. Ellenberg, Benson Farb, Rohit Nagpal, FI-modules over Noetherian rings, Geom. Topol. 18 (2014) 2951-2984 [arXiv:1210.1854, doi:10.2140/gt.2014.18.2951]

Further discussion in relation to representation stability:

• Trevor Hyde, $FI$-Modules and Representation Stability (2016) [pdf, pdf]

• Nir Gadish, Categories of FI type: a unified approach to generalizing representation stability and character polynomials, Journal of Algebra 480 (2017) 450-486 [arXiv:1608.02664, doi:10.1016/j.jalgebra.2017.03.010]

• Joe Moeller, Extensions of representation stable categories [arXiv:2209.03879]

Discussion of FI-representations in the generality of $\infty$-representations in stable $\infty$-categories and their analysis via Goodwillie calculus:

• Kaya Arro, FI-Calculus and Representation Stability (2023) [arXiv:2306.13597]