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.
be a compact group and let 
denote the category of finite-dimensional unitary representations of 
. What is the multiplicity of the trivial representation in 
?
in the ring of cyclotomic integers ![\mathbb{Z}[\zeta_n]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BZ%7D%5B%5Czeta_n%5D&bg=ffffff&fg=333333&s=0&c=20201002)
depends only on the residue class of 
. This is suggestive enough of quadratic reciprocity that now would be a good time to give a full proof.
be an odd prime. Then ![\mathbb{Z}[\zeta_q]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BZ%7D%5B%5Czeta_q%5D&bg=ffffff&fg=333333&s=0&c=20201002)
contains 
.
is a compact Hausdorff space and 
is its ring of functions. From this definition it follows that 
is a 
of compact Hausdorff spaces to the category 
of 
-algebras (which we are assuming have identities). Explicitly, a continuous function
is sent to a continuous function 
. The contravariance may look weird if you’re not used to it, but it’s perfectly natural in the case that 
is an 
with the restriction of 
to the image of 
whose kernel is the set of functions which are zero on 
, so it exhibits 
?
; then 
might count the number of paths from 
to 
. If the path is only constrained by the fact that its steps must come from some set 
of steps containing only up or left steps, then we have the simple identity
.
in closed form?
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.
-graded algebra, and one could argue that the study of homogeneous elements (that is, elements of the same degree) in 
Annoying Precision 