## FI-modules and representation stability, I

Tom ChurchBenson Farb and I have just posted a new paper, “FI-modules:  a new approach to representation stability,” on the arXiv.  This paper has occupied a big chunk of our attention for about a year, so I’m very pleased to be able to release it!

Here is the gist.  Sometimes life hands you a sequence of vector spaces.  Sometimes these vector spaces even come with maps from one to the next.  And when you are very lucky, those maps become isomorphisms far enough along in the sequence; because at that point you can describe the entire picture with a finite amount of information, all the vector spaces after a certain point being canonically the same.  In this case we typically say we have found a stability result for the sequence.

But sometimes life is not so nice.  Say for instance we study the cohomology groups of configuration spaces of points of n distinct ordered points on some nice manifold M.  As one does.  In fact, let’s fix an index i and a coefficient field k and let V_n be the vector space H^i(Conf^n M, k.)

(In the imaginary world where there are people who memorize every word posted on this blog, those people would remember that I also sometimes use Conf^n M to refer to the space parametrizing unordered n-tuples of distinct points.  But now we are ordered.  This is important.)

For instance, you can let M be the complex plane, in which case we’re just computing the cohomology of the pure braid group.  Or, to put it another way, the cohomology of the hyperplane complement you get by deleting the hyperplanes (x_i-x_j) from C^n.

This cohomology was worked out in full by my emeritus colleagues Peter Orlik and Louis Solomon.  But let’s stick to something much easier; what about the H^1?  That’s just generated by the classes of the hyperplanes we cut out, which form a basis for the cohomology group.  And now you see a problem.  If V_n is H^1(Conf^n C, k), then the sequence {V_n} can’t be stable, because the dimensions of the spaces grow with n; to be precise,

dim V_n = (1/2)n(n-1).

But all isn’t lost.  As Tom and Benson explained last year in their much-discussed 2010 paper, “Representation stability and homological stability,” the right way to proceed is to think of V_n not as a mere vector space but as a representation of the symmetric group on n letters, which acts on Conf^n by permuting the n points.  And as representations, the V_n are in a very real sense all the same!  Each one is

“the representation of the symmetric group given by the action on unordered pairs of distinct letters.”

Of course one has to make precise what one means when one says “V_m and V_n are the same symmetric group representation”, when they are after all representations of different groups.  Church and Farb do exactly this, and show that in many examples (including the pure braid group) some naturally occuring sequences do satisfy their condition, which they call “representation stability.”

So what’s in the new paper?  In a sense, we start from the beginning, defining representation stability in a new way (or rather, defining a new thing and showing that it agrees with the Church-Farb definition in cases of interest.)  And this new definition makes everything much cleaner and dramatically expands the range of examples where we can prove stability.  This post is already a little long, so I think I’ll start a new one with a list of examples at the top.