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> representation theory (abstract linear algebra) To me, representation theory means a lot more than abstract linear algebra. Of course, one can define terms
by JadeNB 5y ago
> representation theory (abstract linear algebra)
To me, representation theory means a lot more than abstract linear algebra. Of course, one can define terms however one wishes, including in such a way that these two become synonymous—and, also of course, maybe you were intentionally using the terms informally to give an idea of your meaning without going into too much technicality—but, to me, abstract linear algebra is the study (for each field $k$) of the category whose objects are $k$-vector spaces and whose morphisms are $k$-linear maps, whereas representation theory is the study of the category whose objects are triples $(G, X, \alpha)$ of a group $G$, a space $X$, and an action $\alpha$ of $G$ on $X$ (let's stipulate—though not everyone restricts representation theory to this setting!—that $X$ is a $k$-vector space, and $\alpha$ is a linear action), with morphisms being intertwiners in an obvious sense.
That is, in some sense, I view representation theory as somehow 'reifying' the morphisms of abstract linear algebra to become objects (a perspective which could, and sometimes does, lead one down the path of $n$-categories).
- joppy 5y agoI would say that your stipulation is (if anything) the wrong way around - most people I know would limit representation theory to linear actions on a vector space (or module over a PID), but would not limit it to a group G: there are groups, Lie algebras, algebras in general, etc etc. I think going down the route of trying to identify fields of maths with specific “things that can be stated in category theoretical language” is a bit wrong and bordering on hubris - sure, all group representations can be stated as some kind of functor between categories, but this is just convenient compact language for describing the definition. Any deep theorems that follow can almost certainly not be deduced from the categorical description, and are many times inhibited or obscured by it.
- JadeNB 5y ago> I would say that your stipulation is (if anything) the wrong way around - most people I know would limit representation theory to linear actions on a vector space (or module over a PID), but would not limit it to a group G: there are groups, Lie algebras, algebras in general, etc etc. That's a very good point about other kinds of 'representers' —though I think it doesn't change the central point that, in some sense, linear algebra seems to focus on "spaces as objects", whereas representation theory, loosely, focuses on "morphisms as objects". I agree that most people will mean when they refer to representations to refer to linear actions, but every so often one does encounter references to, say, actions on sets as 'permutation representations' (although that also has a, closely related, linear-action meaning). I just meant to acknowledge that occasional extra inclusiveness.
- JadeNB 5y ago> I think going down the route of trying to identify fields of maths with specific “things that can be stated in category theoretical language” is a bit wrong and bordering on hubris - sure, all group representations can be stated as some kind of functor between categories …. I missed this point in my earlier reply, and it's too late to edit. I certainly agree that it's easy to be misled about the utility or appropriateness of a category-theoretic taxonomy, but I wasn't attempting that here. Rather, I was just looking for a way to capture the intuition I have about the different way in which representation theory looks at what are, in some sense, the same kinds of objects as those considered in abstract linear algebra. (Come to that, I was specifically avoiding looking at group representations as functors; not that it much matters, since, again, the category theory was just a way to try to formalise intuition rather than an attempt to prove anything, but I was actually going the other way by regarding the representations as objects.)
- bigbillheck 5y agoI'm not sure how willing I'd be to say that you can't do 'abstract linear algebra' over rings.
- JadeNB 5y ago> I'm not sure how willing I'd be to say that you can't do 'abstract linear algebra' over rings. Yes, that is a good point.
- poletopole 5y agoRepresentation theory by virtue I feel has more angles than most branches of mathematics, but your definition isn't off at all. I tend to focus in particular in terms of information theory which is where the combinatorial and symmetric groups, to name a few sides of representation theory, are of particular interest, but isn't the entire picture by a long shot.
- JadeNB 5y agoWhat are the combinatorial groups?
- Koshkin 5y agoIndeed, the 'abstract' linear algebra does not include the notion of (ir)reducibility that is crucial to representation theory and its applications.