Character Polynomials

For j ≥ 1, let Xj : Ss→ N be the class function defined by

Xj(σ) = number of j-cycles in σ.

A polynomial in the variables Xj is called a character polynomial. We define the degree

of a character polynomial by setting deg(Xj) = j. The following theorem of Church-

Ellenberg-Farb says that characters of finitely generated FI-modules are eventually described by a single character polynomial, and moreover gives explicit bounds on the degree and the stable range of this polynomial in terms of weight and stability degree of the FI-module.

Theorem 2.3.30 ([7], Theorem 3.3.4). Let V be a finitely generated FI-module of weight ≤ d and stability degree ≤ t. There exists a unique polynomial fV ∈ Q[X1, . . . , Xd]

of degree at most d such that for all n ≥ d + t and all σ ∈ Sn,

ON THE FI-MODULE STRUCTURE OF H (Γn,s)

It is well known that the group of outer automorphisms of the free group of rank n can be described as the space of self-homotopy equivalences of a graph Xn of rank n,

up to homotopy, i.e.,

Out(Fn) ∼= π0(HE(Xn)).

Similarly the full group of automorphisms of the free group of rank n is the space of homotopy equivalences of a graph Xn,1 of rank n with a distinguished basepoint ∂, up

to homotopy,

Aut(Fn) ∼= π0(HE(Xn,1)),

where homotopies are required to fix the basepoint throughout.

X3 X3,1

∂ •

Figure 3.1: Examples of rank 3 graphs that can be used to define Out(F3) and Aut(F3).

There is a natural generalisation then, where we let Xn,s be a graph, by which we

mean a connected finite 1-dimensional CW-complex, of rank n with s marked points ∂ = {x1, · · · , xs}. We should then consider the group of self-homotopy equivalences of

Xn,s fixing ∂ pointwise, modulo homotopies through such maps, i.e.,

Γn,s := π0(HE(Xn,s)).

3.1

The cohomology of Γ

In this section we study the structure of the cohomology Hi_(Γ

n,s), always over a field

of characteristic zero, as a sequence of Ss-modules. The symmetric group Ss acts on

Hi(Γn,s) as follows. A homotopy equivalence h : Xn,s → Xn,s permuting ∂ induces an

automorphism of Γn,s by conjugation. This automorphism depends, a priori, on the

choice of h, however, on the level of cohomology it depends only on the permutation. Indeed, if h fixes ∂ pointwise then the induced automorphism is inner, and thus induces the identity on cohomology.

The groups Γn,s have been used, for example, to show that Out(Fn) and Aut(Fn)

satisfy homological stability in [16, 17], and they appeared in [3] in the proof that Out(Fn) is a virtual duality group. More recently they were used in [10] to investigate

the so called unstable cohomology of Out(Fn) and Aut(Fn) by means of an ‘assembly

map’ Hi(Γn1,s1) ⊗ · · · ⊗ H i_(Γ nk,sk) → H i_(Γ n,s).

In particular, in [10] they compute Hi(Γn,s) as an Ss-module for rank n = 1, 2 and

use these computations to assemble homology classes in the unstable range of Out(Fn)

and Aut(Fn). Moreover these computations show that, in rank n = 1, 2 and for fixed

i ≥ 0 the sequence {Hi_(Γ

n,s)}s≥0satisfies representation stability (see Definition 2.3.7).

In [10] they use an alternate description of Γn,s as a quotient of a certain mapping class

group of a three-manifold, together with general results about representation stability of mapping class groups, to deduce that for any fixed i and n the groups Hi_(Γ

n,s)

satisfy representation stability with stable range s ≥ 3i. However, the calculations made in [10] in rank n = 1, 2 actually adhere to a bound of s ≥ i + n. In this section we improve the stable range to agree with these low rank calculations.

Theorem 3.1.1. For fixed i and n, the sequence, {Hi_(Γ

n,s) : s ∈ N},

is uniformly representation stable with stable range s ≥ n + i. We show this by exhibiting that Hi_(Γ

n,s) defines an FI-module. Building on their

work in [9], and together with Jordan Ellenberg and Rohit Nagpal, the theory of FI- modules was developed [7, 8], facilitating the application of homological techniques to sequences of Ss-modules. We use these techniques to prove the following theorem.

Theorem 3.1.2. The FI-module Hi_(Γ

n,•) is finitely generated of stability degree n and

weight i.

An important feature of finitely generated FI-modules is the existence of character polynomials; integer-valued polynomials in Q[X1, X2, . . .] where Xi : Ss → N is the

class function that counts the number of i-cycles. Let χ_Hi_(Γ

n,s) denote the character of

the Ss-module Hi(Γn,s).

Corollary 3.1.3. There exists a character polynomial f ∈ Q[X1, . . . , Xi] depending on

i and n such that for all s ≥ i + n and all σ ∈ Ss,

χHi_(Γ

n,s)(σ) = f (σ).

In particular, the dimension of Hi(Γn,s) is given by the polynomial f (s, 0, . . . , 0).

One consequence of this result is that, for s sufficiently large, the character χi,n is

insensitive to cycles of length greater than i. We highlight this phenomenon by com- puting examples of these stable character polynomials in Section 2.3.7.

Theorem 3.1.1 and Corollary 3.1.3 follow immediately from Theorem 3.1.2 in light of Proposition 2.3.13 and Proposition 2.3.30.

Recall, for P an Sa-module, and Q an Sb-module, we denote the induced represen-

tation by

P ~ Q := IndSa+b

Sa× SbP ⊗ Q

We denote by V∧kthe Sk-module which is isomorphic as a vector space to V⊗k where

Sk acts by permuting the factors and multiplying by the sign of the permutation. That

is,

V∧k = V⊗k ⊗ k.

With this in hand it is clear that Theorem 3.1.1 is an immediate corollary to The- orem 3.1.2. Another consequence of Theorem 3.1.2 is the existence of stable character polynomials. Corollary 3.1.3 will thus follow immediately from Theorem B. It is worth pointing out that in particular, this shows that the dimension of Hi_(Γ

n,s) is eventually

polynomial (as s grows but i and n remain fixed) given by a single character polynomial.

In [10] Conant-Hatcher-Kassabov-Vogtmann describe the Ss-module structure of

Hi_(Γ

n,s) for n = 1, 2, from which one can read off their irreducible Ss-module decom-

position, that is, a decomposition into terms of the form P (λ)s. We have the following

classical fact, which underpins the theorem above. Fix a partition λ. There exists a unique character polynomial fλ such that for any s ≥ |λ| + λ1, the character of the

Ss-module P (λ)s is given by fλ. In [14] they describe an algorithm constructing fλ

that we will use in conjunction with calculations from [10] to compute some explicit examples of character polynomials of various Hi_(Γ

n,s). It is cleanest to describe these

Example. Fix n = 1 and i = 2. From [10], Proposition 2.7 we obtain the following decomposition of H2_(Γ

1,s) into irreducible Ss-modules.

H2(Γ1,s) = P

 

s

.

Using the algorithm from [14] we obtain the character polynomial f for P
. Corollary 3.1.3 implies that, for s ≥ 3, the character χ2,1 of H2(Γ1,s) is given by the

polynomial,

f2,1(X1, X2) = f (X1, X2) =

1

2 · (X1)2− (X1) − (X2) + 1.

We can use this, for example, to obtain that for s ≥ 3 the dimension of H2(Γ1,s) is

s(s − 1) 2 − s + 1 = s − 1 2  . Notice that this agrees with the description of H2_(Γ

1,s) =

V2_ks−1 _{given in [10].}

Example. Fix n = 2 and i = 4. From [10], Theorem 2.10 we obtain the following stable decomposition of H4_(Γ

2,s) into irreducible Ss-modules. For s ≥ 6,

H4(Γ2,s) = P ( )_s⊕ P   s ⊕ P   s .

Using the algorithm from [14] we obtain the character polynomials f , f and f . Corollary 3.1.3 implies that, for s ≥ 6, the character χ4,2 of H4(Γ2,s) is given by the

sum of these three character polynomials, f4,2(X1, X2, X3, X4) =

1

12(X1)4+ (X2)2− X1· X3. For instance, let τ = (1 2)(3 4)(5 6 · · · 100) ∈ S100. Then χ4,2(τ ) = 2.

Both the stable decomposition of Hi_(Γ

n,s) and the stable character polynomials de-

FI-module. It thus remains to prove Theorem 3.1.2, which we will do by analysing a spectral sequence of FI-modules. It turns out that the E2-page of that spectral sequence

admits a particularly nice description in terms of free FI-modules.