## 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.

## The braid group, analytic number theory, and Weil’s three columns

This post is about a new paper of mine with Akshay Venkatesh and Craig Westerland; but I’m not going to mention that paper in the post. Instead, I want to explain why topological theorems about the stable homology of moduli spaces are relevant to analytic number theory.  If you’ve seen me give a talk about this stuff, you’ve probably heard this spiel before.

“The mathematician who studies these problems has the impression of deciphering a trilingual inscription. In the first column one finds the classical Riemannian theory of algebraic functions. The third column is the arithmetic theory of algebraic numbers.  The column in the middle is the most recently discovered one; it consists of the theory of algebraic functions over finite fields. These texts are the only source of knowledge about the languages in which they are written; in each column, we understand only fragments.”

Let’s see how a classical question of analytic number theory works in Weil’s three languages.  Start with the integers, and ask:  how many of the integers between X and 2X are squarefree?  This is easy:  we have an asymptotic answer of the form

$\frac{6}{\pi^2}X + O(X^{1/2}) = \zeta(2)^{-1} X + O(X^{1/2}).$

(In fact, the best known error term is on order X^{17/54}, and the correct error term is conjectured to be X^{1/4}; see Pappalardi’s “Survey on k-freeness” for more on such questions.)

So far so good.  Now let’s apply the popular analogy between number fields and function fields, going over to Weil’s column 3, and ask: what’s the analogous statement when Z is replaced by F_q[T]?

## Random pro-p groups, braid groups, and random tame Galois groups

I’ve posted a new paper with Nigel Boston, “Random pro-p groups, braid groups, and random tame Galois groups.”

The paper proposes a kind of “non-abelian Cohen-Lenstra heuristic.”   A typical prediction:  if S is a randomly chosen pair of primes, each of which is congruent to 5 mod 8, and G_S(p) is the Galois group of the maximal pro-2 extension of Q unramified away from S, then G_S(p) is infinite 1/16 of the time.

The usual Cohen-Lenstra conjectures — well, there are a lot of them, but the simplest one asks:  given an odd prime p and a finite abelian p-group A, what is the probability P(A) that a randomly chosen quadratic imaginary field K has a class group whose p-primary part is isomorphic to A?  (Note that the existence of P(A) — which we take to be a limit in X of the corresponding probability as K ranges over quadratic imaginary fields of discriminant at most X — is not at all obvious, and in fact is not known for any p!)

Cohen and Lenstra offered a beautiful conjectural answer to that question:  they suggested that the p-parts of class groups were uniformly distributed among finite abelian p-groups.  And remember — that means that P(A) should be proportional to 1/|Aut(A)|.  (See the end of this post for more on uniform distribution in this categorical setting.)

Later, Friedman and Washington observed that the Cohen-Lenstra conjectures could be arrived at by another means:  if you take K to be the function field of a random hyperelliptic curve X over a finite field instead of a random quadratic imaginary field, then the finite abelian p-group you’re after is just the cokernel of F-1, where F is the matrix corresponding to the action of Frobenius on T_p Jac(X).  If you take the view that F should be a “random” matrix, then you are led to the following question:

Let F be a random element of GL_N(Z_p) in Haar measure:  what is the probability that coker(F-1) is isomorphic to A?

And this probability, it turns out, is precisely the P(A) conjectured by Cohen-Lenstra.

(But now you cry out:  but Frobenius isn’t just any old matrix!  It’s in the generalized symplectic group!  Yes — and Jeff Achter has shown that, at least as far as the probability distribution on A/pA goes, the “right” random matrix model gives you the same answer as the Friedman-Washington quick and dirty model.  Phew.)

Now, in place of a random quadratic imaginary field, pick a prime p and a random set S of g primes, each of which is 1 mod p.  As above, let G_S(p) be the Galois group of the maximal pro-p extension of Q unramified away from S; this is a pro-p group of rank g. What can we say about the probability distribution on G_S(p)?  That is, if G is some pro-p group, can we compute the probability that G_S(p) is isomorphic to G?

Again, there are two approaches.  We could ask that G_S(p) be a “random pro-p group of rank g.”  But this isn’t quite right; G_S(p) has extra structure, imparted to it by the images in G_S(p) of tame inertia at the primes of S.  We define a notion of “pro-p group with inertia data,” and for each pro-p GWID G we guess that the probability that G_S(p) = G is proportional to 1/Aut(G); where Aut(G) refers to the automorphisms of G as GWID, of course.

On the other hand, you could ask what would happen in the function field case if the action of Frobenius on — well, not the Tate module of the Jacobian anymore, but the full pro-p geometric fundamental group of the curve — is “as random as possible.”  (In this case, the group from which Frobenius is drawn isn’t a p-adic symplectic group but Ihara’s “pro-p braid group.”)

And the happy conclusion is that, just as in the Cohen-Lenstra setting, these two heuristic arguments yield the same prediction.  For the relatively few pro-p groups G such that we can compute Pr(G_S(p) = G), our heuristic gives the right answer.  For several more, it gives an answer that seems consistent with numerical experiments.

Maybe it’s correct!

## Koberda on dilatation and finite nilpotent covers

One reason dilatation was on my mind was thanks to a very interesting recent paper by Thomas Koberda, a Ph.D. student of Curt McMullen at Harvard.

Recall from the previous post that if f is a pseudo-Anosov mapping class on a surface Σ, there is an invariant λ of f called the dilatation, which measures the “complexity” of f; it is a real algebraic number greater than 1.  By the spectral radius of f we mean the largest absolute value of an eigenvalue of the linear automorphism of $H_1(\Sigma,\mathbf{R})$ induced by f.  Then the spectral radius of f is a lower bound for λ(f), and in fact so is the spectral radius of f on any finite etale cover of Σ preserved by f.

This naturally leads to the following question, which appears as Question 1.2 in Koberda’s paper:

Is λ(f) the supremum of the spectral radii of f on Σ’, as Σ’ ranges over finite etale covers of Σ preserved by f?

It’s easiest to think about variation in spectral radius when Σ’ ranges over abelian covers.  In this case, it turns out that the spectral radii are very far from determining the dilatation.  When Σ is a punctured sphere, for instance, a remark in a paper of Band and Boyland implies that the supremum of the spectral radii over finite abelian covers is strictly smaller than λ(f), except for the rare cases where the dilatation is realized on the double cover branched at the punctures.   It gets worse:  there are pseudo-Anosov mapping classes which act trivially on the homology of every finite abelian cover of Σ, so that the supremum can be 1!  (For punctured spheres, this is equivalent to the statement that the Burau representation isn’t faithful.)  Koberda shows that this unpleasant state of affairs is remedied by passing to a slightly larger class of finite covers:

Theorem (Koberda) If f is a pseudo-Anosov mapping class, there is a finite nilpotent etale cover of Σ preserved by f on whose homology f acts nontrivially.

Furthermore, Koberda gets a very nice purely homological version of the Nielsen-Thurston classification of diffeomorphisms (his Theorem 1.4,) and dares to ask whether the dilatation might actually be the supremum of the spectral radius over nilpotent covers.  I have to admit I would find that pretty surprising!  But I don’t have a good reason for that feeling.

## F_1 and the braid group — a note of skepticism

Since I wrote this post, I’ve become less sure about this assertion that the braid group can be thought of as GL_n(F_1[t]). Here are three reasons to be doubtful:

• As Jim points out in commments, GL_n(F_q) embeds in GL_n(F_q[t]), but S_n doesn’t embed in the braid group. This has to be counted against the braid group, I think. Jim also says that in his version of F_1 geometry, which comes out of lambda-rings, GL_n(F_1[t]) is just S_n.
• Terry Tao observed that one might expect GL_n(F_1[t]) to embed into GL_n(F_q[t]), just as GL_n(F_1) embeds into GL_n(F_q). But this doesn’t appear to be the case, at least not for any obvious reason. Keep in mind, it was quite hard to prove that the braid group had any faithful linear representations at all! The n-dimensional linear representations developed by Lawrence and Krammer, and proven faithful by Bigelow (and then again by Krammer) have coefficients in Z[t,1/t,u,1/u]. So the idea that one might find the braid group inside GL(F_1[t,1/t,u,1/u]) remains, from this point of view, alive! But I wonder whether Jim thinks this latter group is also S_n…?
• Finally, the argument given by Kapranov and Smirnov looks like it’s making a case that the braid group admits a map to GL_n(F_1[[t]]), not so much that it should be thought of as GL_n(F_1[t]).

GL_n(F_1[[t]]), by the way, seems a little easier to get our hands on. Note that the order of the finite group GL_n(F_q[t]/t^k) is just a power of q times |GL_n(F_q)|. So, setting q = 1, one might expect

|GL_n(F_1[t]/t^k)| = |GL_n(F_1)| = n!

and in particular

GL_n(F_1[[t]]) = GL_n(F_1[t]/t^k) = S_n.

(Short version of this argument: “Pro-1 groups are trivial.”) In this case, the braid group certainly does map to GL_n(F_1[[t]])!

By the argument in the previous post, one would then want to say GL_n(F_1((t))) is the affine Weyl group Z^{n-1} semidirect S_n. Which means that the Hecke algebra

GL_n(F_1[[t]]) \ GL_n(F_1((t))) / GL_n(F_1[[t]])

is a pretty standard object — the double cosets above are in bijection with S_n-orbits on Z^{n-1}, which can be identified with the symmetrized monomials in n variables whose degree is a multiple of n, up to multiplication by x_1, … x_n. So the Hecke algebra should just be some version of the algebra of symmetric functions on n variables.

Believers that the braid group is GL_n(F_1[t]) are strongly encouraged to revivify my faith in comments.

## F_1, buildings, the braid group, GL_n(F_1[t,1/t])

It used to be you had to talk about “the field with one element” very quietly, and only among people whose opinion of you was secure. The reason, of course, is that there is no field with one element. Which doesn’t stop people from saying “But if there _were_ a field with one element, what would it be like?”

Nowadays all kinds of people are musing about this odd question in the bright light of day, and no one finds it kooky. John Baez covered the basics in a 2007 issue of This Week’s Finds. And as of a few weeks ago the field with one element has its own blog, “Ceci N’est Pas Un Corps.”

From a recent post on CNPUC, I learned the interesting fact that the braid group on n strands can be thought of as GL_n(F_1[t]).

So here’s a question: what is GL_n(F_1[t,1/t])?