Jason (jcreed) wrote,

Ever since Matus lent me a book on wavelets many months ago, I occasionally find myself returning to thinking about generalizations of fourier analysis and stuff like that.

One thing that keeps coming up whenever I try to find things to read is "representation theory", which, if you ask me, is way up there in the worst-named subfields of mathematics of all time. I mean, what kind of math isn't a theory of representations in some sense? Anyway.

What it actually is is the study of homomorphisms from some group G you found in the gutter to the group of linear maps of (typically) Cn to itself — and the hope is that studying the ways G maps into these groups of linear maps tells you something about G.

The relationship between this and fourier analysis is that the notion of "frequency" or "normal mode of vibration" that you transform into when you do a fourier transform only winds up being nice sensible things like integers or real numbers in the special case that you have a periodic signal coming to you on the real line or over Z or something like that. When you generalize the symmetric structure of your signal to something other than translation groups (the key property of fourier components being that, if you do translate them in time, it's the same as multiplying them by a phase), the corresponding notion of frequency generalizes to representations of the appropriate symmetry group.

But forgetting about that for a moment, there's a separate intuition about what representation theory is about that I've sort of picked up from John Baez's occasional mention of the topic, which is that it's just a more "quantum", more linear-algebraical kind of way of thinking about groups acting on sets.

For a group G acting on a set S of elements — that is, each element of G is mapped to a bijection S → S — I can ask, for instance, which subsets S' of S are left invariant. That is, S' might be scrambled around within itself by the action, but no g ∈ G causes any element of S' to wind up outside of S'. The minimal invariant subsets are just the orbits of the action.

At least for finite-dimensional complex representations, this is pretty much the same story with representations and irreducible representations. The difference is that instead of elements of an n-element set, the representation is thought of as acting on the subspaces of an n-dimensional linear space. This is the new, weird, "quantum" notion of set-hood and cardinality and stuff.

For instance, if we took Z2 acting on a two-point set {x,y}, we'd have one orbit in the set-theoretic perspective, but two orbits in the linear perspective: the subspace generated by scalar multiples of |x>-|y> and the subspace generated by |x>+|y>.

Mostly I've felt a lot of comfort passing back and forth between these two perspectives when trying to understand the linear versions; dimension of finite-dimensional vector spaces very nicely behaves a lot like cardinality. The sum of the dimension of all the invariant subspaces of course has to add up to the dimension of the whole space, stuff like that.

But I just came across Schur's lemma, which says that any map CnCn that commutes with an irreducible representation has to be a scalar — and the obvious set-theoretic analogue just isn't true! The swap operation on {x,y} commutes with Z2 acting on it, and yet it's obviously not the identity. The linear version of this situation gets around the problem precisely because that representation of Z2 isn't irreducible.

So I'm left without any clear intuition for what Schur's lemma is really saying — though I do see from various applications in character theory that it's quite basic and useful.
Tags: math, representation theory
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