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Posts Tagged ‘group actions’

In this post we’ll describe the representation theory of the additive group scheme \mathbb{G}_a over a field k. The answer turns out to depend dramatically on whether or not k has characteristic zero.

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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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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 collect a list of applications of the following theorem, which is perhaps the simplest example of a fixed point theorem.

Theorem: Let G be a finite p-group acting on a finite set X. Let X^G denote the subset of X consisting of those elements fixed by G. Then |X^G| \equiv |X| \bmod p; in particular, if p \nmid |X| then G has a fixed point.

Although this theorem is an elementary exercise, it has a surprising number of fundamental corollaries.

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A common theme in mathematics is to replace the study of an object with the study of some category that can be built from that object. For example, we can

  • replace the study of a group G with the study of its category G\text{-Rep} of linear representations,
  • replace the study of a ring R with the study of its category R\text{-Mod} of R-modules,
  • replace the study of a topological space X with the study of its category \text{Sh}(X) of sheaves,

and so forth. A general question to ask about this setup is whether or to what extent we can recover the original object from the category. For example, if G is a finite group, then as a category, the only data that can be recovered from G\text{-Rep} is the number of conjugacy classes of G, which is not much information about G. We get considerably more data if we also have the monoidal structure on G\text{-Rep}, which gives us the character table of G (but contains a little more data than that, e.g. in the associators), but this is still not a complete invariant of G. It turns out that to recover G we need the symmetric monoidal structure on G\text{-Rep}; this is a simple form of Tannaka reconstruction.

Today we will prove an even simpler reconstruction theorem.

Theorem: A group G can be recovered from its category G\text{-Set} of G-sets.

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Suitably nice groupoids have a numerical invariant attached to them called groupoid cardinality. Groupoid cardinality is closely related to Euler characteristic and can be thought of as providing a notion of integration on groupoids.

There are various situations in mathematics where computing the size of a set is difficult but where that set has a natural groupoid structure and computing its groupoid cardinality turns out to be easier and give a nicer answer. In such situations the groupoid cardinality is also known as “mass,” e.g. in the Smith-Minkowski-Siegel mass formula for lattices. There are related situations in mathematics where one needs to describe a reasonable probability distribution on some class of objects and groupoid cardinality turns out to give the correct such distribution, e.g. the Cohen-Lenstra heuristics for class groups. We will not discuss these situations, but they should be strong evidence that groupoid cardinality is a natural invariant to consider.

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My current top candidate for a mathematical concept that should be and is not (as far as I can tell) consistently taught at the advanced undergraduate / beginning graduate level is the notion of a groupoid. Today’s post is a very brief introduction to groupoids together with some suggestions for further reading.

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