Stable representation theory:
beyond the classical groups

We explain our main ideas and results in the setting of symmetric trilinear forms over the complex numbers to keep the exposition simple. We work more generally in the body of the paper.

Let $X_{n}=\operatorname{Sym}^{3}(\mathbf{C}^{n})^{*}$ be the space of symmetric trilinear forms (i.e., cubic polynomials) in $n$ variables. Also, let $\mathbf{C}^{\infty}=\bigcup_{n\geq 1}\mathbf{C}^{n}$ and $X_{\infty}=\operatorname{Sym}^{3}(\mathbf{C}^{\infty})^{*}$. The set $X_{\infty}$ is the inverse limit of the sets $X_{n}$, and as such carries the inverse limit topology. (Each $X_{n}$ is endowed with the discrete topology). Precisely, a sequence $\{\omega_{i}\}_{i\geq 1}$ in $X_{\infty}$ converges to $\omega$ if for each $n$ we have $\omega_{i}|_{\mathbf{C}^{n}}=\omega|_{\mathbf{C}^{n}}$ for all sufficiently large $i$.

The group $\mathbf{GL}_{n}(\mathbf{C})$ acts on $X_{n}$, and the group $\mathbf{GL}=\bigcup_{n\geq 1}\mathbf{GL}_{n}(\mathbf{C})$ acts on $X_{\infty}$. The group $\mathbf{GL}$ is, in a sense, too small¹¹1The group $\operatorname{Aut}(\mathbf{C}^{\infty})$ is much larger than $\mathbf{GL}$, but it is also too small.. To remedy this, we employ a modification of the concept of orbit: we say that two elements of $X_{\infty}$ belong to the same generalized orbit if each belongs to the closure of the orbit of the other. This idea was introduced in a slightly different way in [BDES]; see §2.5 and Remark 2.3 for details.

We say that an element of $X_{\infty}$ is degenerate if it has the form $\sum_{i=1}^{n}q_{i}\ell_{i}$ where $q_{i}\in\operatorname{Sym}^{2}(\mathbf{C}^{\infty})^{*}$ and $\ell_{i}\in(\mathbf{C}^{\infty})^{*}$, and non-degenerate otherwise. The main theorem of [DES] asserts that the non-degnerate forms constitute a single generalized orbit. (The paper [DES] concerns only symmetric trilinear forms, but this statement was extended to other types of tensorial forms in [BDE, BDES].)

Just as the usual orbits of $\mathbf{GL}$ are too small, so too are the usual stabilizers. One question we sought to answer in this paper is: what is the right notion of “generalized stabilizer”? We have come to the following idea. Let $\omega\in X_{\infty}$ be given. For $n\geq 1$, define $\Gamma_{\omega}(n)$ be the set of elements $g\in\mathbf{GL}$ such that $g^{-1}\omega|_{\mathbf{C}^{n}}=\omega|_{\mathbf{C}^{n}}$. (The inverse here is simply to make some other definitions cleaner.) Note that $\Gamma_{\omega}(n)$ is typically not a subgroup. We define the generalized stabilizer of $\omega$ to be the system $\Gamma_{\omega}=\{\Gamma_{\omega}(n)\}_{n\geq 1}$. One should think of $\Gamma_{\omega}$ as a kind of germ of a neighborhood of the stabilizer of $\omega$. For this reason, we refer to $\Gamma_{\omega}$ as a germinal subgroup; see Definition  7.1 for details. We view $\Gamma_{\omega}$ as an analog of the infinite orthogonal group associated to symmetric trilinear forms.

Let $\omega\in X_{\infty}$ and $\Gamma_{\omega}$ be as above. We define a representation of $\Gamma_{\omega}$ to be a complex vector space $V$ such that each finite dimensional subspace $W\subset V$ is endowed with an action map $\Gamma_{\omega}(n)\times W\to V$, for some $n$ depending on $W$, satisfying certain conditions. A little more precisely, the data defining a representation can be encoded as a linear map

Thus for $v\in V$, one can regard $g\mapsto gv$ as the germ of a function on $\mathbf{GL}$, with respect to the system of neighborhoods $\Gamma_{\omega}$.

Every representation of $\mathbf{GL}$ restricts to a representation of $\Gamma_{\omega}$. We say that a representation of $\Gamma_{\omega}$ is algebraic if it occurs as a subquotient of the restriction of a polynomial representation of $\mathbf{GL}$. In particular, the standard representaiton $\mathbf{C}^{\infty}$ of $\mathbf{GL}$ restricts to an algebraic representation of $\Gamma_{\omega}$, which we call the standard representation of $\Gamma_{\omega}$. We let $\operatorname{Rep}^{\mathrm{alg}}(\Gamma_{\omega})$ denote the category of algebraic representations. This is a Grothendieck abelian category equipped with a tensor product.

The primary purpose of this paper is to understand the algebraic representation theory of $\Gamma_{\omega}$ when $\omega$ is non-degenerate. The following is a summary of our findings.

• •

  Algebraic representations enjoy several finiteness properties:

  • –

    Every algebraic representation is the union of its finite length subrepresentations.

  • –

    The tensor product of two finite length algebraic representations is again finite length.

  • –

    If $V$ and $W$ are finite length algebraic representations then $\operatorname{Hom}_{\Gamma_{\omega}}(V,W)$ is a finite dimensional complex vector space.

• •

  The simple algebraic representations are well-understood:

  • –

    For each partition $\lambda$, there is a simple $L_{\lambda}$, and these exhaust the simples.

  • –

    One can construct $L_{\lambda}$ using a variant of Weyl’s traceless tensor construction. Let $T^{[n]}$ be the intersection of the kernels of the maps $(\mathbf{C}^{\infty})^{\otimes n}\to(\mathbf{C}^{\infty})^{\otimes(n-3)}$ obtained by applying $\omega$ to three tensor factors. This space carries an action of $\mathfrak{S}_{n}\times\Gamma_{\omega}$, where $\mathfrak{S}_{n}$ denotes the symmetric group. The isotypic piece of $T^{[n]}$ corresponding the Specht module $S^{\lambda}$ is exactly $L_{\lambda}$.

• •

  Algebraic representations are well-behaved homologically:

  • –

    The representations $\mathbf{S}_{\lambda}(\mathbf{C}^{\infty})$ are exactly the indecomposable injective algebraic representations; in fact, $\mathbf{S}_{\lambda}(\mathbf{C}^{\infty})$ is the injective envelope of $L_{\lambda}$.

  • –

    Every finite length algebraic representation has finite injective dimension.

• •

  There is a combinatorial description of the entire category $\operatorname{Rep}^{\mathrm{alg}}(\Gamma_{\omega})$: it is equivariant to a category of representations of a certain variant of the upwards Brauer category.

• •

  The category $\operatorname{Rep}^{\mathrm{alg}}(\Gamma_{\omega})$ satisfies a universal property. Let $\mathcal{C}$ be a $\mathbf{C}$-linear abelian category equipped with a tensor product. Then giving a left-exact symmetric monoidal $\mathbf{C}$-linear functor $\operatorname{Rep}^{\mathrm{alg}}(\Gamma_{\omega})^{\mathrm{f}}\to\mathcal{C}$ is equivalent to giving an object of $\mathcal{C}$ equipped with a symmetric trilinear form. The notation $(-)^{\mathrm{f}}$ here denotes the subcategory of finite length objects.

• •

  The symmetric monoidal category $\operatorname{Rep}^{\mathrm{alg}}(\Gamma_{\omega})$ is independent of $\omega$, up to equivalence.

Recall that if a group $G$ acts on a field $K$ then a semi-linear representation of $G$ over $K$ is a $K$-vector space $V$ equipped with an additive action of $G$ such that the equation $g(av)=(ga)(gv)$ holds, for $g\in G$, $a\in K$, and $v\in V$. Semi-linear representations will be a central topic in this paper. To motivate their appearance, we first examine a familiar case.

Let $Y_{n}=\operatorname{Sym}^{2}(\mathbf{C}^{n})^{*}$ be the space of symmetric bilinear forms on $\mathbf{C}^{n}$, regarded as an algebraic variety; explicitly, $Y_{n}=\operatorname{Spec}(S_{n})$ where $S_{n}$ is the polynomial ring $\operatorname{Sym}(\operatorname{Sym}^{2}(\mathbf{C}^{n}))$. Let $Y_{n}^{\circ}$ be the open subvariety of $Y_{n}$ consisting of non-degenerate forms. The algebraic group $\mathbf{GL}_{n}$ acts transitively on $Y_{n}^{\circ}$. Let $y$ be a closed point of $Y_{n}^{\circ}$, and let $\mathbf{O}_{n}$ be its stabilizer. If $\mathcal{F}$ is a $\mathbf{GL}_{n}$-equivariant quasi-coherent sheaf on $Y_{n}^{\circ}$ then its fiber $\mathcal{F}(y)$ at $y$ is an algebraic representation of $\mathbf{O}_{n}$, and this construction gives an equivalence of categories

In fact, we can get a similar equivalence using the generic point of $Y_{n}^{\circ}$. If $\mathcal{F}$ is a $\mathbf{GL}_{n}$-equivariant quasi-coherent sheaf on $Y_{n}^{\circ}$ then its generic fiber is a semi-linear representation of $\mathbf{GL}_{n}$ over $\operatorname{Frac}(S_{n})$ that is algebraic (in the sense that it is spanned by an algebraic subrepresentation). Moreover, letting $\mathcal{C}_{n}$ be the category of such semi-linear representations, this construction defines an equivalence

Thus, combined with the previous equivalence, we obtain an equivalence

This gives us a way of studying representations of $\mathbf{O}_{n}$ (or, at least, the representation category) even if we do not understand the group $\mathbf{O}_{n}$ very well.

We adopt this approach in this paper to replace representations of $\Gamma_{\omega}$ with more familiar objects. Let $R$ be the infinite variable polynomial ring $\operatorname{Sym}(\operatorname{Sym}^{3}(\mathbf{C}^{\infty}))$ and let $K=\operatorname{Frac}(R)$. We show (Theorem 8.11) that $\operatorname{Rep}^{\mathrm{alg}}(\Gamma_{\omega})$ is equivalent to a certain category of semi-linear representations of $\mathbf{GL}$ over $K$ (precisely, the category of “$K$-modules” introduced below). The proof is similar to the one outlined above, but technically more involved, and relies on some non-trivial results from [BDDE] and [BDES]. We find the semi-linear perspective to be technically much easier to work with, so most of the paper is carried out in this setting.

We now explain some of our results on semi-linear representations in more detail. We first introduce some fundamental definitions. A $\mathbf{GL}$-algebra²²2In characteristic 0, $\mathbf{GL}$-algebras are equivalent, under Schur–Weyl duality, to twisted commutative algebras; see [SS2, §8.1]. is an algebra object in the category of polynomial representations of $\mathbf{GL}$; in other words, it is a commutative ring equipped with an action of $\mathbf{GL}$ under which it forms a polynomial representation. For example, the ring $\operatorname{Sym}(\operatorname{Sym}^{3}(\mathbf{C}^{\infty}))$ appearing above is a $\mathbf{GL}$-algebra. If $R$ is a $\mathbf{GL}$-algebra then an $R$-module is a module object; in other words, it is a $\mathbf{GL}$-equivariant $R$-module that forms a polynomial representation.

A $\mathbf{GL}$-field is a field equipped with an action of $\mathbf{GL}$ that can be obtained as the fraction field of an integral $\mathbf{GL}$-algebra. If $K$ is a $\mathbf{GL}$-field then a $K$-module is a semi-linear representation of $\mathbf{GL}$ over $K$ that is generated by a polynomial subrepresentation. The basic example of a $K$-module is $K^{\oplus\lambda}=K\otimes_{\mathbf{C}}\mathbf{S}_{\lambda}(\mathbf{C}^{\infty})$. While $K^{\oplus\lambda}$ is typically not projective, every $K$-module is a quotient of a sum of ones of this form. We let $\operatorname{Mod}_{K}$ denote the category of $K$-modules. This is the fundamental object of study in this paper.

We prove two main technical results about $K$-modules. To state the first one, we must introduce the shift operation. Let $G(n)$ be the subgroup of $\mathbf{GL}$ consisting of block matrices of the form

where the top left block is $n\times n$. This group is isomorphic to $\mathbf{GL}$. If $X$ is a set equipped with an action of $\mathbf{GL}$, we define its $n$th shift, denoted $\operatorname{Sh}_{n}(X)$, to be the set $X$ equipped with the action of $\mathbf{GL}$ coming from restricting the given action to $G(n)\cong\mathbf{GL}$. The shift operation preserves all structure introduced so far (polynomial representations, $\mathbf{GL}$-fields, etc.). Our first theorem is:

This theorem is an instance of the general principle in representation stability that objects can be made “nice” after shifting. The first theorem of this sort was Nagpal’s shift theorem for $\mathbf{FI}$-modules [Na]. The above shift theorem is closely related to the shift theorem for $\mathbf{GL}$-varieties [BDES, Theorem 5.1], and follows a similar proof.

Our second main result about $K$-modules is the following:

This theorem follows rather easily from the shift theorem. It is a very important theorem for us: indeed, all the statements in §1.2 have analogs for $\operatorname{Mod}_{K}$, and can be deduced from the embedding thoerem by comparitively easy arguments. The corresponding results for $\operatorname{Rep}^{\mathrm{alg}}(\Gamma_{\omega})$ are deduced from those for $\operatorname{Mod}_{K}$.

Let $\mathfrak{U}$ be the analog of the upwards Brauer category for symmetric trilinear forms (see §5.2), let $R=\operatorname{Sym}(\operatorname{Sym}^{3}(\mathbf{C}^{\infty}))$, let $K=\operatorname{Frac}(R)$, and let $\omega\in\operatorname{Sym}^{3}(\mathbf{C}^{\infty})^{*}$ be non-degenerate. We show that the following categories are equivalent:

1. (a)

   The category $\operatorname{Mod}_{\mathfrak{U}}^{\mathrm{lf}}$ of $\mathfrak{U}$-modules that are locally of finite length.

2. (b)

   The category $\operatorname{Mod}_{R}^{\mathrm{lf}}$ of $R$-modules that are locally of finite length.

3. (c)

   The generic category $\operatorname{Mod}_{R}^{\mathrm{gen}}$, i.e., the Serre quotient of $\operatorname{Mod}_{R}$ by the subcategory of torsion modules.

4. (d)

   The category $\operatorname{Mod}_{K}$.

5. (e)

   The category $\operatorname{Rep}^{\mathrm{alg}}(\Gamma_{\omega})$.

The equivalence between (a) and (b) is straightforward, as is the equivalence between (c) and (d). The equivalence of (b) and (c) is much more difficult, and relies upon the embedding theorem. The equivalence between (d) and (e) is also difficult, and relies on non-trivial results from [BDDE] and [BDES].

The equivalence between (b) and (c) above has a long history: see Remark 5.9.

The categories (a)–(e) above are $\mathbf{C}$-linear tensor categories. However, only in (e) are the objects $\mathbf{C}$-vector spaces (with extra structure), with the tensor product being the usual one on the underlying vector space. One can therefore view the equivalence $\operatorname{Mod}_{K}\cong\operatorname{Rep}^{\mathrm{alg}}(\Gamma_{\omega})$ as a fiber functor on $\operatorname{Mod}_{K}$. We thus get one such fiber functor for each choice of $\omega$. We show (§6) that all fibers functors are obtained in essentially this manner.

This paper is closely related to four threads of recent work:

• •

  The papers [BDE, BDDE, BDES, Dr, DES] develop aspects of infinite dimensional $\mathbf{GL}$-equivariant algebraic geometry. These theories are based on $\mathbf{GL}$-algebras, which is the main connection to this paper. A few key arguments in this paper are in fact modeled on those from [BDES]. The work of Kazhdan–Ziegler [KaZ1, KaZ2, KaZ3, KaZ4] is closely related.

• •

  The papers [NSS, NSS2, NSS3, SS1, SS5, SS6] study the module theory of a handful of specific $\mathbf{GL}$-algebras (and similar objects). The results of this paper generalize many of the results from those papers.

• •

  The papers [DPS, GS, PSe, PSt, Se, SS3] study the stable representation theory of classical (super)groups. The results summarized in §1.2 are all analogs of results from these papers (especially [SS3]).

• •

  The papers [NS, Ro, Ro2, Ro3] study the semi-linear representation theory of the infinite symmetric group, which is thematically similar to much of the work in this paper.

In this paper, we give a fairly complete description of $\operatorname{Mod}_{K}$ when $K$ is a rational $\mathbf{GL}$-field. While we do prove some results for more general $\mathbf{GL}$-fields (see Theorem 4.12), there is still much left to be done in this direction. We hope to treat this in a future paper.

In the study of modules over $\mathbf{GL}$-algebras, it is also important to understand the generic categories $\operatorname{Mod}_{R}^{\mathrm{gen}}$ when $R$ is a “$\mathbf{GL}$-domain” (this means $\mathfrak{a}\mathfrak{b}=0$ implies $\mathfrak{a}=0$ or $\mathfrak{b}=0$ when $\mathfrak{a}$ and $\mathfrak{b}$ are $\mathbf{GL}$-ideals, which is a weaker condition than being a domain). In [Sn], we gave a useful way of understanding the $\mathbf{GL}$-domain condition in terms of super mathematics, and we believe this should allow us to say something about these generic categories. We hope to return to this topic too.

We list a few questions or problems raised by this work:

1. (a)

   How much of standard Lie theory can be carried over to the generalized stabilizers $\Gamma_{\omega}$? Is there a Dynkin diagram, Cartan matrix, Weyl group, etc.?

2. (b)

   Is there a Tannakian perspective that allows one to recover the generalized stabilizer $\Gamma_{\omega}$ from the fiber functor $\Phi_{\omega}\colon\operatorname{Mod}_{K}\to\mathrm{Vec}_{k}$?

3. (c)

   Prove Theorem 8.11 for general $K$.

4. (d)

   What are the derived specializations of simple objects of $\operatorname{Mod}_{K}$? (See Remark 5.13.)

5. (e)

   We introduce the concept of “germinal subgroup” to define generalized stabilizers. While our definitions work for the purposes of this paper, we are not sure if they are optimal. For instance, our conditions do not say anything about inverses. It would be good to have more clarity on this point.

6. (f)

   In this paper, we consider generalized stabilizers for actions of $\mathbf{GL}$ on infinite dimensional varieties. Are there other situations where generalized stabilizers are interesting? For example, one could consider generalized stabilizers arising from actions of the infinite symmetry group on infinite dimensional varieties.

In §2 we provide background about $\mathbf{GL}$-algebras and related concepts. In §3 we prove our two main technical theorems on $K$-modules, the shift and embedding theorems. We apply these results in §4 to deduce our main structural results on semi-linear representations. These results are in turn used in §5 to obtain the connection to an analog of the Brauer category, which yields an analog of Weyl’s construction and a universal property for $\operatorname{Mod}_{K}$. In §6, we classify the fiber functors of $\operatorname{Mod}_{K}$. In §7 we introduce germinal subgroups and generalized stabilizers in the abstract. Finally, in §8, we apply these concepts to $\mathbf{GL}$-varieties.

We thank Arthur Bik, Jan Draisma, Rob Eggermont, Nate Harman, Steven Sam, and David Treumann for helpful conversations. In particular, Proposition 2.2 came from an e-mail exchange with Bik, Draisma, and Eggermont, and the material in §4.1 came from unpublished notes with Sam.

Fix, for the entirety of the paper, a field $k$ of characteristic 0. Put $\mathbf{GL}=\bigcup_{n\geq 1}\mathbf{GL}_{n}(k)$, regarded as a discrete group. We let $\mathbf{V}=\bigcup_{n\geq 1}k^{n}$ be the standard representation of $\mathbf{GL}$. We say that a representation of $\mathbf{GL}$ on a $k$-vector space is polynomial if it appears as a subquotient of a (possibly infinite) direct sum of tensor powers of $\mathbf{V}$. We let $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL})$ denote the category of polynomial representations. It is a semi-simple Grothendieck abelian category that is closed under tensor product.

For a partition $\lambda$, we let $\mathbf{S}_{\lambda}$ denote the corresponding Schur functor. The simple polynomial representations are exactly those of the form $\mathbf{S}_{\lambda}(\mathbf{V})$. Thus every polynomial representation decomposes as a (perhaps infinite) direct sum of $\mathbf{S}_{\lambda}(\mathbf{V})$’s.

Every polynomial representation of $\mathbf{GL}$ carries a natural grading, with $\mathbf{S}_{\lambda}(\mathbf{V})$ concentrated in degree $|\lambda|$, the size of the partition $\lambda$. This grading is compatible with tensor products: $\mathbf{S}_{\lambda}(\mathbf{V})\otimes\mathbf{S}_{\mu}(\mathbf{V})$ is concentrated in degree $|\lambda|+|\mu|$. The degree 0 piece of a polynomial representation $V$ is exactly the invariant subspace $V^{\mathbf{GL}}$.

We now introduce some non-standard notation that will be convenient for working with these objects. We write $k^{\oplus\lambda}$ in place of $\mathbf{S}_{\lambda}(\mathbf{V})$. More generally, for a $k$-vector space $V$ we put $V^{\oplus\lambda}=V\otimes_{k}k^{\oplus\lambda}$; note that if $R$ is a $k$-algebra then $R^{\oplus\lambda}$ is naturally a free $R$-module. A tuple of partitions (often simply called a tuple) is a tuple $\underline{\smash{\lambda}}=[\lambda_{1},\ldots,\lambda_{r}]$, where each $\lambda_{i}$ is a partition. We put $k^{\oplus\underline{\smash{\lambda}}}=\bigoplus_{i=1}^{r}k^{\oplus\lambda_{i}}$, and define $V^{\oplus\underline{\smash{\lambda}}}$ similarly. We say that $\underline{\smash{\lambda}}$ is pure if it does not contain the empty partition. (This terminology comes from [BDES].)

The category of polynomial representations is equivalent to the category of polynomial functors, with the representation $\mathbf{S}_{\lambda}(\mathbf{V})$ corresponding to the functor $\mathbf{S}_{\lambda}$. Given a polynomial representation $V$ and a vector space $U$, we let $V\{U\}$ be the result of regarding $V$ as a polynomial functor and evaluating on $U$. In the important special case where $U=k^{n}$, we can identify $V\{U\}$ with the invariant space $V^{G(n)}$, where $G(n)$ is defined in §2.3. For example, if $V=k^{\oplus\lambda}$ then $V\{k^{n}\}=\mathbf{S}_{\lambda}(k^{n})$.

Suppose that $V$ is an arbitrary $k$-linear representation of $\mathbf{GL}$. We say that an element $x\in V$ is polynomial if the subrepresentation it generates is a polynomial representation. We let $V^{\mathrm{pol}}$ be the set of all polynomial elements in $V$. It can be characterized as the maximal polynomial subrepresentation of $V$. Moreover, if $\operatorname{REP}(\mathbf{GL})$ denotes the category of all $k$-linear representations of $\mathbf{GL}$ then $V\mapsto V^{\mathrm{pol}}$ is the right adjoint of the inclusion functor $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL})\to\operatorname{REP}(\mathbf{GL})$. As such, $(-)^{\mathrm{pol}}$ is left-exact and continuous; it is not exact.

Recall that $G(n)$ is the subgroup of $\mathbf{GL}$ consisting of block matrices of the form

where the top left block has size $n\times n$. We have a group isomorphism

Given some kind of object $X$ equipped with an action of $\mathbf{GL}$, we define its $n$th shift, denoted $\operatorname{Sh}_{n}(X)$, to be the same object $X$ but with $\mathbf{GL}$ acting through the self-embedding $\mathbf{GL}\cong G(n)\subset\mathbf{GL}$.

One easily sees that if $V$ is a polynomial representation of $\mathbf{GL}$ then $\operatorname{Sh}_{n}(V)$ is also such a representation. From the polynomial functor point of view, we have

If $V$ has finite length then so does $\operatorname{Sh}_{n}(V)$. It follows that if $\underline{\smash{\lambda}}$ is a tuple then there is another tuple, which we denote by $\operatorname{sh}_{n}(\underline{\smash{\lambda}})$, such that $\operatorname{Sh}_{n}(k^{\oplus\underline{\smash{\lambda}}})=k^{\oplus\operatorname{sh}_{n}(\underline{\smash{\lambda}})}$. If $\underline{\smash{\lambda}}=[\lambda]$ consists of a single partition, we write $\operatorname{sh}_{n}(\lambda)$ in place of $\operatorname{sh}_{n}(\underline{\smash{\lambda}})$. In this case, $\operatorname{sh}_{n}(\lambda)$ contains $\lambda$ exactly once, and all other partitions in it are strictly smaller.

A $\mathbf{GL}$-algebra (over $k$) is a commutative algebra object in the tensor category $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL})$; thus, it is a commutative (and associative and unital) $k$-algebra equipped with an action of the group $\mathbf{GL}$ by algebra automorphisms, under which it forms a polynomial representation. Let $R$ be a $\mathbf{GL}$-algebra. By an $R$-module we mean a module object in $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL}$). Explicitly, this is an ordinary $R$-module $M$ equipped with a compatible action of $\mathbf{GL}$ under which $M$ forms a polynomial representation. We let $\operatorname{Mod}_{R}$ denote the category of modules, which is easily seen to be a Grothendieck abelian category.

We say that $R$ is $\mathbf{GL}$-generated (over $k$) by a set of elements if $R$ is generated as a $k$-algebra by the orbits of these elements. We say that $R$ is finitely $\mathbf{GL}$-generated if it is $\mathbf{GL}$-generated by a finite set. We similarly speak of $\mathbf{GL}$-generation for $R$-modules.

We say that a $\mathbf{GL}$-algebra is integral if it is integral in the usual sense (i.e., it is a domain). We will require the following important shift theorem from [BDES].

An affine $\mathbf{GL}$-scheme is an affine scheme $X$ over $k$ equipped with an action of the discrete group $\mathbf{GL}$ such that $\Gamma(X,\mathcal{O}_{X})$ forms a polynomial representation of $\mathbf{GL}$. Every affine $\mathbf{GL}$-scheme has the form $\operatorname{Spec}(R)$ where $R$ is a $\mathbf{GL}$-algebra. An affine $\mathbf{GL}$-variety is a reduced affine $\mathbf{GL}$-scheme $X$ such that $\Gamma(X,\mathcal{O}_{X})$ is finitely $\mathbf{GL}$-generated over $k$.

For a tuple $\underline{\smash{\lambda}}$, let $\mathbf{A}^{\underline{\smash{\lambda}}}$ be the spectrum of the ring $\operatorname{Sym}(k^{\oplus\underline{\smash{\lambda}}})$. This is an affine $\mathbf{GL}$-variety. Moreover, every affine $\mathbf{GL}$-variety is isomorphic to a closed $\mathbf{GL}$-subvariety of some $\mathbf{A}^{\underline{\smash{\lambda}}}$. Thus, in the theory of $\mathbf{GL}$-varieties, the $\mathbf{A}^{\underline{\smash{\lambda}}}$ play the same role as the ordinary affine spaces $\mathbf{A}^{n}$ in ordinary algebraic geometry.

Let $X$ be an affine $\mathbf{GL}$-variety and let $x$ be a (scheme-theoretic) point of $X$. We let $\overline{O}_{x}$ be the Zariski closure of the orbit $\mathbf{GL}\cdot x$ of $x$ (see [BDES, §3.1]). We say that $x$ is $\mathbf{GL}$-generic if $\overline{O}_{x}=X$. Such points play a similar role to generic points in ordinary algebraic geometry. We define the generalized orbit of $x$, denoted $O_{x}$, to be the set of all points $y$ such that $\overline{O}_{x}=\overline{O}_{y}$ (see [BDES, §3.2]).

Write $X=\operatorname{Spec}(R)$ where $R$ is a $\mathbf{GL}$-algebra. Recall that for a vector space $U$ we let $R\{U\}$ be the result of treating $R$ as a polynomial functor and evaluating on $U$; this is a $k$-algebra equipped with an action of $\mathbf{GL}(U)$. We put $X\{U\}=\operatorname{Spec}(R\{U\})$. The standard inclusion $k^{n}\to\mathbf{V}$ induces a ring homomorphism $R\{k^{n}\}\to R$, and thus a map of $k$-schemes $\pi_{n}\colon X\to X\{k^{n}\}$. Since $R$ is the union of the $R\{k^{n}\}$, it follows that $X$ is the inverse limit of the $X\{k^{n}\}$. We define the $\Pi$-topology on $X$ to be the inverse limit topology, where each $X\{k^{n}\}$ is given the discrete topology. The $\Pi$-topology is actually quite concrete: if $k$ is algebraically closed then the set of closed points of $\mathbf{A}^{\underline{\smash{\lambda}}}$ is identified with a product of $k$’s, and the $\Pi$-topology is just the usual product topology; thus a sequence of $k$-points of $\mathbf{A}^{\underline{\smash{\lambda}}}$ converges if each coordinate is eventually constant. One easily sees that any Zariski closed set is $\Pi$-closed (see [NS2, Proposition 2.3]).

We require the following result that relates the Zariski and $\Pi$-topologies:

A $\mathbf{GL}$-field over $k$ is a field extension $K/k$ equipped with an action of $\mathbf{GL}$ by $k$-automorphisms such that every element of $K$ can be expressed in the form $a/b$ with $a,b\in K^{\mathrm{pol}}$. If $K$ is a $\mathbf{GL}$-field then $K^{\mathrm{pol}}$ is an integral $\mathbf{GL}$-algebra over $k$, and $K=\operatorname{Frac}(K^{\mathrm{pol}})$. Thus every $\mathbf{GL}$-field can be realized as the fraction field of an integral $\mathbf{GL}$-algebra.

Let $K$ be a $\mathbf{GL}$-field. A $K$-module is a semi-linear representation $M$ of $\mathbf{GL}$ over $K$ such that every element of $M$ has the form $ax$ with $a\in K$ and $x\in M^{\mathrm{pol}}$. One easily sees that the category $\operatorname{Mod}_{K}$ of $K$-modules is an abelian category satisfying the (AB5) condition. Moreover, if $M$ is any $K$-module then there is a surjection $K\otimes V\to M$ for some polynomial representation $V$ (take $V=M^{\mathrm{pol}}$), which shows that the objects $K^{\oplus\lambda}$ form a generating set; thus $\operatorname{Mod}_{K}$ is a Grothendieck abelian category.

We say that $K$ is finitely $\mathbf{GL}$-generated over $k$ if it is generated as a field extension by the $\mathbf{GL}$-orbits of finitely many elements. We say that $K$ is rational over $k$ if it has the form $\operatorname{Frac}(\operatorname{Sym}(k^{\oplus\underline{\smash{\sigma}}}))$ for some tuple $\underline{\smash{\sigma}}$. The invariant subfield of $K$, denoted $K^{\mathbf{GL}}$, is the subfield of $K$ consisting of all elements that are invariant under $\mathbf{GL}$. It is an extension of $k$. If $K$ is finitely $\mathbf{GL}$-generated over $k$ then $K^{\mathbf{GL}}$ is finitely generated over $k$ ([BDES, Proposition 5.8]).

Let $R$ be an integral $\mathbf{GL}$-algebra. We say that an $R$-module $M$ is torsion if every element of $M$ is annihilated by a non-zero element of $R$. The category $\operatorname{Mod}_{R}^{\mathrm{tors}}$ of torsion $R$-modules is a Serre subcategory of $\operatorname{Mod}_{R}$. We define the generic category of $R$, denoted $\operatorname{Mod}_{R}^{\mathrm{gen}}$, to be the Serre quotient $\operatorname{Mod}_{R}/\operatorname{Mod}_{R}^{\mathrm{tors}}$.

The generic category can be described in terms of semi-linear representations. Let $K=\operatorname{Frac}(R)$. We have a functor

We also have a functor

Indeed, if $N$ is a $K$-module then $R\otimes N^{\mathrm{pol}}$ is a polynomial representation, so its image under the natural map $R\otimes N^{\mathrm{pol}}\to N$ consists of polynomial elements, and is therefore contained in $N^{\mathrm{pol}}$; this shows that $N^{\mathrm{pol}}$ is stable under multiplication by $R$, and is thus an $R$-module.

We say that an $R$-module $M$ is saturated if the natural map $M\to S(T(M))$ is an isomorphism. We will require the following result concerning this concept:

The following proposition is the key input needed for the shift theorem proven in the subsequent subsection. It is a linear analog of [BDES, Theorem 4.2], a result that was essentially taken from arguments in [Dr].

We require some preparation before giving the proof. A weight of $\mathbf{GL}$ is a tuple $\lambda=(\lambda_{1},\lambda_{2},\ldots)$ where $\lambda_{i}\in\mathbf{Z}$ for all $i$ and $\lambda_{i}=0$ for $i\gg 0$. For a finite subset $A$ of $[\infty]=\{1,2,\ldots\}$, we let $1^{A}$ be the weight that is 1 at the coordinates in $A$, and 0 away from $A$. We also write $1^{n}$ in place of $1^{A}$ when $A=[n]$.

Suppose that $V$ is a polynomial representation and $\lambda$ is a weight. We say that $v\in V$ is a weight vector of weight $\lambda$ if whenever $g=\operatorname{diag}(a_{1},a_{2},\ldots)$ we have

We let $V_{\lambda}$ be the space of all weight vectors of weight $\lambda$; this is the $\lambda$ weight space. The space $V$ is the direct sum of its weight spaces $V_{\lambda}$ over all $\lambda$. Moreover, if $V_{\lambda}$ is non-zero then $\lambda$ is non-negative in the sense that $\lambda_{i}\geq 0$ for all $i$.

The weight space $V_{1^{n}}$ carries a representation of $\mathfrak{S}_{n}\subset\mathbf{GL}$. Let $\operatorname{Rep}^{\mathrm{pol},n}(\mathbf{GL})$ be the subcategory of $\operatorname{Rep}^{\mathrm{pol}}(\mathbf{GL})$ spanned by representations of degree $n$. One formulation of Schur–Weyl duality states that the functor

is an equivalence of categories.

We now prove the first main result of this section. It is an analog of [BDES, Theorem 5.1].