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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.

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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!

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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.

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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.)

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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.

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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.

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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.

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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?

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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.

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The goal of this post is to explain something that the cool kids all understood ages ago (David Speyer, John Baez) but that I hadn’t internalized until recently.

Let G be a group and let X and Y be transitive G-sets, so X = G/H and Y = G/K for some subgroups H, K of G. In “geometric” situations (in the sense of the Erlangen program), G is the symmetry group of some kind of geometry (for example, affine geometry, or Euclidean geometry), and X and Y are spaces of “figures” in the geometry (for example, points, lines, or triangles). We’ll call the points of XX-figures” and similarly for Y.

Now, figures in a geometry can be in various “relative positions” (or “incidence relations”) with respect to each other: for example, a point can be contained in a line, or two lines can intersect at right angles. What makes these geometrically meaningful is that they are invariant under the symmetry group G of the geometry: for example, the condition that a point is contained in a line is invariant under affine symmetries, and the condition that two lines intersect at right angles is invariant under Euclidean symmetries. This motivates the following.

Definition: A relative position of X-figures and Y-figures is a G-invariant subset of X \times Y, or equivalently a G-invariant relation R : X \to Y.

Any G-invariant subset of X \times Y decomposes into a disjoint union of G-orbits: these are the atomic relative positions.

Proposition: G-orbits of the action of G on G/H \times G/K (equivalently, the atomic relative positions of X-figures and Y-figures) can canonically be identified with double cosets H \backslash G/K, via the map

\displaystyle G/H \times G/K \ni ([g_1], [g_2]) \mapsto [g_1^{-1} g_2] \in H \backslash G/K

where [g] \in G/H means the image of g \in G under G \to G/H.

This is the conceptual interpretation of double cosets. It took an annoyingly long time between the first time I was introduced to double cosets (which I believe was in 2010) and the time I internalized the above fact (which was this year, 2015). Unlike the usual definition, this interpretation naturally generalizes to a notion of “triple cosets” (G-orbits on a triple product X \times Y \times Z), and so forth.

Example. Let G = \text{Isom}(\mathbb{R}^n) be the group of isometries of Euclidean space, which more explicitly is the semidirect product \mathbb{R}^n \rtimes O(n). If X = Y are both the G-space of points in \mathbb{R}^n, then the atomic relative positions have the form “a point has distance r from another point,” where r is any nonnegative real.

Example. Let G = GL_n(k) be the general linear group over a field k and let H = K = B be the Borel subgroup of upper triangular matrices. G/B is the space of complete flags in V = k^n. As it turns out, there are exactly n! atomic relative positions of a pair of complete flags. When k is a finite field these form a basis of a Hecke algebra. In general they label the Bruhat decomposition of G.

For example, when n = 2, a complete flag is just a line in V = k^2, and there are two atomic relative positions: the lines can be identical or they can be different. When n = 3, a complete flag is a line V_1 contained in a plane V_2 in V = k^3, and there are six atomic relative positions. Letting W_1 \subset W_2 denote a second complete flag, they are

  • V_1 = W_1, V_2 = W_2,
  • V_1 = W_1, V_2 \neq W_2,
  • V_1 \neq W_1, V_2 = W_2,
  • V_1 \neq W_1, V_1 \subset W_2, W_1 \not\subset V_2, V_2 \neq W_2,
  • V_1 \neq W_1, W_1 \subset V_2, V_1 \not\subset W_2, V_2 \neq W_2,
  • V_1 \neq W_2, V_1 \not\subset W_2, W_1 \not\subset V_2, V_2 \neq W_2.

Instead of thinking about relations as conditions on a pair of complete flags, we can also think about them as partial multi-valued functions from complete flags to complete flags. In those terms the six atomic relative positions are

  • Do nothing,
  • Pick a different plane,
  • Pick a different line,
  • Pick a different plane still containing the original line, then pick a different line not contained in the original plane,
  • Pick a different plane not containing the original line, then pick a different line contained in the original plane,
  • Pick a different plane not containing the original line, then pick a different line not contained in the original plane.

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