nLab FI-representation




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

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

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


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.


Original articles on the notion of FI-modules:

Further discussion in relation to representation stability:

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

Last revised on September 13, 2023 at 16:54:53. See the history of this page for a list of all contributions to it.