### representation theory

Apologies for the lack of posts here lately; I’ve been meaning to say many things that I simply have not gotten around to doing. I’ve been taking a course on infinite-dimensional Lie algebras this semester. There are a number of important examples here, most of which I’ve never seen before. This post will set down two of the most basic.

1. The Heisenberg algebra

The Heisenberg Lie algebra ${\mathcal{A}}$ is the Lie algebra with generators ${\left\{a_j, j \in \mathbb{Z}\right\}}$ and another generator ${K}$. The commutation relations are

$\displaystyle [a_j, a_k] = \begin{cases} 0 & \text{ if } j + k \neq 0 \\ j K & \text{ if } j = -k . \end{cases} ,$

and we require ${K}$ to be central. This is a graded Lie algebra with ${a_j}$ in degree ${j}$ and ${K}$ in degree zero.

The Heisenberg algebra is a simple example of a nilpotent Lie algebra: in fact, it has the property that its center contains the commutator subalgebra ${\mathbb{C} K}$.

The factor of ${j}$ in the relation for ${[a_j, a_k]}$ is of course a moot point, as we could choose a different basis so that the relation read ${[a_j, a_k] = \delta_{j, -k}}$. (The exception is ${a_0}$: that has to stay central.) However, there is a geometric interpretation of ${\mathcal{A}}$ with the current normalization. We have

$\displaystyle [a_j, a_k] = \mathrm{Res}( t^j d t^k)_{t = 0} K.$

Here ${\mathrm{Res}}$ denotes the residue of the differential form ${t^j dt^k = k t^j t^{k-1} dt}$ at ${t = 0}$. Since only terms of the form ${t^{-1} dt}$ contribute to the residue, this is easy to check.

As a result, we can think of as the Lie algebra of Laurent polynomials plus a one-dimensional space:

$\displaystyle \mathcal{A}= \mathbb{C}[t, t^{-1}] \oplus \mathbb{C}K$

where ${K}$ is central, and where the Lie bracket of Laurent polynomials ${f, g }$ is

$\displaystyle \mathrm{Res}_{t =0 } (f dg) K.$

Note that any exact form has residue zero, so ${\mathrm{Res}_{t = 0}(fdg) = -\mathrm{Res}_{t=0}(g df)}$ (by comparing with ${d(gf)}$). This explains the antisymmetry of the above form. (more…)

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A little earlier, we studied invariant theory for the general linear group ${GL(V)}$ for a finite-dimensional vector space ${V}$ over ${\mathbb{C}}$. We considered the canonical representation on ${V^{\otimes p} \otimes V^{* \otimes q}}$ and studied “invariant polynomials” on this space: that is, polynomials ${P: V^{\otimes p} \otimes V^{* \otimes q} \rightarrow \mathbb{C}}$ constant on orbits. We showed that these formed a finitely generated ${\mathbb{C}}$-algebra, and indeed gave a set of generators: these were given by pairing a factor of ${V}$ with a factor of ${V^*}$ with respect to the evaluation pairing. This is not, of course, a linear map, but it is a well-defined polynomial map of ${p}$ vector and ${q}$ covector variables.

1. Introduction

Now we want to consider a more general question. Let ${G}$ be an (affine) algebraic group over ${\mathbb{C}}$, acting on the finite-dimensional vector space ${V}$. We’d like to ask what the invariant polynomials on ${V}$ are, or in other words what is ${(\mathrm{Sym} V^*)^G}$. It was a Hilbert problem to show that this “ring of invariants” is finitely generated. The general answer turns out to be no, but we will show that it is the case when ${G}$ is reductive.

What is a reductive group? For our purposes, a reductive group over ${\mathbb{C}}$ is an algebraic group ${G}$ such that the category ${\mathrm{Rep}(G)}$ of (algebraic) finite-dimensional representations is semisimple. In other words, the analog of Maschke’s theorem is true for ${G}$. The “classical groups” (the general linear, special linear, orthogonal, and symplectic groups) are all reductive. There is a geometric definition (which works in characteristic ${p}$ too), but we will just take this semisimplicity as the definition.

The semisimplicity is quite a surprising phenomenon, because the method of proof of Maschke’s theorem—the averaging process—fails for reductive groups, which are never compact in the complex topology (as then they would not be affine varieties). However, it turns out that a reductive group ${G}$ over ${\mathbb{C}}$ contains a maximal compact Lie subgroup ${K}$ (which is not algebraic, e.g. the unitary group in ${GL_n}$), and the category of algebraic representations of ${G}$ is equivalent (in the natural way) to the category of continuous representations of ${K}$. Since the category of continuous representations is always semisimple (by the same averaging idea as in Maschke’s theorem, with a Haar measure on ${K}$), ${\mathrm{Rep}(K)}$ is clearly semisimple. But this is ${\mathrm{Rep}(G)}$.

Anyway, here’s what we wish to prove:

Theorem 1 Let ${G}$ be a reductive group over ${\mathbb{C}}$ acting on the finite-dimensional vector space ${V}$. Then the algebra of invariant polynomials on ${V}$ is finitely generated. (more…)

With the semester about to start, I have been trying to catch up on more classical material. In this post, I’d like to discuss a foundational result on the ring of invariants of the general linear group acting on polynomial rings: that is, a description of generators for the ring of invariants.

1. The Aronhold method

Let ${G}$ be a group acting on a finite-dimensional vector space ${V}$ over an algebraically closed field ${k}$ of characteristic zero. We are interested in studying the invariants of the ring of polynomial functions on ${V}$. That is, we consider the algebra ${\mathrm{Sym} V^*}$, which has a natural ${G}$-action, and the subalgebra ${(\mathrm{Sym} V^*)^G}$. Clearly, we can reduce to considering homogeneous polynomials, because the action of ${G}$ on polynomials preserves degree.

Proposition 1 (Aronhold method) There is a natural ${G}$-isomorphism between homogeneous polynomial functions of degree ${m}$ on ${V}$ and symmetric, multilinear maps ${V \times \dots \times V \rightarrow k}$ (where there are ${m}$ factors).

Proof: It is clear that, given a multilinear, symmetric map ${g: V \times \dots \times V \rightarrow k}$, we can get a homogeneous polynomial of degree ${m}$ on ${V}$ via ${v \mapsto g(v, v, \dots, v)}$ by the diagonal imbedding. The inverse operation is called polarization. I don’t much feel like writing out, so here’s a hand-wavy argument.

Or we can think of it more functorially. Symmetric, multilinear maps ${V \times \dots \times V \rightarrow k}$ are the same thing as symmetric ${k}$-linear maps ${V^{\otimes m} \rightarrow k}$; these are naturally identified with maps ${\mathrm{Sym}^m V \rightarrow k}$. So what this proposition amounts to saying is that we have a natural isomorphism

$\displaystyle \mathrm{Sym}^m V^* \simeq (\mathrm{Sym}^m V)^*.$

But this is eminently reasonable, since there is a functorial isomorphism ${(V^{\otimes m})^* \simeq (V^{*})^{\otimes m}}$ functorially, and replacing with the symmetric algebra can be interpreted either as taking invariants or coinvariants for the symmetric group action. Now, if we are given the ${G}$-action on ${V}$, one can check that the polarization and diagonal imbeddings are ${G}$-equivariant. $\Box$

2. Schur-Weyl duality

Let ${V}$ be a vector space. Now we take ${G = GL(V)}$ acting on a tensor power ${V^{\otimes m}}$; this is the ${m}$th tensor power of the tautological representation on ${V}$. However, we have on ${V^{\otimes m}}$ not only the natural action of ${GL(V)}$, but also the action of ${S_m}$, given by permuting the factors. These in fact commute with each other, since ${GL(V)}$ acts by operators of the form ${ A\otimes A \otimes \dots \otimes A}$ and ${S_m}$ acts by permuting the factors.

Now the representations of these two groups ${GL(V)}$ and ${S_m}$ on ${V^{\otimes m}}$ are both semisimple. For ${S_m}$, it is because the group is finite, and we can invoke Maschke’s theorem. For ${GL(V)}$, it is because the group is reductive, although we won’t need this fact. In fact, the two representations are complementary to each other in some sense.

Proposition 2 Let ${A \subset \mathrm{End}(V^{\otimes m})}$ be the algebra generated by ${GL(V)}$, and let ${B \subset \mathrm{End}(V^{\otimes m})}$ be the subalgebra generated by ${S_m}$. Then ${A, B}$ are the centralizers of each other in the endomorphism algebra. (more…)

[Minor corrections made, 6/21]

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.

We have now discussed some of the basic properties of Deligne’s categories ${\mathrm{Rep}(S_t)}$, and some of the rich structure that they have. It turns out, as I have already mentioned, that Deligne did the same for representation categories of the other classical groups.

Knop described how to do it for the wreath products, obtaining categories ${\mathrm{Rep}(S_t \ltimes G^t)}$ for ${t \in \mathbb{C}}$; here the central object is the “standard representation” ${\mathfrak{h}_G}$ of ${G}$-invariant functions ${G \rightarrow \mathbb{C}oprod_n G}$, which has a natural action of ${S_n \rtimes G^n}$. The representation ${\mathfrak{h}_G}$ is faithful, and again one uses its tensor powers and a combinatorial parametrization of its morphisms to interpolate. For the details in much more generality, see Knop’s paper; he actually constructs tensor categories via the calculus of relations out of arbitrary “regular categories.” (My paper has a brief exposition of how things play out in the special case of ${\mathrm{Rep}(S_t \ltimes G^t)}$.) These categories, like Deligne’s, are semisimple symmetric tensor categories.

It turns out, however, that many families of algebraic objects of interest in representation theory depend on a parameter ${n \in \mathbb{Z}_{\geq 0}}$, and are built out of the corresponding (i.e., depending on ${n}$) classical groups (i.e. symmetric, orthogonal, etc.). One example is the family of algebras ${S_n \ltimes A^{\otimes n}}$ for ${A}$ an associative algebra. This is a rather simple one; a more complicated one is given by the family of Hecke algebras. The additional relations and generators corresponding to the part of these objects not in the classical groups can, however, often be stated in a uniform, categorical manner independent of ${n}$.

Using this, Etingof proposed a program of studying the representation categories of these objects in complex rank, which he constructed out of Deligne’s categories. I will briefly explain what this is all about. Consider the example of the family of semidirect product algebras; it’s simpler than what Etingof focuses on, but I’d be horrendously unqualified to really say anything about any of them. (more…)

[This post, a continuation of the series on representation theory in complex rank, discusses the irreducibles in Deligne’s category $\mathrm{Rep}(S_t)$ for $t \notin \mathbb{Z}_{\geq 0}$ and what one can do with them.]

OK, so we now know that Deligne’s categories ${\mathrm{Rep}(S_t)}$ are semisimple when ${t \notin \mathbb{Z}_{\geq 0}}$. But, this is a paradox. Deligne’s categories, a family of categories constructed to interpolate the semisimple categories of representations of ${S_n, n \in \mathbb{Z}_{\geq 0}}$ are semisimple precisely at the complement of the nonnegative integers!

The problem is, when ${t \in \mathbb{Z}_{\geq 0}}$, ${\mathrm{Rep}(S_t)}$ is not equivalent to the ordinary category ${\mathrm{Rep}^{\mathrm{ord}}(S_t)}$. The problem is that not all relations correspond to actual morphisms. Deligne in fact shows that the ordinary category can be obtained as a quotient of his ${\mathrm{Rep}(S_t)}$ (via the tensor ideal of “neglligible morphisms”) but this isn’t really important for the story I’m telling.

1. Motivation and remarks

Today, I want to talk about what the simple objects in ${\mathrm{Rep}(S_t), t \notin \mathbb{Z}_{\geq 0}}$, look like. We know what the simple ${S_n}$-representations are; they are the Specht modules, parametrized by the Young diagrams of size ${n}$. It turns out that the simple objects in ${\mathrm{Rep}(S_t)}$ are parametrized by the Young diagrams of arbitrary size. There is an interesting way of thinking about this that Etingof explains in his talk, and which I will try to motivate here now.

OK. So, just as we defined a filtration on Deligne’s categories yesterday, let’s define a filtration on the ordinary representation categories ${\mathrm{Rep}^{\mathrm{ord}}(S_n), n \in \mathbb{Z}_{\geq 0}}$. Namely, we let ${\mathrm{Rep}^{\mathrm{ord}}(S_n)^{(N)}}$ denote the category generated by ${\mathfrak{h}^{\otimes p}, p \leq N}$ for ${\mathfrak{h}}$ the regular representation. When ${N}$ is large enough, this becomes the full category, so we will always pretend that ${n}$ is really really large relative to ${N}$ (which is kinda ironic when you think about the notation…).

Anyhow, we want to look at the simple objects in ${\mathrm{Rep}^{\mathrm{ord}}(S_n)^{(N)}}$. Well, these are going to have to correspond to some Young diagrmas of size ${n}$, but the question is which ones?

I claim that the Young diagrams that arise are precisely those where the rows below the top have ${\leq N}$ boxes.

In particular, as ${n}$ gets large, the top row must get really long, but the number of simple objects stays bounded. (more…)

This post continues my series on representation theory in complex rank, begun here with a discussion of Deligne’s interpolation of the representation categories of the symmetric group, introduced in his 2004 paper.

Semisimplicity is the basic structure theorem for Deligne’s categories, and I would be extremely remiss in my discussion of representation theory in complex rank if I did not say something about it.

So, let’s review. In the first post, I explained and motivated the definition of Deligne’s categories ${\mathrm{Rep}(S_t)}$. Incidentally, Deligne did the same for the other classical groups, i.e. ${GL_n, O_n, Sp_{2n}}$, but I shall not discuss them. The categories ${\mathrm{Rep}(S_t)}$, are defined as the pseudo-abelian envelope of the ${\mathbb{C}}$-linear category generated by objects ${\mathfrak{h}^{\otimes p}}$, where the hom-spaces ${\hom( \mathfrak{h}^{\otimes p}, \mathfrak{h}^{\otimes r})}$ are free on the equivalence relations on ${\mathbf{ p+r}}$, and composition is given by a combinatorial expression which is polynomial in the rank ${t}$ (hence interpolable).

Now, we just have an abstract category with formal objects and morphisms corresponding in no obvious way to anything concrete. To prove it is semisimple, we cannot use therefore techniques such as those in the proof of Maschke’s theorem of Weyl’s complete reducibility theorem.

But we can do it by appealing to what I discussed in the second post of this series: by proving that the endomorphism rings are semisimple and the category is nonnilpotent. In fact, since direct products and factor rings of semisimple rings are semisimple, we only need to prove that the algebras ${\hom_{\mathrm{Rep}(S_t)}(\mathfrak{h}^{\otimes p}, \mathfrak{h}^{\otimes p})}$ are semisimple (in addition to nonnilpotence). This endomorphism ring (depending on the size ${p}$ and the rank ${t}$) is an important object, called the partition algebra, and you can look it up e.g. here. But I don’t know how to prove directly that the partition algebra is semisimple. So I will follow Deligne (and Knop) in the (inductive) proof (which will also imply semisimplicity of the partition algebra).

I will do this in two steps. First, I will use a little bit of combinatorics to show that when ${t \notin \mathbb{Z}_{\geq 0}}$, the category ${\mathrm{Rep}(S_t)}$ is nonnilpotent. Next, I will use this to prove semisimplicity.

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