Tag Archives: symmetric groups

FI-modules and representation stability, III

So how does this paper work?  The main idea is quite simple.  Let’s come back to the example of V_n = H^i(Conf^n M,Q), with i fixed and n ranging over nonnegative integers.  Then we have a sequence of vector spaces

V_0, V_1, V_2, …

But more than a sequence.  You have a map Conf^{n+1} -> Conf^n which is “forget the n+1 st point” — which functorially hands you a map V_n -> V_{n+1}.  So you have a diagram

V_0 -> V_1 -> V_2 -> …..

But in fact you have even more than this!  There’s no reason you have to forget just the n+1 st point.  You have tons of maps from Conf^n to Conf^m for all m <= n; one for each m-element subset of 1..n.  And there are lots of natural identifications between the compositions of these maps.  When you keep track of all the maps at your disposal, what you find is that the vector spaces V_n have a very rigid structure.

Definition:  FI is the category of finite sets with injections.  An FI-module over a ring R is a functor from FI to R-modules.

So V is an FI-module over Q!  (The vector space V_n is revealed as the image of the finite set [1..n] under the functor V.) And the main work of our paper is the study of the category of FI-modules, which sheds a great deal of light on representation stability.  For instance, we show that an FI-module over Q yields a representation-stable sequence in Church-Farb’s original sense if and only if it is finitely generated in the natural sense.  Moreover, the category of FI-modules over Q is Noetherian, in the sense that subobjects of finitely generated FI-modules are again finitely generation.  (The Noetherianness was proven independently by Snowden in a different form.)  Theorems like this very easy to show that tons of examples in nature (like the ones in the previous post) yield representation-stable sequences.  The work is all in the definitions and basic properties; once you have that, proving stability in particular examples is often a matter of a few lines.  For instance, you get a fairly instant proof of Murnaghan’s theorem on stability of Kronecker products; from this point of view, this becomes a theorem about the finite generation of a single object in an abelian category, rather than a theorem about a list of coefficients eventually setting down to constancy.

Sometimes there is more structure still.  Suppose, for example, that the manifold M above has nonempty boundary.  Then there are not only maps from Conf^{n+1} to Conf^n, but maps going the other way; you can add a new point in a little neighborhood of a boundary component.  (This is familiar from the configuration space of the complex plane, where you add new points at “the west pole” in the infinite negative real direction.)  These maps don’t quite compose on the nose, but they’re OK up to homotopy, and so the cohomology groups acquire a system of maps going both up and down.  It turns out that the right structure to describe such systems is given by the category of finite sets with partial injections; i.e. a map from A to B is an isomorphism from a subset of A to a subset of B.  We call this category FI#, and we call a functor from FI# to R-modules an FI#-module over R.

When your vector spaces carry an FI#-module structure you can really go to town.  It turns out that all the “eventuallies” disappear; when M is an open manifold, the dimension of H^i(Conf^n M) is a polynomial in n on the nose, for all n.  What’s more, if you want to show finite generation for FI#-modules, it suffices to show that dim V_n is bounded by some polynomial in n.  Once it’s less than a polynomial, it is a polynomial!  This stuff, unlike some other results in our paper, works in any characteristic and in fact is even fine with integral coefficients.

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