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## Writing a blog post every day is hard and possibly not a good idea

So: I’m happy that I’ve kept up MaBloWriMo for 13 days so far, but I’m running out of steam. I’ve gone through essentially all of the posts in my backlog that were relatively easy to write, and the things I’d like to write about at this point either

• really should be done with diagrams (and it’s not easy to finish a blog post with diagrams in a day) or
• might take more time than I allot for blogging in a day to work through the relevant concepts.

Sticking to one post a day at this point is likely to drive down quality, so I think I am going to stop doing it. It was a good goal for awhile in that it got me to write some posts that I’d wanted to write for a long time now, but unfortunately it is now doing the opposite of that.

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

## Epi-mono factorizations

In many familiar categories, a morphism $f : a \to b$ admits a canonical factorization, which we will write

$a \xrightarrow{e} c \xrightarrow{m} b$,

as the composite of some kind of epimorphism $e$ and some kind of monomorphism $m$. Here we should think of $c$ as something like the image of $f$. This is most familiar, for example, in the case of $\text{Set}, \text{Grp}, \text{Ring}$, and other algebraic categories, where $c$ is the set-theoretic image of $f$ in the usual sense.

Today we will discuss some general properties of factorizations of a morphism into an epimorphism followed by a monomorphism, or epi-mono factorizations. The failure of such factorizations to be unique turns out to be closely related to the failure of epimorphisms or monomorphisms to be regular.

## 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.

## Small factors in random polynomials over a finite field

Previously I mentioned very briefly Granville’s The Anatomy of Integers and Permutations, which explores an analogy between prime factorizations of integers and cycle decompositions of permutations. Today’s post is a record of the observation that this analogy factors through an analogy to prime factorizations of polynomials over finite fields in the following sense.

Theorem: Let $q$ be a prime power, let $n$ be a positive integer, and consider the distribution of irreducible factors of degree $1, 2, ... k$ in a random monic polynomial of degree $n$ over $\mathbb{F}_q$. Then, as $q \to \infty$, this distribution is asymptotically the distribution of cycles of length $1, 2, ... k$ in a random permutation of $n$ elements.

One can even name what this random permutation ought to be: namely, it is the Frobenius map $x \mapsto x^q$ acting on the roots of a random polynomial $f$, whose cycles of length $k$ are precisely the factors of degree $k$ of $f$.

Combined with our previous result, we conclude that as $q, n \to \infty$ (with $q$ tending to infinity sufficiently quickly relative to $n$), the distribution of irreducible factors of degree $1, 2, ... k$ is asymptotically independent Poisson with parameters $1, \frac{1}{2}, ... \frac{1}{k}$.

## Short cycles in random permutations

Previously we showed that the distribution of fixed points of a random permutation of $n$ elements behaves asymptotically (in the limit as $n \to \infty$) like a Poisson random variable with parameter $\lambda = 1$. As it turns out, this generalizes to the following.

Theorem: As $n \to \infty$, the number of cycles of length $1, 2, ... k$ of a random permutation of $n$ elements are asymptotically independent Poisson with parameters $1, \frac{1}{2}, ... \frac{1}{k}$.

This is a fairly strong statement which essentially settles the asymptotic description of short cycles in random permutations.

## Groupoid cardinality

Suitably nice groupoids have a numerical invariant attached to them called groupoid cardinality. Groupoid cardinality is closely related to Euler characteristic and can be thought of as providing a notion of integration on groupoids.

There are various situations in mathematics where computing the size of a set is difficult but where that set has a natural groupoid structure and computing its groupoid cardinality turns out to be easier and give a nicer answer. In such situations the groupoid cardinality is also known as “mass,” e.g. in the Smith-Minkowski-Siegel mass formula for lattices. There are related situations in mathematics where one needs to describe a reasonable probability distribution on some class of objects and groupoid cardinality turns out to give the correct such distribution, e.g. the Cohen-Lenstra heuristics for class groups. We will not discuss these situations, but they should be strong evidence that groupoid cardinality is a natural invariant to consider.

## Fixed points of random permutations

The following two results are straightforward and reasonably well-known exercises in combinatorics:

1. The number of permutations on $n$ elements with no fixed points (derangements) is approximately $\frac{n!}{e}$.
2. The expected number of fixed points of a random permutation on $n$ elements is $1$.

As it turns out, it is possible to say substantially more about the distribution of fixed points of a random permutation. In fact, the following is true.

Theorem: As $n \to \infty$, the distribution of the number of fixed points of a random permutation on $n$ elements is asymptotically Poisson with rate $\lambda = 1$.

## String diagrams, duality, and trace

Previously we introduced string diagrams and saw that they were a convenient way to talk about tensor products, partial compositions of multilinear maps, and symmetries. But string diagrams really prove their use when augmented to talk about duality, which will be described topologically by bending input and output wires. In particular, we will be able to see topologically the sense in which the following four pieces of information are equivalent:

• A linear map $U \to V$,
• A linear map $U \otimes V^{\ast} \to 1$,
• A linear map $V^{\ast} \to U^{\ast}$,
• A linear map $1 \to U^{\ast} \otimes V$.

Using string diagrams we will also give a diagrammatic definition of the trace $\text{tr}(f)$ of an endomorphism $f : V \to V$ of a finite-dimensional vector space, as well as a diagrammatic proof of some of its basic properties.

Below all vector spaces are finite-dimensional and the composition convention from the previous post is still in effect.

Below the composition of a map $f : a \to b$ with a map $g : b \to c$ will be denoted $f \circ g : a \to c$ (rather than the more typical $g \circ f$). This will make it easier to translate between diagrams and non-diagrams. All diagrams were drawn in Paper.