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In the proof of Maschke's theorem, we magically transform linear transformations $T$ into maps that preserve the group ring structure. The method is to define

$ \bar T(v) = \frac{1}{|G|} \sum\limits_{g \in G} g^{-1} T(g(v))$

Note for this to make sense, we have to be able to divide by $ |G|$. In other words, the characteristic of $ F$ does not divide $ |G|$. When $ T$ is already an $ FG$-module homomorphism:

$ \bar T(v) = \frac{1}{|G|} \sum\limits_{g \in G} g^{-1} T(g(v)) = \frac{1}{|G|} \sum\limits_{g \in G} g^{-1} gT(v) = \frac{1}{|G|} \sum\limits_{g \in G} T(v) = T(v)$

In essence, we would like $ g^{-1}T(g(v))$ to just be $T(v)$ for any linear transformation $ T$, but it's not. So we form a function by taking averages and trying to water down the deviation. And it works! We end up with an honest $ FG$-module morphism.

I saw another example of this in Guillemin/Pollack's Differential topology. Here they start with a n-tensor $T$ and form an alternating tensor:

$Alt(T) = \frac{1}{n!} \sum\limits_{g \in S_n} sgn(g) T^g$

My question is: Where does this kind of "averaging to pick up structure" work? Can we turn set maps into equivariant maps in the G-set setting? Or something different like averaging continuous maps to get smooth ones. The obvious obstruction is being able to divide, so is there a way around this? So a second question is: when you can't do this averaging, what kind of workarounds exist?

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    Sure, lots of ways. See http://en.wikipedia.org/wiki/Ergodic_theory#Ergodic_theorems .2011-08-16
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    @Qiaochu Is that a generalization of the method in Maschke's theorem, or is it an essentially different type of averaging?2011-08-16
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    I don't know what you're looking for. I would consider it morally the same and don't see any reason not to.2011-08-16
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    I haven't looked at the link suggested by Qiaochu, but this happens in many different ways in representation theory, and the averaging in finite groups also becomes integrating when dealing with Lie groups (eg, the continuous functions on $S^{1}$ have Fourier series, obtained by integration, which can be seen as a limiting case of using orthogonality relations for finite cyclic groups of order $n$ as $n \to \infty$).2011-08-16
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    The technique is also useful for constructing G-invariant metrics on Lie groups and homogeneous spaces, and to construct the Haar measure on compact groups.2011-08-17

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In both cases you mention, the "averaging" construction is projection onto the $G$-invariant subspace.

Namely, suppose you are working with the category of (usually finite-dimensional) representations of some algebraic object: an algebra, a group, a Lie algebra, etc. Usually there is a trivial representation of some sort, which is one-dimensional and has an uninteresting representation structure. For instance, given a group, then there is the representation on the one-dimensional space with the trivial action. Given a Lie algebra, the analog is the one-dimensional representation where everything acts by zero.

Let $A$ be some algebraic object that can act on $k$-vector spaces, so we can form the category of $A$-representations, and suppose $A$ has a trivial representation $\mathbf{1}$. Suppose all the representations of interest are finite-dimensional vector spaces. Let $V$ be an object of our category (of $A$-representations); then $V$ contains a largest subobject $V' \subset V$ on which the $A$-action is trivial: that is, $V'$ is a sum of copies of $\mathbf{1}$. In the case of a group, the largest trivial subobject is the fixed point space of the entire group. In the case of a Lie algebra, the largest trivial subobject consists of vectors that are annihilated by everything in the Lie algebra.

Now the question is whether $V'$ admits a complement in $V$: that is, does the inclusion $A' \to A$ split? If it does, then there is an $A$-equivariant projection $V \to V'$, which you should think of the generalization of averaging. In the case of finite groups (where the characteristic of $k$ is prime to the order), there is such a projection: this follows by Maschke's theorem (semisimplicity of the group algebra), and the idea is the averaging argument. Given a $k$-linear projection $V \to V'$ (which always exists for vector spaces), one "averages" it to get an $A$-linear projection. One case where the analogy with averaging is less obvious is where $A$ is a semisimple Lie algebra, where it is also true that such projection operators exist. (Admittedly one can associate to $A$ a compact Lie group and then for that an averaging argument does work, with $A$ replaced by said Lie group.)

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    should it be "in the case of groups, there is a projection iff the characteristic of k doesn't divide the order of the group"? Thanks for answering.2011-08-17
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    @Matt: Dear Matt, you're right.2011-08-17