Michi’s blog » archive for April, 2007

Interview with a blogger

• April 30th, 2007
• 4:40 pm

The website/forumsite Mathetreff, run by the Bezirksregierung (region government) D

Modular representation theory – when Maschke breaks down

• April 21st, 2007
• 1:43 pm

This post is dedicated to Janine K

• April 18th, 2007
• 9:34 am
nextLookSay = foldr (\xs -> ([length xs, head xs]++)) [] . group
lookSay = iterate nextLookSay [1]

Conway’s Look-and-say sequence

iTeX2MML not activated

• April 6th, 2007
• 2:26 pm

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
• 3:56 pm

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.