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More random updates

Posted in Uncategorized, tagged , , on May 24, 2010| 4 Comments »

I’ve been reading a lot of mathematics lately, but I don’t feel capable of explaining most of what I’ve been reading about, so I’m not sure what to blog about these days. Fortunately, SPUR will be starting soon, so I’ll start focusing on relevant material for my project eventually. Until then, here are some more random updates.

  • Martin Gardner and Walter Rudin both recently passed away. They will be sorely missed by the mathematical community, although I can’t say I’m particularly qualified to eulogize about either.
  • For my number theory seminar with Scott Carnahan I wrote a paper describing an important corollary of the Eichler-Shimura relation in the theory of modular forms. The actual relation is somewhat difficult to state, but the important corollary relates the number of points on certain elliptic curves E over finite fields to the Fourier coefficients of certain modular forms of weight 2. You can find the paper here. Although the class is over, corrections and comments are of course welcome. (Though I hope Scott doesn’t change my grade if someone spots a mistake he missed!)
  • If you’re at all interested in the kind of mathematics where planar diagrams are used instead of traditional algebraic notation for computation, you should read Joachim Kock’s excellent book Frobenius Algebras and 2D Topological Quantum Field Theories. The book is much less intimidating than its title might suggest, and it is full of enlightening pictures and discussions. You might also be interested in a related MO question.

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The McKay correspondence I

Posted in math.CO, math.GR, math.RT, tagged , , on April 27, 2010| 5 Comments »

Today we’re going to relate the representation graphs introduced in this blog post to something I blogged about in the very first and second posts in this blog! The result will be a beautiful connection between the finite subgroups of \text{SU}(2), the Platonic solids, and the ADE Dynkin diagrams. This connection has been written about in several other places on the internet, for example here, but I don’t know that any of those places have actually gone through the proof of the big theorem below, which I’d like to (as much for myself as for anyone else who is reading this).

Let G be a finite subgroup of \text{SL}_2(\mathbb{C}). Since any inner product on \mathbb{C}^2 can be averaged to a G-invariant inner product, every finite subgroup of \text{SL}_2(\mathbb{C}) is conjugate to a finite subgroup of \text{SU}(2), so we’ll suppose this without loss of generality. The two-dimensional representation V of G coming from this description is therefore faithful and self-dual. Consider the representation graph \Gamma(V), whose vertices are the irreducible representations of G and where the number of edges between V_i and V_j is the multiplicity of V_j in V_i \otimes V. We will see that \Gamma(V) is a connected undirected loopless graph whose spectral radius \lambda(\Gamma(V)) is 2. Today our goal is to prove the following.

Theorem: The connected undirected loopless graphs of spectral radius 2 are precisely the affine Dynkin diagrams \tilde{A}_n, \tilde{D}_n, \tilde{E}_6, \tilde{E}_7, \tilde{E}_8.

We’ll describe these graphs later; for now, just keep in mind that they are graphs with a number of vertices which is one greater than their subscript. In a later post we’ll see how these give us a classification of the Platonic solids, and we’ll also discuss other connections.

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