Michi’s blog

Interview with a blogger

• April 30th, 2007

The website/forumsite Mathetreff, run by the Bezirksregierung (region government) Düsseldorf, just performed a mail interview with me.

Here it is, translated to english, for your enjoyment.

MT: Dear Mr. Johansson, you are an expert on mathematics blogs. Thus first off a double question: What is a blog, and what do you do in your algebra blog?

MJ: A blog, or weblog as the name started, is basically just a comfortable way to publish texts sequentially on the web. As such blogs aren’t much more than websites with administration aids. The interesting starts when you involve interactions - comments on the texts or easy linking between different blogs.

For this, there are both many different blog softwares, that make these aspects easy accessible and automated, and many websites that network blogs. Thus, it happens that farreaching blog debates occur, where through a networking site several blog discuss or even fight over some question.

Modular representation theory - when Maschke breaks down

• April 21st, 2007

This post is dedicated to Janine Kühn and her Proseminar-lecture.

We had, in my first representation theory post, a mention of Maschke’s theorem. This states that if the characteristic of our field doesn’t divide the group order, then simple and irreducible mean the same thing.

Now, obviously, the actual proof you normally see first deals with a construction that works for when the characteristic doesn’t divide the group order - which uses 1/|G| at one point. So, what happens when this is impossible to work with? When the conditions of Maschke simply do not hold?

The very simplest answer is that then we can get modules that are glued together by simple modules with some meshing. Such that they aren’t direct sums any more. The ways we can glue together modules are through extensions - i.e. we can glue together A,C by forming a short exact sequence
0 → C → B → A → 0
and we’ll have that B is a module such that B/C=A. Now, the typical such module is the direct sum of A and C - and if Maschke holds, this is indeed all there is.

looksay - today’s Haskell snippet

• April 18th, 2007

Conway’s Look-and-say sequence

iTeX2MML not activated

• April 6th, 2007

I just tried installing the iTeX2MML plugin from Jacques Distler. This is what the n-Category Cafe use for their mathematics, and it gives a neated display than the LaTeXrender plugin I’ve been using so far.

It turns out, though, that
1. The plugin jumps on quoted perl code, interpreting it as mathematics. Bad things ensue.
2. It needs valid XHTML, which has not been a priority so far - and trying to validate it, the validator chokes on the &’s in my LaTeX array expressions for LaTeXrender.

Oh bugger. No iTeX and MathML for me.

Modular representation theory: Simple and semisimple objects

• April 2nd, 2007

Representations of categories

The basic tenet of representation theory is that we have some entity - the group representation theory takes a group, the algebra representation theory most often a quiver, and we look at ways to view 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 surrounding the origin.

There is a way, suggested to me by Miles Gould in my last post, to unify all these various ways of looking at things in one: groups - and quivers - and most other things we care to look at are in fact very small category. For groups, it’s a category with one object, and one morphism/arrow for each group element, and such that the arrow you get by composing two arrows is the one corresponding to a suitable product in the group. A quiver is just a category corresponding to the way the quiver looks - with objects for all vertices, and arrows for all edges.

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