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## The Picard groups

Let $R$ be a commutative ring. From $R$ we can construct the category $R\text{-Mod}$ of $R$-modules, which becomes a symmetric monoidal category when equipped with the tensor product of $R$-modules. Now, whenever we have a monoidal operation (for example, the multiplication on a ring), it’s interesting to look at the invertible things with respect to that operation (for example, the group of units of a ring). This suggests the following definition.

Definition: The Picard group $\text{Pic}(R)$ of $R$ is the group of isomorphism classes of $R$-modules which are invertible with respect to the tensor product.

By invertible we mean the following: for $L \in \text{Pic}(R)$ there exists some $L^{-1}$ such that the tensor product $L \otimes_R L^{-1}$ is isomorphic to the identity for the tensor product, namely $R$.

In this post we’ll meander through some facts about this Picard group as well as several variants, all of which capture various notions of line bundle on various kinds of spaces (where the above definition captures the notion of a line bundle on the affine scheme $\text{Spec } R$).

## The p-group fixed point theorem

The goal of this post is to collect a list of applications of the following theorem, which is perhaps the simplest example of a fixed point theorem.

Theorem: Let $G$ be a finite $p$-group acting on a finite set $X$. Let $X^G$ denote the subset of $X$ consisting of those elements fixed by $G$. Then $|X^G| \equiv |X| \bmod p$; in particular, if $p \nmid |X|$ then $G$ has a fixed point.

Although this theorem is an elementary exercise, it has a surprising number of fundamental corollaries.

## Operations, pro-objects, and Grothendieck’s Galois theory

Previously we looked at several examples of $n$-ary operations on concrete categories $(C, U)$. In every example except two, $U$ was a representable functor and $C$ had finite coproducts, which made determining the $n$-ary operations straightforward using the Yoneda lemma. The two examples where $U$ was not representable were commutative Banach algebras and commutative C*-algebras, and it is possible to construct many others. Without representability we can’t apply the Yoneda lemma, so it’s unclear how to determine the operations in these cases.

However, for both commutative Banach algebras and commutative C*-algebras, and in many other cases, there is a sense in which a sequence of objects approximates what the representing object of $U$ “ought” to be, except that it does not quite exist in the category $C$ itself. These objects will turn out to define a pro-object in $C$, and when $U$ is pro-representable in the sense that it’s described by a pro-object, we’ll attempt to describe $n$-ary operations $U^n \to U$ in terms of the pro-representing object.

The machinery developed here is relevant to understanding Grothendieck’s version of Galois theory, which among other things leads to the notion of étale fundamental group; we will briefly discuss this.

Previously we described $n$-ary operations on (the underlying sets of the objects of) a concrete category $(C, U)$, which we defined as the natural transformations $U^n \to U$.

Puzzle: What are the $n$-ary operations on finite groups?

Note that $U$ is not representable here. The next post will answer this question, but for those who don’t already know the answer it should make a nice puzzle.

## Connected objects and a reconstruction theorem

A common theme in mathematics is to replace the study of an object with the study of some category that can be built from that object. For example, we can

• replace the study of a group $G$ with the study of its category $G\text{-Rep}$ of linear representations,
• replace the study of a ring $R$ with the study of its category $R\text{-Mod}$ of $R$-modules,
• replace the study of a topological space $X$ with the study of its category $\text{Sh}(X)$ of sheaves,

and so forth. A general question to ask about this setup is whether or to what extent we can recover the original object from the category. For example, if $G$ is a finite group, then as a category, the only data that can be recovered from $G\text{-Rep}$ is the number of conjugacy classes of $G$, which is not much information about $G$. We get considerably more data if we also have the monoidal structure on $G\text{-Rep}$, which gives us the character table of $G$ (but contains a little more data than that, e.g. in the associators), but this is still not a complete invariant of $G$. It turns out that to recover $G$ we need the symmetric monoidal structure on $G\text{-Rep}$; this is a simple form of Tannaka reconstruction.

Today we will prove an even simpler reconstruction theorem.

Theorem: A group $G$ can be recovered from its category $G\text{-Set}$ of $G$-sets.

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

## 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 $1 \to U^{\ast} \otimes V^{\ast}$.

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.