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## The Schrödinger equation on a finite graph

One of the most important discoveries in the history of science is the structure of the periodic table. This structure is a consequence of how electrons cluster around atomic nuclei and is essentially quantum-mechanical in nature. Most of it (the part not having to do with spin) can be deduced by solving the Schrödinger equation by hand, but it is conceptually cleaner to use the symmetries of the situation and representation theory. Deducing these results using representation theory has the added benefit that it identifies which parts of the situation depend only on symmetry and which parts depend on the particular form of the Hamiltonian. This is nicely explained in Singer’s Linearity, symmetry, and prediction in the hydrogen atom.

For awhile now I’ve been interested in finding a toy model to study the basic structure of the arguments involved, as well as more generally to get a hang for quantum mechanics, while avoiding some of the mathematical difficulties. Today I’d like to describe one such model involving finite graphs, which replaces the infinite-dimensional Hilbert spaces and Lie groups occurring in the analysis of the hydrogen atom with finite-dimensional Hilbert spaces and finite groups. This model will, among other things, allow us to think of representations of finite groups as particles moving around on graphs.

## Walks on graphs and tensor products

Recently I asked a question on MO about some computations I’d done with Catalan numbers awhile ago on this blog, and Scott Morrison gave a beautiful answer explaining them in terms of quantum groups. Now, I don’t exactly know how quantum groups work, but along the way he described a useful connection between walks on graphs and tensor products of representations which at least partially explains one of the results I’d been wondering about and also unites several other related computations that have been on my mind recently.

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 $V^{\otimes n}$?

In the previous post we showed that the splitting behavior of a rational prime $p$ in the ring of cyclotomic integers $\mathbb{Z}[\zeta_n]$ depends only on the residue class of $p \bmod n$. This is suggestive enough of quadratic reciprocity that now would be a good time to give a full proof.

The key result is the following fundamental observation.

Proposition: Let $q$ be an odd prime. Then $\mathbb{Z}[\zeta_q]$ contains $\sqrt{ q^{*} } = \sqrt{ (-1)^{ \frac{q-1}{2} } q}$.

Quadratic reciprocity has a function field version over finite fields which David Speyer explains the geometric meaning of in an old post. While this is very much in line with what we’ve been talking about, it’s a little over my head, so I’ll leave it for the interested reader to peruse.

## Functoriality

I wanted to talk about the geometric interpretation of localization, but before I do so I should talk more generally about the relationship between ring homomorphisms on the one hand and continuous functions between spectra on the other. This relationship is of utmost importance, for example if we want to have any notion of when two varieties are isomorphic, and so it’s worth describing carefully.

The geometric picture is perhaps clearest in the case where $X$ is a compact Hausdorff space and $C(X) = \text{Hom}_{\text{Top}}(X, \mathbb{R})$ is its ring of functions. From this definition it follows that $C$ is a contravariant functor from the category $\text{CHaus}$ of compact Hausdorff spaces to the category $\mathbb{R}\text{-Alg}$ of $\mathbb{R}$-algebras (which we are assuming have identities). Explicitly, a continuous function $f : X \to Y$

between compact Hausdorff spaces is sent to an $\mathbb{R}$-algebra homomorphism $C(f) : C(Y) \to C(X)$

in the obvious way: a continuous function $Y \to \mathbb{R}$ is sent to a continuous function $X \xrightarrow{f} Y \to \mathbb{R}$. The contravariance may look weird if you’re not used to it, but it’s perfectly natural in the case that $f$ is an embedding because then one may identify $C(X)$ with the restriction of $C(Y)$ to the image of $f$. This restriction takes the form of a homomorphism $C(Y) \to C(X)$ whose kernel is the set of functions which are zero on $f(X)$, so it exhibits $C(X)$ as a quotient of $C(Y)$.

Question: Does every $\mathbb{R}$-algebra homomorphism $C(Y) \to C(X)$ come from a continuous function $X \to Y$?

## Extracting the diagonal

Suppose you are given a bivariate generating function $\displaystyle F(x, y) = \sum_{m, n \ge 0} f(m, n) x^m y^n$

in “closed form,” where I’ll be vague about what that means. Such a generating function may arise, for example, from counting lattice paths in $\mathbb{Z}_{\ge 0}^2$; then $f(m, n)$ might count the number of paths from $(0, 0)$ to $(m, n)$. If the path is only constrained by the fact that its steps must come from some set $S \subset \mathbb{Z}_{\ge 0}^2$ of steps containing only up or left steps, then we have the simple identity $\displaystyle F_S(x, y) = \frac{1}{1 - \sum_{(a, b) \in S} x^a y^b}$.

Question: When can we determine the generating function $\displaystyle D_F(x) = \sum_{n \ge 0} f(n, n) x^n$ in closed form?

I’d like to discuss an analytic approach to this question that gives concrete answers in at least a few important special cases, including the generating function for the central binomial coefficients, which is our motivating example.

Often in mathematics we work in an algebra with the property that the “degree” of an element has a multiplicative property. For example, in a polynomial ring in $n$ variables we can define the degree of a monomial to be the vector of its degrees with respect to each variable, and the product of monomials corresponds to the sum of vectors. More typically we can define the degree of a monomial to be its total degree (the sum of the components of the above vector); this degree is also multiplicative.
Algebras with this additional property are called graded algebras, and they show up surprisingly often in mathematics. As Alexandre Borovik notes, when schoolchildren work with units such as “apples” and “people” they are really working in a $\mathbb{Z}^n$-graded algebra, and one could argue that the study of homogeneous elements (that is, elements of the same degree) in $\mathbb{Z}^n$-graded algebras is the entire content of dimensional analysis.