Contents

Idea and definition

Classically, a ‘Schur functor’ is a specific sort of functor

where FinDimVect is the category of finite dimensional vector spaces over the complex numbers. Namely, it is a functor where $S(X)$ is obtained by taking the $n$th tensor power of the vector space $X$ and then picking out a subspace that transforms in a certain way with respect to the symmetric group $S_n$. In fact, Schur functors can be defined on a large class of categories resembling FinDimVect: for example, categories of representations of a group, or vector bundles, or coherent sheaves. After an elementary introduction to Schur functors, this article will give a conceptual approach based on the fact that these functors are precisely those that can be ‘naturally’ — or more precisely, ‘pseudonaturally’ defined on all symmetric monoidal linear Cauchy complete categories.

The most famous examples of Schur functors are these:

• For each $n \geq 0$, the $n^{th}$ symmetric power $X \mapsto S^n(X)$ is a Schur functor.

• For each $n \geq 0$, the $n^{th}$ alternating power $X \mapsto \Lambda^n(X)$ is a Schur functor.

More generally, complex irreducible representations of $S_n$ correspond to $n$-box Young diagrams, so Schur functors are usually described with the help of these. An $n$-box Young diagram is simply a pictorial way of writing $n$ as a sum of natural numbers listed in decreasing order. For example, this 17-box Young diagram:

describes the partition of $17$ as $5 + 4 + 4 + 2 + 1 + 1$. However, it also can be used to construct an irreducible complex representation of the permutation group $S_{17}$, and thus a Schur functor.

In general, here is the recipe for constructing the Schur functor $S_\lambda$ associated to an $n$-box Young diagram $\lambda$. For now we only say what this functor does to objects (that is, finite-dimensional vector spaces):

• Given an $n$-box Young diagram $\lambda$ and a vector space $X$, we first form the tensor power $X^{\otimes n}$. We think of each factor in this tensor power as corresponding to a specific box of the Young diagram.

• Then we pick out the subspace of $X^{\otimes n}$ consisting of tensors that are unchanged by any permutation that interchanges two boxes in the same row.

• Then we project this subspace onto the space of tensors that change sign under any permutation that interchanges two boxes in the same column. The result is called $S_\lambda(X)$, where $S_\lambda$ is the Schur functor corresponding to $\lambda$.

It is easy to see from this description that:

• The tall skinny Young diagrams with one column and $n$ rows give the Schur functors $\Lambda^n$.

• The short fat Young diagrams with one row and $n$ columns give the Schur functors $S^n$.

We can also think of this relation between Young diagrams and Schur functors in a slightly more abstract way using the group algebra of the symmetric group, $\mathbb{C}[S_n]$. Suppose $\lambda$ is an $n$-box Young diagram. Then we can think of the operation ‘symmetrize with respect to permutations of the boxes in each row’ as an element $p^S_\lambda \in \mathbb{C}[S_n]$. Similarly, we can think of the operation ‘antisymmetrize with respect to permutations of the boxes in each column’ as an element $p^A_\lambda \in \mathbb{C}[S_n]$. By construction, each of these elements is idempotent:

Now, it is easy to see that the product of commuting idempotents is idempotent. The elements $p^S_\lambda$ and $p^A_\lambda$ do not commute, but amazingly, their product

is idempotent up to a scalar multiple!

This element $p_\lambda \in \mathbb{C}[S_n]$ is called the Young symmetrizer corresponding to the $n$-box Young diagram $\lambda$. There is a functor, called a Schur functor:

defined on any finite-dimensional vector space $X$ as follows:

Here we are using the fact that $S_n$, and thus its group algebra, acts on $X^{\otimes n}$. Thus, $p_\lambda$ acts as an idempotent operator on $X^{\otimes n}$, and $S_\lambda(X)$ as defined above is the range of this operator.

An even deeper approach to Schur functors is based on the relation between Young diagrams and representations of the symmetric groups. The subspace

has an obvious right action of the algebra $\mathbb{C}[S_n]$, and thus becomes a representation of the group $S_n$. In fact, it is an irreducible representation of $S_n$, and every irreducible representation of $S_n$ is isomorphic to one of this form. Even better, this recipe sets up a one-to-one correspondence between $n$-box Young diagrams and isomorphism classes of irreducible representations of $S_n$.

(Strictly speaking, to think of $p_\lambda$ as an element of $\mathbb{C}[S_n]$, we must choose a way to label the boxes of the Young diagram with numbers $1, \dots, n$. Such an identification is called a Young tableau. For our purposes it will cause no harm to randomly choose a Young tableau for each Young diagram, since different choices give isomorphic representations $V_\lambda$. In other contexts, the difference between various choices of Young tableaux can be extremely important.)

This description of irreducible representations of $S_n$ paves the way towards an important generalization of Schur functors. First, note that

It follows that

These isomorphisms here are natural, so there is no harm in defining the Schur functor $S_\lambda$ by

This formula defines the Schur functor not only on objects but also on morphisms. Even better, we can use the same formula to define a functor

for any representation $R$ of $S_n$, as follows:

These more general functors are still called ‘Schur functors’.

Even more generally, any finite direct sum of the Schur functors just described may also be called a Schur functor. In other words, if $R = \{R_n\}_{n \ge 0}$ where $R_n$ is a finite-dimensional representation of $S_n$, and only finitely many of these representations are nonzero, then we define the Schur functor

• For each $n \geq 0$, the $n^{th}$ tensor power $V \mapsto V^{\otimes n}$ is a Schur functor.

• If $F$ and $G$ are Schur functors, the functor $V \mapsto F(V) \oplus G(V)$ is a Schur functor.

• If $F$ and $G$ are Schur functors, the functor $V \mapsto F(V) \otimes G(V)$ is also a Schur functor.

• If $F$ and $G$ are Schur functors, the composite $V \mapsto F(G(V))$ is a Schur functor. This way of constructing Schur functors is known as plethysm.

The category of Schur functors

There is a category $Schur$ with

• Schur functors $S_R \colon FinDimVect \to FinDimVect$ as objects. Here

  $$S_R(-) = \bigoplus_{n \ge 0} R_n \otimes_{\mathbb{C}[S_n]} (-)^{\otimes n} \,$$

  each $R_n$ is a finite-dimensional representation of $S_n$, and only finitely many of these representations are nonzero.

• natural transformations between such functors as morphisms.

In the rest of this article, we would like to give a conceptual explanation of this category.

As a warm-up, let us note that $\Schur$ has has a nice description in terms of groupoid of finite sets and bijections. This groupoid is the core of the category FinSet, so it is denoted $core(FinSet)$. What is the relation between Schur functors and this groupoid. Every Schur functor is a finite direct sum of Schur functors coming from irreducible representations of symmetric groups $S_n$ for various $n$. But what sort of entity is a direct sum of representations of symmetric groups $S_n$ for various $n$? It is nothing but a representation of the permutation groupoid:

where objects are natural numbers, all morphisms are automorphisms, and the automorphisms of $n$ form the group $S_n$. In other words, it is a functor

But conceptually, the importance of $\mathbb{P}$ is that it is a skeleton of the groupoid of finite sets and bijections. So,

As a result, any Schur functor gives a functor

Following Joyal‘s work on combinatorics, such functors are known as $Vect$-valued species, or $Vect$-valued structure types. The idea here is that just as an ordinary Set-valued species

assigns to any finite set a set of structures of some type, a $Vect$-valued species assigns to any finite set a vector space of structures of some type, Thanks to the ‘free vector space on a set’ functor

we can linearize any ordinary species $T$ and obtain a linear species $F \circ T$. This process is extremely important in the work of Marcelo Aguiar and Swapneel Mahajan:

• Marcelo Aguiar and Swapneel Mahajan, Monoidal Functors, Species and Hopf Algebras, 2010.

However, not all $Vect$-valued species corresond to Schur functors, because we have defined Schur functors to arise from finite direct sums of irreducible representations of permutation groups. So, $Schur$ is equivalent to the category where:

• objects are polynomial species: that is, functors $R: core(Fin\Set) \to FinDimVect$ such that $R(n) = \{0\}$ for all sufficiently large finite sets $n$;

• morphisms are natural transformations.

We call this the category of polynomial species. The reason for the term ‘polynomial’ is that any functor of the form

gives rise to a formal power series called its generating function:

and this power series is a polynomial precisely when $R$ is a polynomial species.

The category of representations of any groupoid has many nice features. For example, it is a symmetric monoidal abelian category, by which we mean a symmetric monoidal category that is also abelian, where tensoring with any object is right exact. So, the category of $Vect$-valued species is symmetric monoidal abelian — and it is easy to check that the subcategory of polynomial species inherits this structure. Since $Schur$ is equivalent to the category of polynomial species, it too is symmetric monoidal abelian.

In particular, $Schur$ has two monoidal structures, $\oplus$ and $\otimes$, defined by

and

Since $\otimes$ distributes over $\oplus$, these make $Schur$ into a rig category.

In the literature on species, the operation $\oplus$ is often called addition, since adding species this way corresponds to adding their generating functions. Aguiar and Mahajan call $\otimes$ the Hadamard product (see section 8.1.2).

But the category of representations of a groupoid has even more nice features when the groupoid itself has a monoidal structure: then the representation category acquires a monoidal structure thanks to Day convolution. The groupoid $\mathbb{P}$ has, in fact, two important monoidal structures, coming from the product $\times$ and disjoint union $+$ of finite sets. Since $\times$ distributes over $+$, these make $\mathbb{P}$ into a rig category. Thanks to Day convolution, these give the category of representations of $\mathbb{P}$ two more monoidal structures, making it into a rig category in another way. The same is true for the subcategory $Schur$.

In the literature on species, the monoidal structure coming from $+$ is often called multiplication, since multiplying species in this way corresponds to multiplying their generating functions. Aguiar and Mahajan call this monoidal structure the Cauchy product. The monoidal structure coming from $\times$ has no commonly used name, but it deserves to be called the Dirichlet product.

Aguiar and Mahajan point out that some of the relationships between the “pointwise” products (addition and Hadamard) and the “Day” products (Cauchy and Dirichlet) can be described in terms of duoidal categories. Specifically, the Hadamard and Cauchy products form duoidal structures in both orders.

On top of all this, the composite of Schur functors is again a Schur functor. This gives $Schur$ a fifth monoidal structure: the plethystic tensor product. Unlike the four previous monoidal structures, this one is not symmetric.

Mathematicians often work with a decategorified version of $Schur$: its Grothendieck group, also known as the ring of symmetric functions. The various structures that $Schur$ possesses endow this ring with corresponding structures. Among other things, it is the free lambda-ring on one generator. As we shall see, this corresponds to the fact that $Schur$ is the free symmetric monoidal Cauchy complete linear category on one object.

Schur functors on more general categories

We have described Schur functors as special functors

But in fact, functors such as the $n^{th}$ alternating power, $n^{th}$ symmetric power, etc. make sense in much wider contexts. For starters, we can replace the complex numbers by any field $k$ of characteristic zero, and everything in our discussion still works. More importantly, Schur functors can be applied to any symmetric monoidal Cauchy complete linear category (“tensor category”). Here by linear category we mean a category enriched over Vect, the category of vector spaces over $k$. Such a category is Cauchy complete when:

• it has biproducts, also known as direct sums, and

• idempotentssplit.

To illustrate the full breadth of this generalization, here are a few examples:

• the category Vect, consisting of vector spaces over any field $k$ of characteristic zero

• the category FinDimVect, consisting of finite-dimensional vector spaces over $k$

• the category of representations of any group on vector spaces (or finite-dimensional vector spaces) over $k$

• the category of super vector spaces, graded vector spaces or chain complexes over $k$

• for $k = \mathbb{R}$ or $\mathbb{C}$, the category of finite-dimensional real or complex vector bundles over any topological space, or smooth vector bundles over any smooth manifold

• the category of algebraic vector bundles over any algebraic variety (or more generally, scheme or algebraic stack) over $k$

• the category of coherent sheaves of vector spaces over any algebraic variety (or scheme or algebraic stack) over $k$

These examples can be hybridized, and thus they multiply indefinitely: for example, we could take coherent sheaves of chain complexes, or vector bundles equipped with a group action, and so on.

In the following subsections, we explain how to define Schur functors on any category of this sort. A somewhat novel feature of our treatment is that we do not require the theory of Young diagrams to define and study Schur functors.

Our strategy is as follows. We fix a symmetric monoidal Cauchy complete linear category, $C$. The group algebra $k[S_n]$ begins life as a monoid in the symmetric monoidal category FinDimVect. However, we shall explain how interpret it as living in $C$ by a “change of base” functor going from FinDimVect to $C$. This will let us use the Young symmetrizers $p_\lambda$ to construct idempotents on $V^{\otimes k}$ for any object $V \in C$. Splitting these idempotents, we obtain the Schur functors $S_\lambda : C \to C$.

Change of base

To achieve the desired change of base, let $Mat$ be the linear category whose objects are integers $m \geq 0$ and whose morphisms $m \to n$ are $m \times n$ matrices with entries in $k$. Because $C$ is Cauchy complete and in particular has finite biproducts (direct sums), there is an evident linear functor

which takes $m$ to $I^m$, the direct sum of $m$ copies of the tensor unit $I$. It is the unique linear functor taking $1$ to $I$, up to unique linear isomorphism. In the case $C = FinDimVect$, the linear functor

taking $1$ to $k$, is a linear equivalence (exhibiting $Mat$ as a skeleton of $C$). Because of this equivalence, we could equally well say that there is a linear functor

which, up to unique linear isomorphism, is the unique linear functor taking $k$ to $I$. Notice that a symmetric monoidal functor of this form must take the tensor unit $k$ to $I$ (up to coherent isomorphism, as always), and in fact $i$ is symmetric monoidal, because there is a canonical isomorphism

using the fact that $\otimes$ preserves direct sums in each argument, and the fact that there is a canonical isomorphism $I \otimes I \cong I$.

In summary, we have the following proposition.

The action of Young symmetrizers

Next we explain how given an object $X \in C$, any Young symmetrizer in $k[S_n]$ acts as an idempotent on $X^{\otimes n}$.

For this we only need to know a little bit about the group algebra $k[S_n]$, which we recall here. By Maschke's theorem, for any finite group $G$, the group algebra $k[G]$ decomposes as a direct sum of matrix algebras

where $\lambda$ ranges over isomorphism classes of irreducible representations of $G$. The identity elements of these matrix algebras $hom(V_\lambda, V_\lambda)$ thus correspond to certain special elements $p_\lambda \in k[G]$. Clearly these elements are idempotent:

We are particularly interested in the case $G = S_n$. In this case, we call the idempotents $p_\lambda$ are ‘Young symmetrizers’. However, we will not need the formula for these idempotents.

The key step is to apply base change to $k[S_n]$. Here we exploit the fact that

is a monoid in the monoidal category FinDimVect. Since $i \colon FinDimVect \to C$ is a monoidal functor, it follows that $i$ carries $k[S_n]$ to a monoid object in $C$, which we again call $k[S_n]$ by abuse of notation. As an object of $C$, we have

There is a general concept of what it means for a monoid in a monoidal category to act on an object in that category. In particular, if $X$ is an object of $C$, the monoid $k[S_n]$ acts on the tensor power $X^{\otimes n}$. To see this, note that for each $\sigma \in S_n$, there is a corresponding symmetry isomorphism

Putting these together with the help of (1) we obtain a morphism

which is the desired action.

Finally, we would like to describe how each Young symmetrizer $p_\lambda \in k[S_n]$ acts on $X^{\otimes n}$. Quite generally, any element $x \in k[S_n]$ gives a linear map from $k$ to $k[S_n]$, namely the unique map sending $1$ to $x$. Applying the functor $i$ to this, we obtain a morphism which by abuse of language we call

This then yields an endomorphism

given as the composite

It is easy to check that for any $x,y \in k[S_n]$,

Thus for any Young symmetrizer $p_\lambda$, the morphism

is idempotent, because $p_\lambda$ is.

Constructing Schur functors

By construction, the morphisms

are the components of a natural transformation from the functor $X \mapsto X^{\otimes n}$ to itself. Since idempotents split in $C$, we can form the cokernel of $1 - \widetilde{p}_\lambda$, or in other words, the coequalizer of the pair

More generally we can define a Schur functor

for any finite-dimensional representation $R$ of $S_n$, as follows. We can write $R$ as a finite direct sum of irreducible representations:

and then define

Schur functors are “natural”

Suppose now that we have a symmetric monoidal linear functor $G: C \to D$. We can think of $G$ as a “change of base category”, and Schur functors are “natural” with respect to change of base.

That is to say: if $G$ is a symmetric monoidal linear functor $C \to D$, then by definition $G$ preserves tensor products (at least up to coherent natural isomorphism), and $G$ will automatically preserve both direct sums (by linearity) as well as splittings of idempotents (as all functors do). Therefore, for a Schur functor $S_\lambda(X) = V_\lambda \otimes_{S_n} X^{\otimes n}$, we have natural isomorphisms

where the first isomorphism uses the symmetric monoidal structure of $G$; the second uses the fact that $V_{\lambda, D} \cong G(V_{\lambda, C})$ because there is, up to isomorphism, only one symmetric monoidal linear functor $FinDimVect \to D$; the third uses the symmetric monoidal structure again and preservation of idempotent splittings.

If $R$ is any representation, then by writing $R$ as a direct sum of irreducible representations $V_\lambda$ and using the fact that $G$ preserves direct sums, we have more generally

In summary, Schur functors $S_R$ transfer “naturally” across change of base functors $G: C \to D$.

Conceptual description of Schur functors

As we have seen, Schur functors $S_R$ are definable under fairly mild hypotheses: working over a field of characteristic zero, they can be defined on any symmetric monoidal linear category $C$ which is Cauchy complete. So, for such $C$ we can define a Schur functor

and moreover, if $G: C \to D$ is a symmetric monoidal linear functor, the Schur functors on $C$ and $D$ are “naturally” compatible, in the sense that the diagram

commutes up to a canonical isomorphism $\phi_G: S_{R, D} \circ G \cong G \circ S_{R, C}$, and moreover these $\phi_G$ fit together sensibly when we compose symmetric monoidal linear functors.

In this abstract framework, it may be wondered what significant role is played by the representations $R$ of the symmetric group. The natural isomorphisms $\phi_G$ which relate the Schur functors across change of base $G: C \to D$ are pleasant to observe, but surely this is just some piddling general nonsense in the larger story of Schur functors $S_R$, which are after all deeply studied and incredibly rich classical constructions?

Let us put the question another way. We have seen the Schur functors $S_R$ are constructed in a uniform (or “polymorphic”) way across all symmetric monoidal Cauchy complete linear categories $C$, and this construction is natural with respect to symmetric monoidal change of base functors $G: C \to D$. Or rather: not natural in a strict sense, but pseudonatural in the sense that naturality squares commute up to isomorphism $\phi_G$. Now pseudonaturality is a very general phenomenon in 2-category theory. So the question is: among all such pseudonatural transformations $S$, what is special about the Schur functors $S_R$? What extra properties pick out exactly the Schur functors $S_R$ from the class of all pseudonatural transformations $S$?

The perhaps surprising answer is: no extra properties! That is, the Schur functors $S_R$ are precisely those functors that are defined on all symmetric monoidal Cauchy complete linear $C$ and that are pseudonatural with respect to change of base $G: C \to D$!

Let us now make this precise. Schur functors are defined on certain symmetric monoidal linear categories, but they respect neither the symmetric monoidal structure nor the linear structure. So, we have to forget some of the structure of the objects on which Schur functors are defined. This focuses our attention on the ‘forgetful’ 2-functor

where:

As we shall see, Schur functors correspond to pseudonatural transformations from $U$ to itself, and morphisms between Schur functors correspond to modifications between these pseudonatural transformations. For the reader unaccustomed to these 2-categorical concepts, we recall:

We now propose our conceptual definition of Schur functor:

What this proposed definition makes manifestly obvious is that Schur functors are closed under composition. This provides a satisfying conceptual explanation of plethysm, as we will explore in the next two sections. However, we should first check that this proposed definition gives a category of Schur functors equivalent to the category $Schur$ defined earlier!

Before launching into the proof, it is worth pondering an easier problem where we replace categories by sets, and symmetric monoidal linear Cauchy-complete categories by commutative rings.
So, instead of $Cat$ let us consider $Set$, and instead of $SymMonLinCauch$ let us consider $CommRing$. There is a forgetful functor

What are the natural transformations from this functor to itself? Any polynomial $P \in \mathbb{Z}[x]$ defines such a natural transformation, since for any commutative ring $R$ there is a function $P_R: U(R) \to U(R)$ given by

and this is clearly natural in $R$. But in fact, the set of natural transformations from this functor turns out to be precisely $\mathbb{Z}[x]$. And the reason is that $\mathbb{Z}[x]$ is the free commutative ring on one generator!

To see this, note that the forgetful functor

has a left adjoint, the ‘free commutative ring’ functor

The free commutative ring on a 1-element set is

and homomorphisms from $F(1)$ to any commutative ring $R$ are in one-to-one correspondence with elements of the underlying set of $R$, since

So, we say $F(1)$ represents the functor $U$. This makes it easy to show that the set of natural transformations from $U$ to itself is isomorphic to the underlying set of $\mathbb{Z}[x]$, namely $U(F(1))$:

In the first step here we use the representability $U \cong hom(F(1), -)$; in the second we use the Yoneda lemma, and in the third we use the adjointness between $U$ and $F$.

We shall carry out a categorified version of this argument to prove that $Schur$ is the category of endomorphisms of the 2-functor

The key is that $Schur$ is the free symmetric monoidal linear Cauchy-complete category on one generator.

Representability

To build a bridge from abstract Schur functors as pseudonatural transformations to the more classical descriptions, we start with the following key result. In what follows we use $k \mathbb{P}$ to denote the ‘linearization’ of the permutation groupoid: that is, the linear category formed by replacing the homsets in $\mathbb{P}$ by the free vector spaces on those homsets. We use $\widebar{k \mathbb{P}}$ to denote the Cauchy completion of the linearization of $\mathbb{P}$. As we shall see, $\widebar{k \mathbb{P}}$ is equivalent to the category of Schur functors. But first:

Structure of the representing object

Let us now calculate $\widebar{k \mathbb{P}}$. In general, the linear Cauchy completion of a linear category $C$ consists of the full subcategory of linear presheaves $C^{op} \to Vect$ that are obtained as retracts of finite direct sums of representables $C(-, c): C^{op} \to Vect$. In the case $C = k\mathbb{P}$, these are the functors

where $F(n) = 0$ for large enough $n$. For it is clear that this category contains the representables and is closed under finite direct sums and retracts. On the other hand, every polynomial $F$ is a sum of monomials $F(0) \oplus F(1) \oplus \cdots \oplus F(n)$, and by Maschke’s theorem, each $S_j$-module $F(j)$ is the retract of a finite sum of copies of the group algebra $k[S_j]$ which corresponds to the representable $k\mathbb{P}(-, j)$.

So, inspired by Joyal’s work on combinatorial species, we make the following definition:

As we have mentioned, the category of polynomial species inherits two monoidal structures from $\mathbb{P}$ via Day convolution. Most important is the one coming from the additive monoidal structure on $\mathbb{P}$, which is given on the level of objects by adding natural numbers, and on the morphism level given by group homomorphisms

which juxtapose permutations. This can be linearized to give algebra maps

which give the monoidal category structure of $k\mathbb{P}$. This monoidal structure uniquely extends via Day convolution to the Cauchy completion $\widebar{k\mathbb{P}}$, which is intermediate between $k\mathbb{P}$ and the category of $Vect$-valued presheaves on $k\mathbb{P}$. The general formula for the Day convolution product applied to presheaves $F, G: \mathbb{P}^{op} \to Vect$ is

or, in other notation,

and by restriction this formula gives a tensor product on polynomial species. This tensor product is a kind of categorification of the usual definition of product of ordinary polynomials, where given

the $n^{th}$ Taylor coefficient of the product $F(x)G(x)$ is

So in summary:

Now, having defined Schur functors abstractly as pseudonatural transformations $U \to U$, the representability theorem together with the 2-categorical Yoneda lemma means that the category of Schur functors is equivalent to the category of symmetric monoidal linear functors on $\widebar{k \mathbb{P}}$. Accordingly, we calculate

In other words,

NB: This theorem refers only to the underlying category $U(\overline{k\mathbb{P}})$. In other words, this category certainly has linear tensor category structure as well, but this structure is not respected by Schur functor composition which we consider next.

Composition of Schur functors

Now we consider composition of Schur functors $U \to U$, or equivalently symmetric monoidal linear functors $\widebar{k\mathbb{P}} \to \widebar{k \mathbb{P}}$. Composition endows $[U, U]$ with a monoidal structure, and this monoidal structure transfers across the equivalence of the preceding theorem to a monoidal structure on the underlying category of Schur functors, or equivalently, polynomial species $\mathbb{P}^{op} \to FinDimVect$. We proceed to analyze this monoidal structure.

It may be easier to do this in reverse. Any Schur functor may regarded as a functor

This induces a symmetric monoidal functor, unique up to (unique) symmetric monoidal isomorphism:

Here $F^{\otimes m}$ is a Day convolution product of $m$ copies of $F$. Finally, the functor $F^\sim$ is linearized and extended (uniquely) to the linear Cauchy completion, to give a symmetric monoidal linear functor on $\widebar{k \mathbb{P}}$. The efficient tensor product description is

as this manifestly preserves colimits in the blank argument and therefore all colimits needed for the Cauchy completion. (And since the extension to the Cauchy completion is unique, this formula must be correct! The only question is whether this functor is valued in $\overline{k\mathbb{P}}$.)

In the language of species, this construction is called the substitution product, and is denoted $G \circ F$. This is morally correct because it is indeed an appropriate categorification of polynomial composition. However, to avoid overloading the symbol $\circ$ in ways that might be confusing, we will rename it $G \boxtimes F$. Thus,

In notation which looks slightly less abstract, this is the Schur object given by the formula

It should be noted that $(G \boxtimes F)(n)$ is indeed $0$ for $n \gt (deg G)(deg F)$, so that $G \boxtimes F$ is indeed a polynomial species. It is just the polynomial special case of the substitution product which is defined on general linear species $F, G: \mathbb{P}^{op} \to Vect$.

To summarize: we have equivalences between

• The category of pseudonatural transformations $U \to U$;

• The category of symmetric monoidal linear functors $\overline{k\mathbb{P}} \to \overline{k \mathbb{P}}$;

• The category $Schur = U(\overline{k\mathbb{P}})$.

The equivalence $Schur \to SymMonLinCauch(\overline{k \mathbb{P}}, \overline{k \mathbb{P}})$ takes a polynomial species $F$ to $- \boxtimes F$. Moreover, the associativity isomorphism is precisely a structure

of strong monoidal equivalence from $(Schur, \boxtimes)$ to the monoidal category $SymMonLinCauch(\overline{k\mathbb{P}}, \overline{k\mathbb{P}})$ under endofunctor composition. (The hexagonal coherence condition for a monoidal functor follows from the pentagon; one side of the hexagon is an identity since endofunctor composition is a strict monoidal product.)

The tensor product $\boxtimes$ on $Schur$ goes by another name: it is the plethystic tensor product.

Related concepts

• plethory
• Specht module

References

A nice introduction to Schur functors can be found here:

• William Fulton and Joe Harris, Representation Theory: a First Course, Springer, Berlin, 1991.

For a quick online introduction to Young tableaux and representations of the symmetric groups, try:

• Yufei Zhao, Young tableaux and the representations of the symmetric group, MIT. (web)

For more details on these topics, see:

• William Fulton, Young Tableaux, with Applications to Representation Theory and Geometry, Cambridge U. Press, 1997.

• Bruce E. Sagan, The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, Springer, Berlin, 2001.

A treatment of the category of Schur functors as the free 2-rig on one generator is in

• John Baez, Joe Moeller, Todd Trimble, Schur functors and categorified plethysm, (arXiv:2106.00190)

Species were invented here:

• André Joyal, Une théorie combinatoire des séries formelles, Adv. Math 42 (1981), 1–82.

• André Joyal: Foncteurs analytiques et espèces des structures, in Combinatoire Énumérative, Lecture Notes in Mathematics 1234, Springer, Berlin, 1986, pp. 126–159.

A standard reference on species is:

• François Bergeron, Gilbert Labelle, Pierre Leroux, Combinatorial Species and Tree-like Structures, Cambridge University Press, Cambridge 1998.

Here is an important new book on combinatorics which emphasizes the use of $Vect$-valued species:

• Marcelo Aguiar and Swapneel Mahajan, Monoidal Functors, Species and Hopf Algebras, to be published by Cambridge U. Press. (web)

Characteristic-free approach to Schur functors and the notion of Schur complexes are developed in

• Kaan Akin, David Alvin Buchsbaum, Jerzy Weyman, Schur functors and Schur complexes, Adv. Math. 44:3 (1982) 207–278
• Marcin Chałupnik, Extensions of Weyl and Schur functors, Homology, Homotopy and Applications 11(2), s. 27-48, 2009.

See also

• Martin Brandenburg, Operations on categories of modules are given by Schur functors (arXiv:1610.02180)

Old Stuff

John Baez: this stuff should go somewhere, perhaps:

John Baez: This stuff should get worked into the discussion near the end of how Schur functors are like polynomials…