There’s a new article at Quanta today promoting representation theory, Kevin Hartnett’s The ‘Useless’ Perspective that Transformed Mathematics. Representation theory is a central, unifying theme in modern mathematics, one that deserves a lot more attention than it usually gets, with undergraduate math majors often not exposed to the subject at all. My book on quantum mechanics is very much based on the idea that the subject is best understood in terms of representation theory. Unfortunately, physics students typically get even less exposure to representation theory than math students.
While I think the article is a great idea, and well-worth reading, I do have two quibbles, one minor and one major. The minor quibble is that one example given of a group, the real numbers with multiplication, is not quite right: you need to remove the element 0, since it has no inverse. If the group law is the additive one, then the real number line with nothing removed truly is a group.
The major quibble is with the theme of the article that a group representation can be thought of as a simplification of something more complicated, the group itself. This is a good way of thinking about one aspect of the use of representation theory in number theory, where representations provide a tractable way to get at the much more complicated structure of the absolute Galois group of a number field. The talk by Geordie Williamson linked to in the article (slides here) explains this well, but Williamson also gets right the more general context, where the group can be easy to understand, the representations complicated. For a simple example of this, in the case of the circle group $S^1$ the group is very easy to understand, its representation theory (the theory of Fourier series) is much more complicated (and much more interesting).
As Williamson explains, a good way to think about what is going on is that representation theory does simplify something by linearizing it, but it’s not the group, it’s a group action. When people talk about the importance of the study of “symmetry” in mathematics, physics, and elsewhere, they often make the mistake of only paying attention to the symmetry groups. The structure you actually have is not just a group (the abstract “symmetries”), but an action of that group on some other object, the thing that has symmetries. When you talk about “rotational symmetry” you have a rotation group, but also something else: the thing that is getting rotated. Representation theory is the linearization of this situation, often achieved by going from the group action on an object to the corresponding group action on some version of functions on the object. Once linearized, the group action becomes a problem in linear algebra, with the group elements represented as matrices, which act on the vectors of the linearization.
To further add to the confusion, “symmetry” is often described in popular accounts as meaning “invariance”. In typical examples given, “invariance” just means that you have a group action, since the group is taking elements of the set to other elements of the set (e.g. rotations not of an arbitrary object, but of a sphere). In representation theory, you have a different notion of invariance. For instance, for the representation of rotations on functions on the sphere, the constant functions are a one-dimensional invariant subspace, giving a trivial representation. But, there are lots of more interesting invariant subspaces of higher dimensions. These are the irreducible representations on the sets of spherical harmonics.
As an expert on this topic, at least with regards to QM, could you give your opinion on the most important texts/papers in the field?
I,
The fascinating thing about representation theory is that it is a unifying theme, appearing centrally in very different areas of mathematics. In each such area there’s a huge literature and many different textbooks, so I can’t come up with a short list, and I don’t know of a book that provides a single overview of the many different areas of the subject.
It’s strange they used Burnside as an exemplary representation theory sceptic. He founded it together with Frobenius.
As a popular article explaining representation theory to non-mathematicians, it does a reasonably good job. To a professional representation-theorist, however, it jars in a couple of places. “Mathematicians aim to avoid grappling with the full complexity of a group; instead they gain a sense of its properties by looking at how it behaves when converted into the stripped-down format of linear transformations” is wrong on so many levels. Physicists may behave like this, but mathematicians do not: we strip away the inessentials of linear transformations in order to study the stripped-down abstract group, not the other way around. It is true that to study a group it is often essential to study its representations: but this makes things more complicated, not less. It is also true that applications of group theory almost always involve representations: the group itself is usually too abstract for this purpose, without the mediation of representation theory.
nikita,
You are absolutely right. What about Burnside’s p^a.q^b theorem? The quintessential example of a theorem in group theory that until relatively recently could *only* be proved using representation theory, and not by group theory alone. In the very early days of representation theory in the second half of the 1890s it might have seemed as though the subject was too abstruse to be useful, but this point of view very rapidly became obsolete.
I explain how Burnside changed his attitude toward group representation theory here.
When people say “representation theory is a central, unifying theme in modern mathematics” I wonder to what extent they really mean that the objects to which representation theory can usefully be applied (say, simple Lie groups and all their offshoots) are central. It seems there is still a bit of mystery as to why representation theory is so effective for understanding these kind of algebraic objects and their actions. Certainly Williamson’s work (and the work of his forebears like Lusztig, etc.) explores the boundary of representation theory’s effective reach.
As an undergraduate, and also later, I didn’t understand the point with groups and representation theory. I took courses and could follow the steps, but it didn’t seem very important to me. As a graduate student, I specialized critical phenomena and fractals, in particular in 2D because that was where progress was made. This was in the mid 1980s, when CFT came around, which I noticed but didn’t understand.
Then when preparing for my dissertation, I read the NPB paper by Dotsenko and Fateev, and it suddenly clicked. I realized that what I had studied in grad school could be explained in terms of the representation theory of the Virasoro algebra. This was a complete eye-opener and changed forever how I viewed physics, for better or worse.
Regarding your first quibble, the apparently trivial example of the multiplicative real numbers as a group, i.e. the reals w/0 zero, is actually kind of useful
for understanding the difference between GL(2, C) and GL(2,R). The multiplicative complexes and reals are of course GL(1, C) and GL(1,R) and for both 1 and 2 dimensions, the
complex groups are connected while the real ones have two connected components, with positive and negative determinant, the former a normal subgroup of index 2. I had never understood this very well, and realized only recently that the explanations are the same. Since the exponential function takes addition to multiplication, the Lie algebra (i.e. the additive group of all matrices) is mapped to the positive determinant matrices GL^+(2,R); note that indeed
det(exp(A))= exp(trace(A))), so the image must have positive determinant. Further, to show the image is pathwise connected
you can linearly interpolate in the Lie algebra and push that forward to a path in the image.
(You can’t directly linearly interpolate in the group since you might pass through something with determinant 0).
Geordie Williamson contacted me personally to point out that he is quoted out of context in this article, and in his lecture he discusses Burnside’s change of heart in depth.