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## Meditation on the Sylow theorems II

In Part I we discussed some conceptual proofs of the Sylow theorems. Two of those proofs involve reducing the existence of Sylow subgroups to the existence of Sylow subgroups of $S_n$ and $GL_n(\mathbb{F}_p)$ respectively. The goal of this post is to understand the Sylow $p$-subgroups of $GL_n(\mathbb{F}_p)$ in more detail and see what we can learn from them about Sylow subgroups in general.

## Drawing subgroups of the modular group

Previously we learned how to count the finite index subgroups of the modular group $\Gamma = PSL_2(\mathbb{Z})$. The worst thing about that post was that it didn’t include any pictures of these subgroups. Today we’ll fix that.

The pictures in this post can be interpreted in at least two ways. On the one hand, they are graphs of groups in the sense of Bass-Serre theory, and on the other hand, they are also dessin d’enfants (for the rest of this post abbreviated to “dessins”) in the sense of Grothendieck. But you don’t need to know that to draw and appreciate them.

## Conjugacy classes of finite index subgroups

Previously we learned how to count the number of finite index subgroups of a finitely generated group $G$. But for various purposes we might instead want to count conjugacy classes of finite index subgroups, e.g. if we wanted to count isomorphism classes of connected covers of a connected space with fundamental group $Gi$.

There is also a generating function we can write down that addresses this question, although it gives the answer less directly. It can be derived starting from the following construction. If $X$ is a groupoid, then $LX = [S^1, LX]$, the free loop space or inertia groupoid of $X$, is the groupoid of maps $S^1 \to X$, where $S^1$ is the groupoid $B\mathbb{Z}$ with one object and automorphism group $\mathbb{Z}$. Explicitly, this groupoid has

• objects given by automorphisms $f : x \to x$ of the objects $x \in X$, and
• morphisms $(f_1 : x_1 \to x_1) \to (f_2 : x_2 \to x_2)$ given by morphisms $g : x_1 \to x_2$ in $X$ such that

$x_1 \xrightarrow{f_1} x_1 \xrightarrow{g} x_2 = x_1 \xrightarrow{g} x_2 \xrightarrow{f_2} x_2$.

It’s not hard to see that $L(X \coprod Y) \cong LX \coprod LY$, so to understand this construction for arbitrary groupoids it’s enough to understand it for connected groupoids, or (up to equivalence) for groupoids $X = BG$ with a single object and automorphism group $G$. In this case, $LBG$ is the groupoid with objects the elements of $G$ and morphisms given by conjugation by elements of $G$; equivalently, it is the homotopy quotient or action groupoid of the action of $G$ on itself by conjugation.

In particular, when $G$ is finite, this quotient always has groupoid cardinality $1$. Hence:

Observation: If $X$ is an essentially finite groupoid (equivalent to a groupoid with finitely many objects and morphisms), then the groupoid cardinality of $LX$ is the number of isomorphism classes of objects in $X$.

I promise this is relevant to counting subgroups!

## Projective representations are homotopy fixed points

Yesterday we described how a (finite-dimensional) projective representation $\rho : G \to PGL_n(k)$ of a group $G$ functorially gives rise to a $k$-linear action of $G$ on $\text{Mod}(M_n(k)) \cong \text{Mod}(k)$ such that the Schur class $s(\rho) \in H^2(BG, k^{\times})$ classifies this action.

Today we’ll go in the other direction. Given an action of $G$ on $\text{Mod}(k)$ explicitly described by a 2-cocycle $\eta \in Z^2(BG, k^{\times})$, we’ll recover the category of $\eta$-projective representations, or equivalently the category of modules over the twisted group algebra $k \rtimes_{\eta} G$, by taking the homotopy fixed points of this action. We’ll end with another puzzle.

## Projective representations give categorical representations

Today we’ll resolve half the puzzle of why the cohomology group $H^2(BG, k^{\times})$ appears both when classifying projective representations of a group $G$ over a field $k$ and when classifying $k$-linear actions of $G$ on the category $\text{Mod}(k)$ of $k$-vector spaces by describing a functor from the former to the latter.

(There is a second half that goes in the other direction.)

## Projective representations

Three days ago we stated the following puzzle: we can compute that isomorphism classes of $k$-linear actions of a group $G$ on the category $C = \text{Mod}(k)$ of vector spaces over a field $k$ correspond to elements of the cohomology group

$\displaystyle H^2(BG, k^{\times})$.

This is the same group that appears in the classification of projective representations $G \to PGL(V)$ of $G$ over $k$, and we asked whether this was a coincidence.

Before answering the puzzle, in this post we’ll provide some relevant background information on projective representations.

## Fixed points of group actions on categories

Previously we described what it means for a group $G$ to act on a category $C$ (although we needed to slightly correct our initial definition). Today, as the next step in our attempt to understand Galois descent, we’ll describe what the fixed points of such a group action are.

John Baez likes to describe (vertical) categorification as replacing equalities with isomorphisms, which we saw on full display in the previous post: we replaced the equality $F(g) F(h) = F(gh)$ with isomorphisms $\eta(g, h) : F(g) F(h) \cong F(gh)$, and as a result we found 2-cocycles lurking in this story.

I prefer to describe categorification as replacing properties with structures, in the nLab sense. That is, the real import of what we just did is to replace the property (of a function between groups, say) that $F(g) F(h) = F(gh)$ with the structure of a family of isomorphisms between $F(g) F(h)$ and $F(gh)$. The use of the term “structure” emphasizes, as we also saw in the previous post, that unlike properties, structures need not be unique.

Accordingly, it’s not surprising that being a fixed point of a group action on a category is also a structure and not a property. Suppose $F : G \to \text{Aut}(C)$ is a group action as in the previous post, and $c \in C$ is an object. The structure of a fixed point, or more precisely a homotopy fixed point, is the data of a family of isomorphisms

$\displaystyle \alpha(g) : c \cong F(g) c$

which satisfy the compatibility condition that the two composites

$\displaystyle c \xrightarrow{\alpha(g)} F(g) c \xrightarrow{F(g)(\alpha(h))} F(g) F(h) c \xrightarrow{\eta(g, h)(c)} F(gh) c$

and

$\displaystyle c \xrightarrow{\alpha(gh)} F(gh) c$

are equal, as well as the unit condition that

$\displaystyle \alpha(e) = \varepsilon(c) : c \to F(e) c$

where $\varepsilon$ is the unit isomorphism $\text{id}_C \cong F(e)$. This is, in a sense we’ll make precise below, a 1-cocycle condition, but this time with nontrivial (local) coefficients.

Curiously, when the action $F$ is trivial (meaning both that $F(g) = \text{id}_C$ and that $\eta(g, h) = e \in Z(C)^{\times}$), this reduces to the definition of a group action of $G$ on $c \in C$ in the usual sense. In general, we can think of homotopy fixed point structure as a “twisted” version of a group action on $c \in C$ where the twist is provided by the group action on $C$.

## Group actions on categories

Yesterday we decided that it might be interesting to describe various categories as “fixed points” of Galois actions on various other categories, whatever that means: for example, perhaps real Lie algebras are the “fixed points” of a Galois action on complex Lie algebras. To formalize this we need a notion of group actions on categories and fixed points of such group actions.

So let $G$ be a group and $C$ be a category. For starters, we should probably ask for a functor $F(g) : C \to C$ for each $g \in G$. Next, we might naively ask for an equality of functors

$\displaystyle F(g) F(h) = F(gh) : C \to C$

but this is too strict: functors themselves live in a category (of functors and natural transformations), and so we should instead ask for natural isomorphisms

$\displaystyle \eta(g, h) : F(g) F(h) \cong F(gh)$.

These natural isomorphisms should further satisfy the following compatibility condition: there are two ways to use them to write down an isomorphism $F(g) F(h) F(k) \cong F(ghk)$, and these should agree. More explicitly, the composite

$\displaystyle F(g) F(h) F(k) \xrightarrow{F(g) \eta(h, k)} F(g) F(hk) \xrightarrow{\eta(g, hk)} F(ghk)$

should be equal to the composite

$\displaystyle F(g) F(h) F(k) \xrightarrow{\eta(g, h) F(k)} F(gh) F(k) \xrightarrow{\eta(gh, k)} F(ghk)$.

(There’s also some stuff going on with units which I believe we can ignore here. I think we can just require that $F(e) = \text{id}_C$ on the nose and nothing will go too horribly wrong.)

These natural isomorphisms $\eta(g, h)$ can be regarded as a natural generalization of 2-cocycles, and the condition above as a natural generalization of a cocycle condition. Below the fold we’ll describe this and other aspects of this definition in more detail, and we’ll end with two puzzles about the relationship between this story and group cohomology.

## The puzzle of Galois descent

Suppose we have a system $f_1, f_2, \dots f_n \in k[x_1, x_2, \dots x_m]$ of polynomial equations over a perfect (to keep things simple) field $k$, and we’d like to consider solutions of it over various field extensions $L$ of $k$. Write $V(L)$ for the set of all solutions to this system over $L$.

As it happens, knowing $V(L)$ for any algebraic extension $L$ of $k$ is equivalent to knowing $V(\bar{k})$, where $\bar{k}$ denotes the algebraic closure of $k$, together with the action of the absolute Galois group $G = \text{Gal}(\bar{k}/k)$. After picking an embedding of $L$ into $\bar{k}$, the infinite Galois correspondence says that $L$ is precisely the set of fixed points of the closed subgroup $H$ of $G$ which stabilizes $L$, and it’s not hard to see that this extends to $V(L)$; that is, $G$ naturally acts on $V(\bar{k})$, and we have a natural identification

$\displaystyle V(L) \cong V(\bar{k})^H$.

Now let’s categorify this situation. Before we considered, for each algebraic extension $L$ of $k$, a set $V(L)$. There are many situations in mathematics in which it’s natural to consider instead a category $F(L)$, such that a morphism $L_1 \to L_2$ induces a functor $F(L_1) \to F(L_2)$, and so forth. The basic example is the case that $F(L) = \text{Mod}(L)$ is the category of $L$-vector spaces, and for $f : L_1 \to L_2$ a morphism the corresponding functor is given by extension of scalars

$\displaystyle \text{Mod}(L_1) \ni V \mapsto V \otimes_{L_1} L_2 \in \text{Mod}(L_2)$.

This leads to many other examples coming from equipping vector spaces with extra structure: for example, $F(L)$ might be

• the category of representations of some finite group $G$ over $L$,
• the category of commutative (or associative, or Lie) algebras over $L$, or
• the category of schemes over $L$.

It would be great if understanding all of these categories was in some sense as simple as understanding the category $F(\bar{k})$, which generally tends to be simpler, and the action of the absolute Galois group $G$ on it, whatever that means. For example, the representation theory of finite groups over algebraically closed fields of characteristic zero is well understood, as is, say, the classification of semisimple Lie algebras. The general problem of trying to extract an understanding of $F(L)$ from an understanding of $F(\bar{k})$ is the problem of Galois descent.

We might very optimistically hope that the story here is directly analogous to the story above. This suggests the following puzzle.

Puzzle: In what sense could the statement $F(L) \cong F(\bar{k})^H$ be true for the examples given above?

## Hecke operators are also relative positions

Continuing yesterday’s story about relative positions, let $G$ be a finite group and let $X$ and $Y$ be finite $G$-sets. Yesterday we showed that $G$-orbits on $X \times Y$ can be thought of as “atomic relative positions” of “$X$-figures” and “$Y$-figures” in some geometry with symmetry group $G$, and further that if $X \cong G/H$ and $Y \cong G/K$ are transitive $G$-sets then these can be identified with double cosets $H \backslash G / K$.

Representation theory provides another interpretation of $G$-orbits on $X \times Y$ as follows. First, if $\mathbb{C}[X]$ is any permutation representation, then the $G$-fixed points $\mathbb{C}[X]^G$ have a natural basis given by summing over $G$-orbits. (This is a mild categorification of Burnside’s lemma.) Next, consider the representations $\mathbb{C}[X], \mathbb{C}[Y]$. Because $\mathbb{C}[X]$ is self-dual, we have

$\displaystyle \text{Hom}_G(\mathbb{C}[X], \mathbb{C}[Y]) \cong (\mathbb{C}[X] \otimes \mathbb{C}[Y])^G \cong \mathbb{C}[X \times Y]^G$

and hence $\text{Hom}_G(\mathbb{C}[X], \mathbb{C}[Y])$ has a natural basis given by summing over $G$-orbits of the action on $X \times Y$.

Definition: The $G$-morphism $\mathbb{C}[X] \to \mathbb{C}[Y]$ associated to a $G$-orbit of $X \times Y$ via the above isomorphisms is the Hecke operator associated to the $G$-orbit (relative position, double coset).

Below the fold we’ll write down some details about how this works and see how we can use the idea that $G$-morphisms between permutations have a basis given by Hecke operators to work out, quickly and cleanly, how some permutation representations decompose into irreducibles. At the end we’ll state another puzzle.