## Rant: Matrices Are Not Arrays of Numbers

The following is an excerpt from a current work of mine. I thought I'd share it here, as some people have told me they enjoyed it. As I'll stress repeatedly, a matrix represents a linear map between two vector spaces. Writing it in the form of an $latex {m \times n}&fg=000000$ matrix is merely a… Continue reading Rant: Matrices Are Not Arrays of Numbers

## Representation Theory, Part 4: The Finite Regular Representation

Good luck to everyone taking the January TST for the IMO 2015 tomorrow! Now that we have products of irreducibles under our belt, I'll talk about the finite regular representation and use it to derive the following two results about irreducibles. The number of (isomorphsim classes) of irreducibles $latex {\rho_\alpha}&fg=000000$ is equal to the number… Continue reading Representation Theory, Part 4: The Finite Regular Representation

## Represenation Theory, Part 3: Products of Representations

Happy New Year to all! A quick reminder that $latex {2015 = 5 \cdot 13 \cdot 31}&fg=000000$. This post will set the stage by examining products of two representations. In particular, I'll characterize all the irreducibles of $latex {G_1 \times G_2}&fg=000000$ in terms of those for $latex {G_1}&fg=000000$ and $latex {G_2}&fg=000000$. This will set the… Continue reading Represenation Theory, Part 3: Products of Representations

## Representation Theory, Part 2: Schur’s Lemma

Merry Christmas! In the previous post I introduced the idea of an irreducible representation and showed that except in fields of low characteristic, these representations decompose completely. In this post I'll present Schur's Lemma at talk about what Schur and Maschke tell us about homomorphisms of representations. 1. Motivation Fix a group $latex {G}&fg=000000$ now,… Continue reading Representation Theory, Part 2: Schur’s Lemma

## Representation Theory, Part 1: Irreducibles and Maschke’s Theorem

Good luck to everyone taking the December TST tomorrow! The goal of this post is to give the reader a taste of representation theory, a la Math 55a. In theory, this post should be accessible to anyone with a knowledge of group actions and abstract vector spaces. Fix a ground field $latex {k}&fg=000000$ (for all… Continue reading Representation Theory, Part 1: Irreducibles and Maschke’s Theorem