I’d now like to discuss my paper “Categories parametrized by schemes and representation theory in complex rank.” My RSI project was rather-open ended: to investigate the categories of representation theory in complex rank. Pavel Etingof told me that it would be expected that they would behave similar in some ways to the integral case (at least if “there was justice in the world”). For instance, we know that Deligne’s ${\mathrm{Rep}(S_t)}$ has a comibnatorial parametrization of simple objects similar to the classical case. However, as I discovered when I got there, I don’t actually know representation theory. I had looked through some material on finite groups, and knew (in outline, not usually proofs) the basic facts about the symmetric group. I certainly didn’t know anything about Hecke algebras (the literature of which seems rather inaccessible to beginners), and I don’t think I could define a semisimple Lie algebra. Anyway, so what I did was therefore was the easy case: representation theory in transcendental rank. I sort of ended up stumbling into this by accident, so I’ll try to reconstruct the story below, somewhat. I apologize in advance to readers that know algebraic geometry and will probably find this post rather slow-moving (it’s really addressed with a younger version of myself in mind). Readers may wish, however, to review my earlier posts on this topic.

1. A toy example

To motivate this with something very well-known and elementary, let’s fix ${m,n}$ and say we are considering a “family ${M(t)}$ of ${m}$-by-${n}$ matrices depending polynomially on a parameter ${t \in \mathbb{C}}$, which is to say the ${i,j}$-th entry of ${M(t)}$ is a polynomial in ${t}$, say with integral coefficients. Then the following conditions are equivalent:

1. The rank of ${M(t)}$ for ${t}$ transcendental is ${k}$

2. The rank of ${M(T)}$ for ${T}$ indeterminate is ${k}$

3. The rank of ${M(r)}$ is ${k}$ for almost all integers ${r}$

Why are they equivalent? This is easy to see. Suppose 2. Put the matrix ${M(T)}$ for ${T}$ indeterminate in reduced row-echelon form ${N(T)}$ by row operations in ${\mathbb{Q}(T)}$. Wlog, ${N(T)}$ (by multiplying by some polynomial) has entries in ${\mathbb{C}[T]}$. Then ${N(T)}$ has rank ${k}$. The matrix ${N(r)}$ has rank ${k}$ whenever the leading terms don’t vanish, which is true for almost all ${r}$ by the trivial observation that a polynomial in one variable can have only finitely many roots if it is nonzero. Taking into account the row operations used to transform ${M}$ into ${N}$, and whenever the corresponding rational functions used are nonzero, we must have ${\mathrm{rank} M(r) = \mathrm{rank} N(r)}$. So ${\mathrm{rank} M(r) = k}$ for almost all ${r}$.

So, let’s suppose we are given a “family” of maps ${U(t): V \rightarrow W}$ for ${V,W}$ vector spaces, which depend polynomially on ${t}$ (in an obvious sense) with coefficients in ${\mathbb{Q}[T]}$ (in an appropriate choice of bases). Then if we know that ${U(n)}$ is injective (resp. surjective) at almost all ${n}$ (“the classical case,” which may already be known), the same is true in transcendental rank.

2. Categories parametrized by schemes

Ok, so the next step is to note that Deligne’s categories can be viewed as “categories depending on a parameter.” Why is this? Well, let’s look again at the definition of an object in ${\mathrm{Rep}(S_t)}$. The object is a pair ${[U_1] \oplus [U_2] \oplus \dots \oplus [U_k], e}$, where each ${U_i}$ is a finite set and ${e}$ is an idempotent matrix of recollements. Since composition is given by a law polynomial in ${t}$, while recollements don’t depend on ${t}$, the ${e}$‘s associated prescribed

3. Making it precise

So, around the end of last summer, I showed this to my two mentors, and explained to them how one could deduce interesting results from it. But it doesn’t feel appropriately Bourbakianized. The idea of a “category depending on a parameter” is not very precise, and it’s unclear what we can do with it.

It took a fair bit of soul-searching, but this is the definition used in the paper. Namely, one starts with a ground scheme ${S}$ of parameters, and considers a category internal to the category of ${S}$-schemes; for convenience we shall call this an ${S}$-category or category over ${S}$.

What does this mean? Well, normally, in a (small) category ${\mathbb{C}al{C}}$, we have a set ${Ob}$ of objects, and a set ${Mor}$ of morphisms. There is a map ${Mor \rightarrow Ob \times Ob}$ that sends each morphism to a domain adn codomain, and there is a function called composition ${Mor \times_{Ob} Mor \rightarrow Mor}$. Here the fibered product is used to indicate that the domain of the second map is the codomain of the first. Similarly, there must be a map ${Ob\rightarrow Mor}$ sending each object to the corresponding identity morphism.

This is just the definition of a category, and most readers of this blog probably know it already. Why am I restating it? Because the definition is entirely arrow-theoretic. I’ve talked about “sets” of objects and morphisms, and “functions” on these sets. But this makes sense in any category (where fibered products exist, at least). Instead of an object set, you have an object object ${Ob}$ (not a typo). You also have a morphism object ${Mor}$ in this category, and a morphism ${Mor \times_{Ob} Mor \rightarrow Mor}$ In this way, you can talk about an internal category.

Well, what does all this have to do with families of categories? The point is that an ${S}$-scheme ${X}$ (with morphism ${X \rightarrow S}$) is really a family of schemes ${X_s, s \in S}$, obtained by taking the fiber.

So, similarly, if we have an ${S}$-category ${\mathfrak{C}}$ (which is the collection of an object-scheme, morphism-scheme, composition-morphism), we can base change by the maps ${\mathrm{Spec} \ k(s) \rightarrow S}$ to get ${\mathrm{Spec} \ k(s)}$ categories ${\mathfrak{C}_s}$ for each ${s \in S}$. What’s more, if we have a ${T}$-category for any scheme ${T}$, the ${T}$-valued points form an actual category . So we can look at ${k(s)}$-valued points in ${\mathfrak{C}_s}$ to get from this an actual category ${\mathbb{C}al{C}_s}$.

However, it’s more convenient to consider algebraically closed fields. The reason is that the ${k}$-valued points in a ${k}$-scheme (say of finite type) are not necessarily dense, but they are if ${k}$ is algebraically closed; this is essentially a version of the Hilbert Nullstellensatz.

\renewcommand{\b}[1]{\overline{#1}} So, we have to modify what was just said. Given a ${S}$-category ${\mathfrak{C}}$, we consider it as a family of categories ${\mathbb{C}al{C}_{\b{s}}}$ obtained by base-changing by ${\mathrm{Spec} \ \overline{k(s)}}$ and taking ${\b{k(s)}}$ points.

Here is a basic example. Let ${A}$ be a finitely generated commutative algebra over ${\mathbb{C}[T]}$. (I think commutativity is actually unnecessary, but it’ll make things easier for now.) Then ${A}$ as a ${\mathbb{C}[T]}$-algebra is given by generators ${g_1, \dots, g_k}$ and certain relations ${P_i(T, g_1, \dots, g_k)}$ between them, where the ${P_i}$ are polynomials.

For ${t \in \mathbb{C}}$, consider the algebra ${A_t = A \otimes_{\mathbb{C}[T]} \mathbb{C}}$, where ${\mathbb{C}}$ denotes the module where ${T}$ acts by multiplication by ${t}$. An ${n}$-dimensional representation of ${A_t}$ is the same thing as a choice of ${k}$ commuting ${n}$-by-${n}$ matrices ${M_1, \dots, M_k}$ satisfying ${P_i(t, M_1, \dots, M_k) = 0}$. A morphism between two such collections ${\{M_i, N_i\}}$ is an ${n}$-by-${n}$ matrix ${P}$ that satisfies ${PM_i = N_i P}$ for all ${i}$.

These are polynomial conditions in the matrix coefficients of the ${M_i}$ that depend polynomially on the rank ${t}$. In particular, there is a category ${\mathfrak{C}_{A,n}}$ over ${\mathbb{A}_{\mathbb{C}}^1}$ such that ${\mathbb{C}al{C}_t}$ for ${t \in \mathbb{C}}$ is the category of ${n}$-dimensional representations of ${A}$. We can do the same for ${\mathfrak{C}_{A, \leq n}}$, for representations of dimension ${\leq n}$.

So, that’s all fine, but there is a better example. Deligne’s categories fit into this framework too. There is a category ${\mathfrak{R}ep(S)}$ over ${\mathbb{A}_{\mathbb{C}}^1}$ whose fibers at ${t \in \mathbb{C}}$ are just the Deligne categories ${\mathrm{Rep}(S_t)}$. This is precisely the meaning of the earlier informal remarks that Deligne’s categories “depend polynomially on ${t}$.”

4. Constructibility

So, ok. We have our definition that encapsulates the notion of a family of categories; we need to use this to prove something.

The first part of my paper looks at how the category-theoretic properties of a family of categories ${\mathbb{C}al{C}_{\b{s}}}$ arising from a ${S}$-category ${\mathfrak{C}}$ are actually described by constructible sets, assuming that ${S}$ is noetherian and all schemes in question are of finite type.

Proposition 1 If ${Mor}$ is the morphism-scheme of an ${S}$-category ${\mathfrak{C}}$ of finite type, and ${S}$ is noetherian, then there exists a constructible set ${Mon \subset Mor}$ whose fibers ${Mor_{\b{s}}}$ correspond to the monomorphisms in ${\mathbb{C}al{C}_{\b{s}}}$, for all ${s \in S}$.

Here is an explanation for why this is true. The property of being a monomorphism in a category is definable via first-order logic. ${f}$ is a monomorphism if whenever ${g,h}$ are two morphisms with the same domain and codomain the domain of ${f}$, then ${f \circ g = f\circ h}$ implies ${g = h}$.

But the properties in affine space definable by first-order logic are precisely the constructible ones. This is basically Chevalley’s theorem. This is because adding existential quantifiers corresponds to taking projections down to a subset of coordinates. So it is not that surprising that the monomorphisms are constructible. Granted, this doesn’t quite work for arbitrary noetherian schemes, but the principle is the same (and one just uses Chevalley’s theorem).

Let’s look at what this states for the ${\mathbb{A}_{\mathbb{C}}^1}$-category ${\mathfrak{C}_{A,\leq n}}$; it states that the monomorphisms are described by a constructible set. But this constructible set can be described directly: it is the set of matrices that are admissible morphisms (i.e. commute with the action of ${A_t)}$ and have a nonvanishing minor of the right size.

Things are more difficult for Deligne’s category. Well, first of all, it isn’t finite type over ${\mathbb{A}_{\mathbb{C}}^1}$. Hence, we construct subcategories ${\mathfrak{R}ep(S)^{(N,N)}}$ where the sets ${U_1, \dots, U_k}$ are restricted to have cardinality at most ${N}$, and ${k \leq N}$ as well. There are only finitely many possibilities for the series ${U_1, \dots, U_k}$ under these restrictions, so we have finite type in this case. In particular, the monomorphisms in this category are described by constructible sets. But it is not clear a priori what they are, since these are not concrete categories.

That proposition was one example. There are others necessary:

Proposition 2 The properties of universality, finte limits and/or colimits, and simplicity are describable by constructible sets under noetherian and finite-type hypotheses.

That’s nice. So here’s an example of how we could use it. Let’s say we have a finite-type ${S}$-category ${\mathfrak{C}}$, over a noetherian scheme ${S}$, with an internal additive structure, and we want to prove that ${\mathbb{C}al{C}_{\b{s}}}$ admits kernels for generic ${s}$. If we know that kernels exist for a Zariski dense subset of parameters (e.g., large integers in ${\mathbb{A}_{\mathbb{C}}^1}$), then I claim they exist for ${s}$ in a dense open subset.

Indeed, there is a constructible set ${Ker \subset Mor \times Mor}$ that parametrizes pairs of morphisms ${(f,g)}$ where ${f}$ is the kernel of ${g}$, essentially by the above proposition for limits.

The projection of ${Ker}$ onto the second factor is surjective when ${s}$ belongs to teh aforementioned Zariski dense subset, by assumption. In particular, if we consider ${C:= Mor - p_2(Ker)}$, then ${C_{\b{s}}}$ is empty for ${s}$ in a Zariski dense subset. Hence this must be true generically; as a result, kernels exist for generic ${s}$.

So, how does this apply to representation theory in complex rank? Well, a lot of properties of representation theory can be stated categorically. For instance, consider the semidirect product ${S_n \rtimes \mathbb{C}[x_1, \dots, x_n]}$; the simple objects are parametrized by a partition of ${n = n_1 + \dots + n_k}$ and an assignment of a complex number ${c_i}$ to each block ${n_i}$. The simple object correspnonding to this is obtained via a well-known “parabolic induction” process.

I claim that the simple objects in ${\mathrm{Rep}(S_t \rtimes \mathbb{C}[x]^{\otimes t})}$ have to be obtainable generically by a similar process. The first step is to interpolate this parabolic induction business into complex rank, so that one has objects corresponding to a “partition” ${t = (t-N) + n_2 + \dots + n_k}$ (where ${t-N}$ corresponds to a “long top row”) and ${c_1, \dots, c_k}$ that can be viewed as depending polynomially on ${t, c_1, \dots, c_k}$. The images of these form a constructible set, and since the fibers of this constructible set contain the fibers of the constructible set ${Sim}$ parametrizing simple objects for a Zariski dense subset of parameters, the same holds generically—in particular, for transcendental ${t}$.

So, in outline, this is the sort of thing that happens in my paper. The case of semidirect products is a little easier than the first case I investigate, which is the degenerate affine Hecke algebra. This is the technique used, and it provides generic information about representation theory in complex rank.

There is one more part of my paper that I thought I should mention. Remember how Knop introduced categories ${\mathrm{Rep}(S_t \ltimes G^t)}$ for ${G}$ a finite group, but we also have Etingof categories ${\mathrm{Rep}(S_t \ltimes \mathbb{C}[G]^{\otimes t})}$? I used the same techniques to prove them equivalent (in fact, tensor equivalent—I introduced a tensor structure on the Etingof categories) for transcendental ${t}$.

So yeah, that’s about it. The really hard stuff is to see what happens at algebraic ${t}$. Someone who knows (a lot) more about representation theory than I do should look into it.