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## The p-group fixed point theorem

The goal of this post is to collect a list of applications of the following theorem, which is perhaps the simplest example of a fixed point theorem.

Theorem: Let $G$ be a finite $p$-group acting on a finite set $X$. Let $X^G$ denote the subset of $X$ consisting of those elements fixed by $G$. Then $|X^G| \equiv |X| \bmod p$; in particular, if $p \nmid |X|$ then $G$ has a fixed point.

Although this theorem is an elementary exercise, it has a surprising number of fundamental corollaries.

## Moments, Hankel determinants, orthogonal polynomials, Motzkin paths, and continued fractions

Previously we described all finite-dimensional random algebras with faithful states. In this post we will describe states on the infinite-dimensional $^{\dagger}$-algebra $\mathbb{C}[x]$. Along the way we will run into and connect some beautiful and classical mathematical objects.

A special case of part of the following discussion can be found in an old post on the Catalan numbers.

## More about the Schrödinger equation on a finite graph

It looks like the finite graph model is not just a toy model! It’s called a continuous-time quantum random walk and is used in quantum computing in a way similar to how random walks on graphs are used in classical computing. The fact that quantum random walks mix sooner than classical random walks relates to the fact that certain quantum algorithms are faster than their classical counterparts.

I learned this from a paper by Lin, Lippner, and Yau, Quantum tunneling on graphs, that was just posted on the arXiv; apparently the idea goes back to a 1998 paper. I have an idea about another sense in which the finite graph model is not just a toy model, but I have not yet had time to work out the details.

## Test your intuition: consecutive tails

Something very unfortunate has happened: several things I have recently written that could have been blog entries are instead answers on math.SE! In the interest of exposition beyond the Q&A format I am going to “rescue” one of these answers. It is an answer to the following question, which I would like you to test your intuition about:

Flip $150$ coins. What is the probability that, at some point, you flipped at least $7$ consecutive tails?

Jot down a quick estimate; see if you can get within a factor of $2$ or so of the actual answer, which is below the fold.

## Walks on graphs and statistical mechanics

I finally learned the solution to a little puzzle that’s been bothering me for awhile.

The setup of the puzzle is as follows. Let $G$ be a weighted undirected graph, e.g. to each edge $i \leftrightarrow j$ is associated a non-negative real number $a_{ij}$, and let $A$ be the corresponding weighted adjacency matrix. If $A$ is stochastic, one can interpret the weights $a_{ij}$ as transition probabilities between the vertices which describe a Markov chain. (The undirected condition then means that the transition probability between two states doesn’t depend on the order in which the transition occurs.) So one can talk about random walks on such a graph, and between any two vertices the most likely walk is the one which maximizes the product of the weights of the corresponding edges.

Suppose you don’t want to maximize a product associated to the edges, but a sum. For example, if the vertices of $G$ are locations to which you want to travel, then maybe you want the most likely random walk to also be the shortest one. If $E_{ij}$ is the distance between vertex $i$ and vertex $j$, then a natural way to do this is to set

$a_{ij} = e^{- \beta E_{ij}}$

where $\beta$ is some positive constant. Then the weight of a path is a monotonically decreasing function of its total length, and (fudging the stochastic constraint a bit) the most likely path between two vertices, at least if $\beta$ is sufficiently large, is going to be the shortest one. In fact, the larger $\beta$ is, the more likely you are to always be on the shortest path, since the contribution from any longer paths becomes vanishingly small. As $\beta \to \infty$, the ring in which the entries of the adjacency matrix lives stops being $\mathbb{R}$ and becomes (a version of) the tropical semiring.

That’s pretty cool, but it’s not what’s been puzzling me. What’s been puzzling me is that matrix entries in powers of $A$ look an awful lot like partition functions in statistical mechanics, with $\beta$ playing the role of the inverse temperature and $E_{ij}$ playing the role of energies. So, for awhile now, I’ve been wondering whether they actually are partition functions of systems I can construct starting from the matrix $A$. It turns out that the answer is yes: the corresponding systems are called one-dimensional vertex models, and in the literature the connection to matrix entries is called the transfer matrix method. I learned this from an expository article by Vaughan Jones, “In and around the origin of quantum groups,” and today I’d like to briefly explain how it works.

## The McKay correspondence I

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.

Let $G$ be a compact group and let $\text{Rep}(G)$ denote the category of finite-dimensional unitary representations of $G$. As in the finite case, due to the existence of Haar measure, $\text{Rep}(G)$ is semisimple (i.e. every unitary representation decomposes uniquely into a sum of irreducible representations), and via the diagonal action it comes equipped with a tensor product with the property that the character of the tensor product is the product of the characters of the factors.
Question: Fix a representation $V \in \text{Rep}(G)$. What is the multiplicity of the trivial representation in $V^{\otimes n}$?