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## The representation theory of the additive group scheme

In this post we’ll describe the representation theory of the additive group scheme $\mathbb{G}_a$ over a field $k$. The answer turns out to depend dramatically on whether or not $k$ has characteristic zero.

## Hecke operators are also relative positions

Continuing yesterday’s story about relative positions, let $G$ be a finite group and let $X$ and $Y$ be finite $G$-sets. Yesterday we showed that $G$-orbits on $X \times Y$ can be thought of as “atomic relative positions” of “$X$-figures” and “$Y$-figures” in some geometry with symmetry group $G$, and further that if $X \cong G/H$ and $Y \cong G/K$ are transitive $G$-sets then these can be identified with double cosets $H \backslash G / K$.

Representation theory provides another interpretation of $G$-orbits on $X \times Y$ as follows. First, if $\mathbb{C}[X]$ is any permutation representation, then the $G$-fixed points $\mathbb{C}[X]^G$ have a natural basis given by summing over $G$-orbits. (This is a mild categorification of Burnside’s lemma.) Next, consider the representations $\mathbb{C}[X], \mathbb{C}[Y]$. Because $\mathbb{C}[X]$ is self-dual, we have

$\displaystyle \text{Hom}_G(\mathbb{C}[X], \mathbb{C}[Y]) \cong (\mathbb{C}[X] \otimes \mathbb{C}[Y])^G \cong \mathbb{C}[X \times Y]^G$

and hence $\text{Hom}_G(\mathbb{C}[X], \mathbb{C}[Y])$ has a natural basis given by summing over $G$-orbits of the action on $X \times Y$.

Definition: The $G$-morphism $\mathbb{C}[X] \to \mathbb{C}[Y]$ associated to a $G$-orbit of $X \times Y$ via the above isomorphisms is the Hecke operator associated to the $G$-orbit (relative position, double coset).

Below the fold we’ll write down some details about how this works and see how we can use the idea that $G$-morphisms between permutations have a basis given by Hecke operators to work out, quickly and cleanly, how some permutation representations decompose into irreducibles. At the end we’ll state another puzzle.

## A transcript of my qualifying exam

I passed my qualifying exam last Friday. Here is a copy of the syllabus and a transcript.

Although I’m sure there are more, I’m only aware of two other students at Berkeley who’ve posted transcripts of their quals, namely Christopher Wong and Eric Peterson. It would be nice if more people did this.

## 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) $\mathfrak{u}(V), \text{U}(V)$.

## The double commutant theorem

Let $A$ be an abelian group and $T = \{ T_i : A \to A \}$ be a collection of endomorphisms of $A$. The commutant $T'$ of $T$ is the set of all endomorphisms of $A$ commuting with every element of $T$; symbolically,

$\displaystyle T' = \{ S \in \text{End}(A) : TS = ST \}$.

The commutant of $T$ is equal to the commutant of the subring of $\text{End}(A)$ generated by the $T_i$, so we may assume without loss of generality that $T$ is already such a subring. In that case, $T'$ is just the ring of endomorphisms of $A$ as a left $T$-module. The use of the term commutant instead can be thought of as emphasizing the role of $A$ and de-emphasizing the role of $T$.

The assignment $T \mapsto T'$ is a contravariant Galois connection on the lattice of subsets of $\text{End}(A)$, so the double commutant $T \mapsto T''$ may be thought of as a closure operator. Today we will prove a basic but important theorem about this operator.

## 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.

## The quaternions and Lie algebras II

We now know what a Lie algebra is and we know they are abstractions of infinitesimal symmetries, which are given by derivations. Today we will see what we can say about associating infinitesimal symmetries to continuous symmetries: that is, given a matrix Lie group $G$, we will describe its associated Lie algebra $\mathfrak{g}$ of infinitesimal elements and the exponential map $\mathfrak{g} \to G$ which promotes infinitesimal symmetries to real ones.

As in the other post, I will be ignoring some technical details for the sake of exposition. For example, I am generally not specifying how I’m topologizing various objects, and this is because of the general fact that a finite-dimensional real vector space has a unique Hausdorff topology compatible with addition and scalar multiplication. Whenever I talk about limits in such a vector space, I therefore don’t need to specify how I’m imposing a topology, although it will generally be convenient to induce it via a norm (which I am also not specifying).

Someone who has just read the previous post on how exponentiating quaternions gives a nice parameterization of $\text{SO}(3)$ might object as follows: “that’s nice and all, but there has to be a general version of this construction for more general Lie groups, right? You can’t always depend on the nice properties of division algebras.” And that someone would be right. Today we’ll begin to describe the appropriate generalization, the exponential map from a Lie algebra to its Lie group. To simplify the exposition, we’ll restrict to the case of matrix groups; that is, nice subgroups of $\text{GL}_n(\mathbb{F})$ for $\mathbb{F} = \mathbb{R}$ or $\mathbb{C}$, which will allow us to mostly avoid differential geometry.
For studying the hydrogen atom, our interest in Lie algebras comes from the following. If a Lie group $G$ acts smoothly on a smooth manifold $M$, its Lie algebra acts by differential operators on the space $C^{\infty}(M)$ of smooth functions, and these differential operators are the “infinitesimal generators” which give us conserved quantities for the evolution of a quantum system on $M$ (in the case that $G$ consists of symmetries of the Hamiltonian). Despite the fact that Lie algebras are commonly sold as a tool for understanding Lie groups, arguably in quantum mechanics the Lie algebra of symmetries of a Hamiltonian is more fundamental. This is important in sitations where the Lie algebra can sometimes exist without an associated Lie group.