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## Four flavors of Schur-Weyl duality

If $V$ is a finite-dimensional complex vector space, then the symmetric group $S_n$ naturally acts on the tensor power $V^{\otimes n}$ by permuting the factors. This action of $S_n$ commutes with the action of $\text{GL}(V)$, so all permutations $\sigma : V^{\otimes n} \to V^{\otimes n}$ are morphisms of $\text{GL}(V)$-representations. This defines a morphism $\mathbb{C}[S_n] \to \text{End}_{\text{GL}(V)}(V^{\otimes n})$, and a natural question to ask is whether this map is surjective.

Part of Schur-Weyl duality asserts that the answer is yes. The double commutant theorem plays an important role in the proof and also highlights an important corollary, namely that $V^{\otimes n}$ admits a canonical decomposition

$\displaystyle V^{\otimes n} = \bigoplus_{\lambda} V_{\lambda} \otimes S_{\lambda}$

where $\lambda$ runs over partitions, $V_{\lambda}$ are some irreducible representations of $\text{GL}(V)$, and $S_{\lambda}$ are the Specht modules, which describe all irreducible representations of $S_n$. This gives a fundamental relationship between the representation theories of the general linear and symmetric groups; in particular, the assignment $V \mapsto V_{\lambda}$ can be upgraded to a functor called a Schur functor, generalizing the construction of the exterior and symmetric products.

The proof below is more or less from Etingof’s notes on representation theory (Section 4.18). We will prove four versions of Schur-Weyl duality involving $\mathfrak{gl}(V), \text{GL}(V)$, and (in the special case that $V$ is a complex inner product space) $\mathfrak{u}(V), \text{U}(V)$.

## Noncommutative probability and group theory

There are, roughly speaking, two kinds of algebras that can be functorially constructed from a group $G$. The kind which is covariantly functorial is some variation on the group algebra $k[G]$, which is the free $k$-module on $G$ with multiplication inherited from the multiplication on $G$. The kind which is contravariantly functorial is some variation on the algebra $k^G$ of functions $G \to k$ with pointwise multiplication.

When $k = \mathbb{C}$ and when $G$ is respectively either a discrete group or a compact (Hausdorff) group, both of these algebras can naturally be endowed with the structure of a random algebra. In the case of $\mathbb{C}[G]$, the corresponding state is a noncommutative refinement of Plancherel measure on the irreducible representations of $G$, while in the case of $\mathbb{C}^G$, the corresponding state is by definition integration with respect to normalized Haar measure on $G$.

In general, some nontrivial analysis is necessary to show that the normalized Haar measure exists, but for compact groups equipped with a faithful finite-dimensional unitary representation $V$ it is possible to at least describe integration against Haar measure for a dense subalgebra of the algebra of class functions on $G$ using representation theory. This construction will in some sense explain why the category $\text{Rep}(G)$ of (finite-dimensional continuous unitary) representations of $G$ behaves like an inner product space (with $\text{Hom}(V, W)$ being analogous to the inner product); what it actually behaves like is a random algebra, namely the random algebra of class functions on $G$.

The Artin-Wedderburn theorem shows that the definition of a semisimple ring is enormously restrictive. Even $\mathbb{Z}$ fails to be semisimple! A less restrictive notion, but one that still captures the notion of a ring which can be understood by how it acts on simple (left) modules, is that of a semiprimitive or Jacobson semisimple ring, one with the property that every element $r \in R$ acts nontrivially in some simple (left) module $M$.

Said another way, let the Jacobson radical $J(R)$ of a ring consist of all elements of $r$ which act trivially on every simple module. By definition, this is an intersection of kernels of ring homomorphisms, hence a two-sided ideal. A ring $R$ is then semiprimitive if it has trivial Jacobson radical.

The goal of this post will be to discuss some basic properties of the Jacobson radical. I am again working mostly from Lam’s A first course in noncommutative rings.

## The representation theory of SU(2)

Today we will give four proofs of the classification of the (finite-dimensional complex continuous) irreducible representations of $\text{SU}(2)$ (which you’ll recall we assumed way back in this previous post). As a first step, it turns out that the finite-dimensional representation theory of compact groups looks a lot like the finite-dimensional representation theory of finite groups, and this will be a major boon to three of the proofs. The last proof will instead proceed by classifying irreducible representations of the Lie algebra $\mathfrak{su}(2)$.

At the end of the post we’ll briefly describe what we can conclude from all this about electrons orbiting a hydrogen atom.

## 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.