Comments on: Representation theory – basics

By: Michi’s blog » Blog Archive » Modular representation theory - when Maschke breaks down

Sat, 21 Apr 2007 12:43:17 +0000

[...] had, in my first representation theory post, a mention of Maschke’s theorem. This states that if the characteristic of our field divides [...]

By: Michi’s blog » Blog Archive » Modular representation theory: Simple and semisimple objects

Mon, 02 Apr 2007 14:57:01 +0000

[...] the elements of the structure as endomorphisms of some vectorspace. The attentive reader remembers my last post on the subject, where was given a group action by the rotations and reflections of a polygon [...]

By: John Armstrong

John Armstrong — Sun, 01 Apr 2007 19:50:43 +0000

Well, did you read the slides for my bracket extension talk that I posted March 17? That's a case study right there.

By: Michi

Michi — Sun, 01 Apr 2007 19:06:38 +0000

John: Please, please, please give me more details about that! Sounds very interesting – and I -want- to know more about it!

By: John Armstrong

John Armstrong — Sun, 01 Apr 2007 18:54:31 +0000

Miles: of course what you say is true, and it extends from there. Tangles form yet another kind of category, and the skein-theoretic approach to knot theory is nothing but the representation theory of those tangle categories.

By: Michi

Michi — Sun, 01 Apr 2007 18:45:04 +0000

Sorry for the late comment approval – I had a short while without internet access over the weekend.

You have a very good point with the generality of representations: I’m probably going to put more emphasis on that in my next post – where I plan to talk about structure theorems for simple and semisimple algebras of various sorts – associative and Lie algebras end up having a very similar theory of simple modules and algebras.

The “sticking together” interpretation is closely coupled with a very graphical way to reason about representations that Benson &al have written a paper about. I need to understand that paper at some point, but the “sticking together” is what stuck so far.

As for monads/operads – I saw the post, but was listening to Jiri Matousek while reading it, so didn’t get around to commenting.

By: Miles Gould

Miles Gould — Fri, 30 Mar 2007 17:19:57 +0000

I’d have said that groups, partially ordered sets and (the free categories on) quivers are just special categories, and so the case of representations of categories both covers them all and is especially simple: a representation of a category C is just a functor C -> Vect_k (or Set, if you prefer).

But then, I would say that

Good article – I particularly liked the bit about higher Ext groups measuring ways of sticking simple modules together. I’m always on the lookout for intuitive interpretations of homological gadgets like that.

By the way, did you see my recent post about monads and operads?