For some reason several classes at MIT this year involve Fourier analysis. I was always confused about this as a high schooler, because no one ever gave me the ``orthonormal basis'' explanation, so here goes. As a bonus, I also prove a form of Arrow's Impossibility Theorem using binary Fourier analysis, and then talk about… Continue reading Things Fourier
Tag: representation theory
Tannakian Reconstruction
These notes are from the February 23, 2016 lecture of 18.757, Representations of Lie Algebras, taught by Laura Rider. Fix a field $latex {k}&fg=000000$ and let $latex {G}&fg=000000$ be a finite group. In this post we will show that one can reconstruct the group $latex {G}&fg=000000$ from the monoidal category of $latex {k[G]}&fg=000000$-modules (i.e. its… Continue reading Tannakian Reconstruction
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
Power Overwhelming 