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Archive for April, 2010

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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Let G be a group and let

\displaystyle V = \bigoplus_{n \ge 0} V_n

be a graded representation of G, i.e. a functor from G to the category of graded vector spaces with each piece finite-dimensional. Thus G acts on each graded piece V_i individually, each of which is an ordinary finite-dimensional representation. We want to define a character associated to a graded representation, but if a character is to have any hope of uniquely describing a representation it must contain information about the character on every finite-dimensional piece simultaneously. The natural definition here is the graded trace

\displaystyle \chi_V(g) = \sum_{n \ge 0} \chi_{V_n}(g) t^n.

In particular, the graded trace of the identity is the graded dimension or Hilbert series of V.

Classically a case of particular interest is when V_n = \text{Sym}^n(W^{*}) for some fixed representation W, since V = \text{Sym}(W^{*}) is the symmetric algebra (in particular, commutative ring) of polynomial functions on W invariant under G. In the nicest cases (for example when G is finite), V is finitely generated, hence Noetherian, and \text{Spec } V is a variety which describes the quotient W/G.

In a previous post we discussed instead the case where V_n = (W^{*})^{\otimes n} for some fixed representation W, hence V is the tensor algebra of functions on W. I thought it might be interesting to discuss some generalities about these graded representations, so that’s what we’ll be doing today.

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Nationals 2010

MIT is hosting the United States Rubik’s Cube Championships this summer, August 6-8. All ages welcome! Normally I wouldn’t post about such things, but

  • I happen to be a member of the Rubik’s Cube club here, and
  • Some people use the Rubik’s cube group to motivate group theory. I’m a fan of hands-on mathematics, and there’s a lot to learn from the cube; for example, you quickly understand that groups are not in general commutative. The Rubik’s cube itself is also a good example of a torsor.

Actually, just so this post has some mathematical content, there’s something about the Rubik’s cube group that is probably very simple to explain, but which I don’t completely understand. It’s a common feature of Rubik’s cube algorithms that they need to switch around some parts of the cube without disturbing others; in other words, the corresponding permutation needs to have a lot of fixed points. This seems to be done by writing down a lot of commutators, but I’m not familiar with any statements in group theory of the form “commutators tend to have fixed points.” Can anyone explain this?

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