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## SU(2) and the quaternions

The simplest compact Lie group is the circle $S^1 \cong \text{SO}(2)$. Part of the reason it is so simple to understand is that Euler’s formula gives an extremely nice parameterization $e^{ix} = \cos x + i \sin x$ of its elements, showing that it can be understood either in terms of the group of elements of norm $1$ in $\mathbb{C}$ (that is, the unitary group $\text{U}(1)$) or the imaginary subspace of $\mathbb{C}$.

The compact Lie group we are currently interested in is the $3$-sphere $S^3 \cong \text{SU}(2)$. It turns out that there is a picture completely analogous to the picture above, but with $\mathbb{C}$ replaced by the quaternions $\mathbb{H}$: that is, $\text{SU}(2)$ is isomorphic to the group of elements of norm $1$ in $\mathbb{H}$ (that is, the symplectic group $\text{Sp}(1)$), and there is an exponential map from the imaginary subspace of $\mathbb{H}$ to this group. Composing with the double cover $\text{SU}(2) \to \text{SO}(3)$ lets us handle elements of $\text{SO}(3)$ almost as easily as we handle elements of $\text{SO}(2)$.

## SO(3) and SU(2)

In order to study the hydrogen atom, we’ll need to know something about the representation theory of the special orthogonal group $\text{SO}(3)$. This post consists of a few preliminaries along the way to doing this. I’ll be somewhat vague about a few things that 1) I don’t have much experience with, and 2) that would detract from the main narrative anyway.

## 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.

## Lattice paths and the quadratic coefficient of Kazhdan-Lusztig polynomials

SPUR is finally over! Instead of continuing my series of blog posts, I thought I’d just link to my paper, Lattice paths and the quadratic coefficient of Kazhdan-Lusztig polynomials, and the first few blog posts should more or less provide enough background to read it.

My project ended up changing direction. The formula I was working with for the quadratic coefficient was so unwieldy that I ended up spending the whole time trying to simplify it, and instead of saying anything about non-negativity I ended up saying something about combinatorial invariance. The combinatorial invariance conjecture, which goes back to Lusztig and, independently, Dyer, says that the Kazhdan-Lusztig polynomial $P_{u,v}(q)$ depends only on the poset structure of $[u, v]$. In the special case that $u = e$ this was proven in 2006 by Brenti, Caselli, and Marietti. However, the conjecture is still open in general.

In particular, explicit nonrecursive formulas in which each term only depends on poset-theoretic data are not known in general. They are known in the case that the length $\ell(u, v)$ of the interval $[u, v]$ is less than or equal to $4$, and there is also such a formula for the coefficient of $q$ of $P_{e,v}(q)$ where $e$ is the identity. The main result of the paper is a formula for the coefficient of $q^2$ of $P_{u,v}(q)$ in which all but three of the terms depend only on poset data, which is a simplification of a general formula due to Brenti for $P_{u,v}(q)$ in terms of lattice paths. It reduces to

• a formula for the coefficient of $q^2$ of $P_{e,v}(q)$ in which all but one of the terms depends only on poset data,
• a formula for the coefficient of $q^2$ of $P_{u,v}(q)$ where $\ell(u, v) = 5$ in which all but one of the terms (but a different term) depends only on poset data (not in the paper), and
• a formula for the coefficient of $q^2$ of $P_{e,v}(q)$ where $\ell(u, v) = 6$ in which every term depends only on poset data.

I believe these formulas are known in some form, but the method of proof is likely to be novel. In any case, the troublesome terms in the above are all essentially coefficients of R-polynomials. If I revisit this project in the future, I will be focusing my attention on these coefficients, and my goal will be to find a poset-theoretic formula for $P_{u,v}(q)$ in the length $5$ case, the smallest-length case where (to my knowledge) combinatorial invariance is open.

## Hecke algebras and the Kazhdan-Lusztig polynomials

The Hecke algebra attached to a Coxeter system $(W, S)$ is a deformation of the group algebra of $W$ defined as follows. Take the free $\mathbb{Z}[q^{ \frac{1}{2} }, q^{ - \frac{1}{2} }]$-module $\mathcal{H}_W$ with basis $T_w, w \in W$, and impose the multiplicative relations

$T_w T_s = T_{ws}$

if $\ell(sw) > \ell(w)$, and

$T_w T_s = q T_{ws} + (q - 1) T_w$

otherwise. (For now, ignore the square root of $q$.) Humphreys proves that these relations describe a unique associative algebra structure on $\mathcal{H}_W$ with $T_e$ as the identity, but the proof is somewhat unenlightening, so I will skip it. (Actually, the only purpose of this post is to motivate the definition of the Kazhdan-Lusztig polynomials, so I’ll be referencing the proofs in Humphreys rather than giving them.)

The motivation behind this definition is a somewhat long story. When $W$ is the Weyl group of an algebraic group $G$ with Borel subgroup $B$, the above relations describe the algebra of functions on $G(\mathbb{F}_q)$ which are bi-invariant with respect to the left and right actions of $B(\mathbb{F}_q)$ under a convolution product. The representation theory of the Hecke algebra is an important tool in understanding the representation theory of the group $G$, and more general Hecke algebras play a similar role; see, for example MO question #4547 and this Secret Blogging Seminar post. For example, replacing $G$ and $B$ with $\text{SL}_2(\mathbb{Q})$ and $\text{SL}_2(\mathbb{Z})$ gives the Hecke operators in the theory of modular forms.

## Coxeter groups

At SPUR this summer I’ll be working on the Kazhdan-Lusztig polynomials, although my mentor and I haven’t quite pinned down what problem I’m working on. I thought I’d take the chance to share some interesting mathematics and also to write up some background for my own benefit. I’ll mostly be following the second half of Humphreys.

A Coxeter system $(W, S)$ is a group $W$ together with a generating set $S$ and presentation of the form

$\langle s_1, ... s_n | (s_i s_j)^{m(i, j)} = 1 \rangle$

where $m(i, j) = m(j, i), m(i, i) = 1$, and $m(i, j) \ge 2, i \neq j$. (When there is no relation between $s_i, s_j$, we write this as $m(i, j) = \infty$.) The group $W$ is a Coxeter group, and is usually understood to come with a preferred presentation, so we will often abuse terminology and use “group” and “system” interchangeably. $S$ is also referred to as the set of simple reflections in $W$, and $n$ the rank. (We will only consider finitely-generated Coxeter groups.)

Historically, Coxeter groups arose as symmetry groups of regular polytopes and as Weyl groups associated to root systems, which in turn are associated to Lie groups, Lie algebras, and/or algebraic groups; the former are very important in understanding the latter. John Armstrong over at the Unapologetic Mathematician has a series on root systems. In addition, for a non-technical overview of Coxeter groups and $q$-analogues, I recommend John Baez’s week184 through week187. The slogan you should remember is that Weyl groups are “simple algebraic groups over $\mathbb{F}_1$.”

## 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 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.

## 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 $V^{\otimes n}$?

## The Jacobi-Trudi identities

Today I’d like to introduce a totally explicit combinatorial definition of the Schur functions. Let $\lambda \vdash n$ be a partition. A semistandard Young tableau $T$ of shape $\lambda$ is a filling of the Young diagram of $\lambda$ with positive integers that are weakly increasing along rows and strictly increasing along columns. The weight of a tableau $T$ is defined as $\mathbf{x}^T = x_1^{T_1} x_2^{T_2} ...$ where $T_i$ is the total number of times $i$ appears in the tableau.

Definition 4: $\displaystyle s_{\lambda}(x_1, x_2, ...) = \sum_T \mathbf{x}^T$

where the sum is taken over all semistandard Young tableaux of shape $\lambda$.

As before we can readily verify that $s_{(k)} = h_k, s_{(1^k)} = e_k$. This definition will allow us to deduce the Jacobi-Trudi identities for the Schur functions, which describe among other things the action of the fundamental involution $\omega$. Since I’m trying to emphasize how many different ways there are to define the Schur functions, I’ll call these definitions instead of propositions.

Definition 5: $\displaystyle s_{\lambda}= \det(h_{\lambda_i+j-i})_{1 \le i, j \le n}$.

Definition 6: $\displaystyle s_{\lambda'} = \det(e_{\lambda_i+j-i})_{1 \le i, j \le n}$.