FI-modules and representation stability, II

Here are some sequences of vector spaces.  In each case, the sequence is indexed by n, and all other variables are understood to be constant.  So suppose V_n is the space

• H^i(Conf^n M, Q) for M a connected oriented manifold of dimension at least 2.
• The (j_1, .. j_r)-multidegree piece of the diagonal coinvariant algebra on r sets of n variables.
• H^i(M_{g,n},Q), the cohomology of the moduli space of curves of genus g with n marked points.
• The tautological subring of the above.
• The space of degree-d polynomials on the rank variety parametrizing nxn matrices of rank at most r.

By a character polynomial we mean a polynomial with integral coefficients in variables X_1, X_2, X_3, … .  We interpret these symbols (and thus character polynomials) as class functions on the symmetric group by S_n by taking

X_i(s) = number of i-cycles in s

for each permutation s.

Then we show that, in each of the examples above, there’s a character polynomial P such that the character of the action of S_n on V_n is given by P, for all sufficiently large n.  This is one way in which one can say that a sequence of representations of larger and larger symmetric groups are “all the same.”  In particular, by plugging in the identity we find that dim V_n is a polynomial in n, for n large enough.

For many of these examples, almost nothing is known about dimensions of individual spaces!  So a strong regularity theorem like this is perhaps surprising.  Even more surprising (to us at any rate) is that theorems like this require only very meager input from whatever context generate the vector spaces.  You get this stability (and many others) almost for free.

More about how it all works tomorrow!