John Baez
Schur functors I

Contents

Preface

This is a paper that John Baez and Todd Trimble are writing. There is a followup, Schur functors II.

Introduction

Classically, a Schur functor is a specific sort of functor

F:FinVectFinVectF: \Fin\Vect \to \Fin\Vect

where FinVectFin\Vect is the category of finite-dimensional complex space. Namely, it is one where F(V)F(V) is obtained by taking a tensor power of VV, say V nV^{\otimes n}, and then picking out the subspace that transforms according to a particular irreducible representation of the symmetric group S nS_n, which acts on V nV^{\otimes n} by permuting the tensor factors.

Irreducible complex representations of S nS_n correspond to nn-box Young diagrams, so Schur functors are usually described with the help of these. An nn-box Young diagram is simply a handy notation for a way of writing nn as a sum of natural numbers written in decreasing order. For example, this 17-box Young diagram:

<!-- Created with SVG-edit - http://svg-edit.googlecode.com/ --> Young diagram (5,4,4,2,1,1) Layer 1

describes the partition of 1717 as 5+4+4+2+1+15 + 4 + 4 + 2 + 1 + 1. However, it also can be used to construct an irreducible complex representation of the permutation group S 17S_{17}, and thus a Schur functor.

The relation between Young diagrams and Schur functors can be described using the group algebra of the symmetric group, [S n]\mathbb{C}[S_n]. Given an nn-box Young diagram λ\lambda, we can think of the operation ‘symmetrize with respect to permutations of the boxes in each row’ as an element p λ S[S n]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 λ A[S n]p^A_\lambda \in \mathbb{C}[S_n]. By construction, each of these elements is idempotent:

(p λ A) 2=p λ A,(p λ S) 2=p λ S (p^A_\lambda)^2 = p^A_\lambda \, , \; \; (p^S_\lambda)^2 = p^S_\lambda

Now, it is easy to see that the product of commuting idempotents is idempotent. The elements p Sp_S and p Ap_A do not commute, but amazingly, their product

p λ=p λ Ap λ S p_\lambda = p^A_\lambda p^S_\lambda

is still idempotent! The reason is that p λp_\lambda commutes with all permutations, so it lies in the center of the group algebra. This ensures that

p λ 2=p λ Ap λ Sp λ Ap λ S=p λ Ap λ Ap λ Sp λ S=p λ p_\lambda^2 = p^A_\lambda p^S_\lambda p^A_\lambda p^S_\lambda = p^A_\lambda p^A_\lambda p^S_\lambda p^S_\lambda = p_\lambda

Checking that p λp_\lambda commutes with all permutations is a calculation everyone must do for themselves.

This element p λ[S n]p_\lambda \in \mathbb{C}[S_n] is called the Young symmetrizer corresponding to the nn-box Young diagram λ\lambda. Since the group algebra k[S n]k[S_n] acts on V nV^{\otimes n} by permuting the factors, the Young symmetrizer gives a projection

p λ:V nV n p_\lambda : V^{\otimes n} \to V^{\otimes n}

whose range is a subspace called S λ(V)S_\lambda(V). Since p λp_\lambda commutes with everything in k[S n]k[S_n], this subspace is invariant under the action of S nS_n, and — as already mentioned — it is a direct sum of copies of a specific irreducible representation of S nS_n. But the point is this: there is a functor, called a Schur functor:

S λ:FinVectFinVect S_\lambda : \Fin\Vect \to \Fin\Vect

which has precisely this effect on objects:

VS λ(V) V \mapsto S_\lambda(V)

More generally, any finite direct sum of the Schur functors just described may also be called a Schur functor. For example:

  • For each n0n \geq 0, the n thn^{th} tensor power VV nV \mapsto V^{\otimes n} is a Schur functor.

  • If FF and GG are Schur functors, of course the functor VF(V)G(V)V \mapsto F(V) \oplus G(V) is a Schur functor.

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

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

There is a category SchurSchur with these more general Schur functors as objects and natural transformations between them as morphisms. In this article we would like to give a conceputal explanation of this category and some of its generalizations.

For starters, we can replace the complex numbers by any field kk of characteristic zero, and everything in our discussion still works. Even better, the resulting category Schur\Schur has has a nice description in terms of groupoid of finite sets and bijections. We will find it convenient to work with a skeleton of this groupoid, namely the permutation groupoid:

= n0S n, \mathbb{P} = \bigsqcup_{n \ge 0} S_n \, ,

where objects are natural numbers, all morphisms are automorphisms, and the automorphisms of the object nn form the group S nS_n. In these terms, SchurSchur is equivalent to the category where:

  • objects are functors F:FinVectF: \mathbb{P} \to \Fin\Vect such that F(n)={0}F(n) = \{0\} for all sufficiently large nn;

  • morphisms are natural transformations.

This allows us to think of SchurSchur as a subcategory of the category of representations of \mathbb{P}, by which we simply mean functors F:VectF : \mathbb{P} \to \Vect. Every irreducible representation of \mathbb{P} is finite-dimensional: it is really just an irreducible representation of some group S nS_n. Every representation of \mathbb{P} is a direct sum of these irreducibles. SchurSchur may be identified with the full subcategory consisting of finite direct sums of irreducibles.

Schur functors on more general categories

We have described Schur functors as special functors

F:FinVectFinVect F: \Fin\Vect \to \Fin\Vect

But in fact, these functors make sense much more broadly. In fact they can be applied to any ‘symmetric monoidal Cauchy complete linear category’. Here by linear category we mean a category enriched over VectVect, the category of vector spaces over a fixed field kk of characteristic zero. A symmetric monoidal linear category is a symmetric monoidal category where the tensor product distributes over the linear structure on morphisms. Such a category is Cauchy complete when:

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

  • idempotents split.

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

  • the category VectVect, consisting of vector spaces over any field kk of characteristic zero

  • the category FinVect\Fin\Vect, consisting of finite-dimensional vector spaces over kk

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

  • the category of super vector spaces, graded vectors or chain complexes over kk

  • for k=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 kk

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

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, CC. The group algebra k[S n]k[S_n] begins life as a monoid in the symmetric monoidal category FinVect\Fin\Vect. However, we shall explain how interpret it as living in CC by a “change of base” functor going from FinVect\Fin\Vect to CC. This will let us use the Young symmetrizers p λp_\lambda to construct idempotents on V kV^{\otimes k} for any object VCV \in C. Splitting these idempotents, we obtain the Schur functors S λ:CCS_\lambda : C \to C.

Change of base

To achieve the desired change of base, let MatMat be the linear category whose objects are integers m0m \geq 0 and whose morphisms mnm \to n are m×nm \times n matrices with entries in kk. Because CC is Cauchy complete and in particular has finite biproducts (direct sums), there is an evident linear functor

MatCMat \to C

which takes mm to I mI^m, the direct sum of mm copies of the tensor unit II. It is the unique linear functor taking 11 to II, up to unique linear isomorphism. In the case C=FinVectC = \Fin\Vect, the linear functor

MatFinVect,Mat \to \Fin\Vect,

taking 11 to kk, is a linear equivalence (exhibiting MatMat as a skeleton of CC). Because of this equivalence, we could equally well say that there is a linear functor

i:FinVectCi: \Fin\Vect \to C

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

I mI nI mn,I^m \otimes I^n \cong I^{m n},

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

In summary, we have the following proposition.

Proposition

There is exactly one symmetric monoidal linear functor i:FinVectCi: \Fin\Vect \to C, up to symmetric monoidal linear isomorphism.

The action of Young symmetrizers

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

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

λhom(V λ,V λ)\bigoplus_{\lambda} hom(V_\lambda, V_\lambda)

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

p λ 2=p λ. p_\lambda^2 = p_\lambda \, .

We are particularly interested in the case G=S nG = S_n. In this case, we call the idempotents p λp_\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]k[S_n]. Here we exploit the fact that

k[S n]= σS nk k[S_n] = \bigoplus_{\sigma \in S_n} k

is a monoid in the monoidal category FinVect\Fin\Vect. Since i:FinVectCi : \Fin\Vect \to C is a monoidal functor, it follows that ii carries k[S n]k[S_n] to a monoid in CC, which we again call k[S n]k[S_n] by abuse of notation. As an object of CC, we have

(1)k[S n] σS nI k[S_n] \cong \bigoplus_{\sigma \in S_n} I

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 XX is an object of CC, the monoid k[S n]k[S_n] acts on the tensor power X nX^{\otimes n}. To see this, note that for each σS n\sigma \in S_n, there is a corresponding symmetry isomorphism

σ:X nX n \sigma : X^{\otimes n} \to X^{\otimes n}

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

k[S n]X nX nk[S_n] \otimes X^{\otimes n} \to X^{\otimes n}

which is the desired action.

Finally, we would like to describe how each Young symmetrizer p λk[S n]p_\lambda \in k[S_n] acts on X nX^{\otimes n}. Quite generally, any element xk[S n]x \in k[S_n] gives a linear map from kk to k[S n]k[S_n], namely the unique map sending 11 to xx. Applying the functor ii to this, we obtain a morphism which by abuse of language we call

x:Ik[S n] x : I \to k[S_n]

This then yields an endomorphism

x˜:X nX n \widetilde{x}: X^{\otimes n} \to X^{\otimes n }

given as the composite

X nIX nx1k[S n]X nX n X^{\otimes n} \stackrel{\cong}{\longrightarrow} I \otimes X^{\otimes n} \stackrel{x \otimes 1}{\longrightarrow} k[S_n] \otimes X^{\otimes n} \longrightarrow X^{\otimes n}

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

xy˜=x˜y˜ \widetilde{x y} = \widetilde{x} \widetilde{y}

Thus for any Young symmetrizer p λp_\lambda, the morphism

p˜ λ:X nX n \widetilde{p}_\lambda : X^{\otimes n} \to X^{\otimes n}

is idempotent, because p λp_\lambda is.

Constructing Schur functors

By construction, the morphisms

p˜ λ:X nX n \widetilde{p}_\lambda : X^{\otimes n} \to X^{\otimes n}

are the components of a natural transformation from the functor XX nX \mapsto X^{\otimes n} to itself. Since idempotents split in CC, we can form the cokernel of 1p˜ λ1 - \widetilde{p}_\lambda, or in other words, the coequalizer of the pair

X n1p˜ λX n X^{\otimes n} \stackrel{\overset{\widetilde{p}_\lambda}{\to}}{\underset{1}{\to}} X^{\otimes n}
Definition

For any Young diagram λ\lambda, the Schur functor S λ:CCS_\lambda: C \to C is defined as follows. Given an object XX of CC, let S λ(X)S_\lambda(X) be the cokernel of p˜ λ:X nX n \widetilde{p}_\lambda : X^{\otimes n} \to X^{\otimes n}. Given a morphism f:XYf: X \to Y in CC, let S λ(f)S_\lambda(f) be the unique map S λ(X)S λ(Y)S_\lambda(X) \to S_\lambda(Y) such that

X n S λ(X) f n S λ(f) Y n S ν(Y)\array{ X^{\otimes n} & \to & S_\lambda(X) \\ f^{\otimes n} \downarrow \; \; & & \downarrow S_\lambda(f) \\ Y^{\otimes n} & \to & S_\nu(Y) }

commutes, where the horizontal arrows are the cokernel maps.

More generally we can define a Schur functor

S R:CC S_R : C \to C

for any finite-dimensional representation RR of S nS_n, as follows. We can write RR as a finite direct sum of irreducible representations:

R= iV λ i R = \bigoplus_i V_{\lambda_i}

and then define

S R()= iS λ i(). S_R(-) = \bigoplus_i S_{\lambda_i}(-) \, .

Schur functors are “natural”

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

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

S λ,D(GX)=V λ,D S n(GX) n V λ,D S nG(X n) G(V λ,C) S nG(X n) G(V λ,C S nX n) G(S λ,C(X))\array{ S_{\lambda, D} (G X) = V_{\lambda, D} \otimes_{S_n} (G X)^{\otimes n} & \cong & V_{\lambda, D} \otimes_{S_n} G(X^{\otimes n}) \\ & \cong & G(V_{\lambda, C}) \otimes_{S_n} G(X^{\otimes n}) \\ & \cong & G(V_{\lambda, C} \otimes_{S_n} X^{\otimes n}) \\ & \cong & G(S_{\lambda, C}(X)) }

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

If RR is any representation, then by writing RR as a direct sum of irreducible representations V λV_\lambda and using the fact that GG preserves direct sums, we have more generally

S R,DGGS R,C.S_{R, D} \circ G \cong G \circ S_{R, C}.

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

Conceptual description of Schur functors

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

S R:CCS_R: C \to C

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

C G D S R,C S R,D C G D\array{ C & \stackrel{G}{\to} & D \\ S_{R, C} \downarrow & & \downarrow S_{R, D} \\ C & \underset{G}{\to} & D }

commutes up to a canonical isomorphism ϕ G:S R,DGGS R,C\phi_G: S_{R, D} \circ G \cong G \circ S_{R, C}, and moreover these ϕ G\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 RR of the symmetric group. The natural isomorphisms ϕ G\phi_G which relate the Schur functors across change of base G:CDG: C \to D are pleasant to observe, but surely this is just some piddling general nonsense in the larger story of Schur functors S RS_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 RS_R are constructed in a uniform (or “polymorphic”) way across all symmetric monoidal Cauchy complete linear categories CC, and this construction is natural with respect to symmetric monoidal change of base functors G:CDG: C \to D. Or rather: not natural in a strict sense, but pseudonatural in the sense that naturality squares commute up to isomorphism ϕ G\phi_G. Now pseudonaturality is a very general phenomenon in 2-category theory. So the question is: among all such pseudonatural transformations SS, what is special about the Schur functors S RS_R? What extra properties pick out exactly the Schur functors S RS_R from the class of all pseudonatural transformations SS?

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

U:SymMonLinCauchCatU: \Sym\Mon\Lin\Cauch \to Cat

where:

Definition

SymMonLinCauch is the 2-category with

  • small symmetric monoidal Cauchy complete linear categories as objects,

  • symmetric monoidal linear functors as morphisms,

  • symmetric monoidal linear natural transformations as 2-morphisms.

As we shall see, Schur functors correspond to pseudonatural transformations from UU 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:

Definition

Given two 2-functors U,V:SCU, V: S \stackrel{\to}{\to} C between 2-categories, a pseudonatural transformation ϕ:UV\phi: U \to V is a rule that assigns to each 0-cell ss of SS a 1-cell ϕ(s):U(s)V(s)\phi(s): U(s) \to V(s) of CC, and to each 1-cell f:rsf: r \to s of SS an invertible 2-cell ϕ(f)\phi(f) of CC:

U(r) U(f) U(s) ϕ(r) ϕ(f) ϕ(s) V(r) V(f) V(s)\array{ U(r) & \stackrel{U(f)}{\to} & U(s) \\ \phi(r) \downarrow & \phi(f) \swArrow & \downarrow \phi(s) \\ V(r) & \underset{V(f)}{\to} & V(s) }

such that the following pasting diagram equalities hold:

U(r) U(f) U(s) U(g) U(t) U(r) U(gf) U(t) ϕ(r) ϕ(f) ϕ(s) ϕ(g) ϕ(t) = ϕ(r) ϕ(gf) ϕ(t) V(r) V(f) V(s) V(g) V(t) V(r) V(gf) V(t) \array{ U(r) & \stackrel{U(f)}{\to} & U(s) & \stackrel{U(g)}{\to} & U(t) & & U(r) & \stackrel{U(g f)}{\to} & U(t) & & \\ \phi(r) \downarrow & \phi(f) \swArrow & \downarrow \phi(s) & \phi(g) \swArrow & \downarrow \phi(t) & = & \phi(r) \downarrow & \phi(g f) \swArrow & \downarrow \phi(t) & & & \\ V(r) & \underset{V(f)}{\to} & V(s) & \underset{V(g)}{\to} & V(t) & & V(r) & \underset{V(g f)}{\to} & V(t) & & & }

and

U(r) U(1 r) U(r) ϕ(r) ϕ(1 r) ϕ(r) = 1 ϕ(r) V(r) V(1 r) V(r) \array{ U(r) & \stackrel{U(1_r)}{\to} & U(r) & & \\ \phi(r) \downarrow & \phi(1_r) \swArrow & \downarrow \phi(r) & = & 1_{\phi(r)} \\ V(r) & \underset{V(1_r)}{\to} & V(r) & & }
Definition

With notation as above, let ϕ,ψ:UV\phi, \psi: U \to V be two pseudonatural transformations. A modification x:ϕψx: \phi \to \psi is a rule which associates to each 0-cell ss of SS a 2-cell x(s):ϕ(s)ψ(t)x(s): \phi(s) \to \psi(t) of CC, such that the following compatibility condition holds:

tincan.png:pic

We now propose our conceptual definition of Schur functor:

Definition

An (abstract) Schur functor is a pseudonatural transformation S:UUS: U \to U, where

U:SymMonLinCauchCatU: \Sym\Mon\Lin\Cauch \to Cat

is the forgetful 2-functor. A morphism of Schur functors is a modification between such pseudonatural transformations.

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 SchurSchur 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 CatCat let us consider SetSet, and instead of SymMonLinCauch\Sym\Mon\Lin\Cauch let us consider CommRing\Comm\Ring. There is a forgetful functor

U:CommRingSet. U : \Comm\Ring \to Set \, .

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

P R:xP(x) P_R : x \mapsto P(x)

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

To see this, note that the forgetful functor

U:CommRingSet U : \Comm\Ring \to Set

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

F:SetCommRing. F : Set \to \Comm\Ring \, .

The free commutative ring on a 1-element set is

F(1)[x] F(1) \cong \mathbb{Z}[x]

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

U(R)hom(1,U(R))hom(F(1),R). U(R) \cong hom(1, U(R)) \cong hom(F(1), R) \, .

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

[U,U][hom(F(1),),hom(F(1),)]hom(F(1),F(1))U(F(1))[U,U] \cong [hom(F(1), -), hom(F(1), -)] \cong hom(F(1), F(1)) \simeq U(F(1))

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

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

U:SymMonLinCauchCatU: \Sym\Mon\Lin\Cauch \to Cat

The key is that SchurSchur 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 kk \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 k¯\widebar{k \mathbb{P}} to denote the Cauchy completion of the linearization of \mathbb{P}. As we shall see, k¯\widebar{k \mathbb{P}} is equivalent to the category of Schur functors. But first:

Theorem

The underlying 2-functor

U:SymMonLinCauchCatU: \Sym\Mon\Lin\Cauch \to Cat

is represented by k¯\widebar{k \mathbb{P}}. In other words:

U()hom(,k¯) U(-) \simeq hom(-, \widebar{k \mathbb{P}})
Proof (Sketch)

It is well-known that the permutation category \mathbb{P}, whose objects are integers m0m \geq 0 and whose morphisms are precisely automorphisms mmm \to m given by permutation groups S mS_m, is the representing object for the underlying 2-functor

U 0:SymMonCatCat.U_0: \Sym\Mon\Cat \to \Cat \, .

Let SymMonLin\Sym\Mon\Lin denote the 2-category of small symmetric monoidal linear (but not necessarily Cauchy complete) categories, and let Lin\Lin denote the 2-category of small linear categories. Let k():CatLink(-): Cat \to Lin denote linearization, given by change of base

k:SetVectk \cdot - : Set \to Vect

applied to a SetSet-enriched category (C 0,hom:C 0×C 0Set)(C_0, hom: C_0 \times C_0 \to Set) to yield a linear category (C 0,C 0×C 0Vect)(C_0, C_0 \times C_0 \to Vect). k():CatLink(-): Cat \to Lin is left 2-adjoint to the underlying 2-functor U 0:LinCatU_0: Lin \to Cat. For this, we use the fact that if VV is a nice closed category (here VectVect) – in particular cocomplete – then the lax monoidal functor hom(I,):VSet\hom(I, -): V \to Set has a left adjoint I:SetV- \cdot I: Set \to V (here linearization) which is strong (symmetric) monoidal. This induces a 2-functor Cat=SetCat = Set-CatVCat \to V-CatCat which is strong 2-symmetric monoidal. It therefore sends symmetric pseudomonoids in SetSet-CatCat to symmetric pseudomonoids in VV-CatCat. In other words, it sends symmetric monoidal categories to symmetric monoidal linear categories. Therefore, the 2-adjunction k()U 0k(-) \dashv U_0 between CatCat and LinCat\Lin\Cat lifts to one between SymMonCat\Sym\Mon\Cat and SymMonLinCat\Sym\Mon\Lin\Cat:

(k():SymMonCatSymMonLin)(U 1:SymMonLinSymMonCat)(k(-): \Sym\Mon\Cat \to Sym\Mon\Lin) \dashv (U_1: \Sym\Mon\Lin \to \Sym\Mon\Cat)

Finally, let LinCauch\Lin\Cauch denote the 2-category of small Cauchy complete linear categories. The linear Cauchy completion gives a 2-reflector ()¯:LinLinCat\widebar{(-)}: \Lin \to \Lin\Cat which is left 2-adjoint to the 2-embedding i:LinCauchLini: \Lin\Cauch \to \Lin, and again the 2-adjunction ()¯i\widebar{(-)} \dashv i lifts to the level of symmetric monoidal structure to give a 2-adjunction

(()¯:SymMonLinSymMonLinCauch)(U 2:SymMonLinCauchSymMonLin)(\widebar{(-)}: \Sym\Mon\Lin \to \Sym\Mon\Lin\Cauch) \dashv (U_2: \Sym\Mon\Lin\Cauch \to \Sym\Mon\Lin)

For this, the key fact is that if ABA \otimes B denotes the tensor product of two VV-enriched categories, then there is a canonical enriched functor A¯B¯AB¯\overline{A} \otimes \overline{B} \simeq \overline{A \otimes B} making Cauchy completion into a lax 2-monoidal functor on VV-CatCat. Even better, it is lax 2-symmetric monoidal. So, it sends symmetric pseudomonoids to symmetric pseudomonoids. In this case, then, it sends symmetric monoidal linear categories to symmetric monoidal linear Cauchy-complete categories.

Putting this all together, the underlying functor U:SymMonLinCauchCatU: \Sym\Mon\Lin\Cauch \to Cat is the evident composite

SymMonLinCauchU 2SymMonLinU 1SymMonCatU 0Cat\Sym\Mon\Lin\Cauch \stackrel{U_2}{\to} \Sym\Mon\Lin \stackrel{U_1}{\to} \Sym\Mon\Cat \stackrel{U_0}{\to} \Cat

and therefore we have pseudonatural equivalences

SymMonLinCauch(k()¯,) SymMonLin(k(),U 2) SymMonCat(,U 1U 2) U 0U 1U 2 U\array{ \Sym\Mon\Lin\Cauch(\widebar{k(\mathbb{P})}, -) & \cong & \Sym\Mon\Lin(k(\mathbb{P}), U_2 -) \\ & \cong & \Sym\Mon\Cat(\mathbb{P}, U_1 U_2 -) \\ & \cong & U_0 U_1 U_2 \\ & \cong & U }

so that k¯\widebar{k \mathbb{P}} is the representing object.

Structure of the representing object

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

F: opFinVectF: \mathbb{P}^{op} \to \Fin\Vect

where F(n)=0F(n) = 0 for large enough nn. 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 FF is a sum of monomials F(0)F(1)F(n)F(0) \oplus F(1) \oplus \cdots \oplus F(n), and by Maschke’s theorem, each S jS_j-module F(j)F(j) is the retract of a finite sum of copies of the group algebra k[S j]k[S_j] which corresponds to the representable k(,j)k\mathbb{P}(-, j).

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

Definition

A polynomial species is a functor F: opFinVectF: \mathbb{P}^{op} \to \Fin\Vect where F(n)=0F(n) = 0 for all sufficiently large nn. A morphism of polynomial species is a natural transformation between such functors.

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

S m×S nS m+nS_m \times S_n \to S_{m+n}

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

k[S m]k[S n]k[S m+n]k[S_m] \otimes k[S_n] \to k[S_{m+n}]

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

(FG)(n)= j+k=n(F(j)G(k)) S j×S kk[S n](F G)(n) = \sum_{j+k = n} (F(j) \otimes G(k)) \otimes_{S_j \times S_k} k[S_n]

or, in other notation,

(FG)(n)= j+k=nInd S j×S k S nF(j)G(k)(F G)(n) = \sum_{j+k = n} Ind_{S_j \times S_k}^{S_n} F(j) \otimes G(k)

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

F(x)= 0jMf jx jj!G(x)= 0kNg kx kk!F(x) = \sum_{0 \leq j \leq M} \frac{f_j x^j}{j!} \qquad G(x) = \sum_{0 \leq k \leq N} \frac{g_k x^k}{k!}

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

j+k=nn!j!k!f jg k\sum_{j+k = n} \frac{n!}{j! k!} f_j g_k

So in summary:

Theorem

k¯\widebar{k \mathbb{P}} is equivalent to the symmetric monoidal category of polynomial species.

Now, having defined Schur functors abstractly as pseudonatural transformations UUU \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 k¯\widebar{k \mathbb{P}}. Accordingly, we calculate

[U,U]SymMonLinCauch(k¯,k¯)U(k¯)[U, U] \cong \Sym\Mon\Lin\Cauch(\widebar{k\mathbb{P}}, \widebar{k\mathbb{P}}) \cong U(\widebar{k\mathbb{P}})

In other words,

Theorem

The category SchurSchur is equivalent to the category of polynomial species opFinVect\mathbb{P}^{op} \to \Fin\Vect.

NB: This theorem refers only to the underlying category U(k¯)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 UUU \to U, or equivalently symmetric monoidal linear functors k¯k¯\widebar{k\mathbb{P}} \to \widebar{k \mathbb{P}}. Composition endows [U,U][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 opFinVect\mathbb{P}^{op} \to \Fin\Vect. We proceed to analyze this monoidal structure.

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

1Fk¯.1 \stackrel{F}{\to} \overline{k\mathbb{P}} \, .

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

F k¯:mF m\mathbb{P} \stackrel{F^\sim}{\to}\overline{k\mathbb{P}}: m \mapsto F^{\otimes m}

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

F :k¯k¯- \otimes_{\mathbb{P}} F^\sim: \overline{k\mathbb{P}} \to \overline{k\mathbb{P}}

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 k¯\overline{k\mathbb{P}}.)

In the language of species, this construction is called the substitution product, and is denoted GFG \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 GFG \boxtimes F. Thus,

GF=G F G \boxtimes F = G \otimes_{\mathbb{P}} F^\sim

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

(GF)(n)= k0G(k) S kF k(n)(G \boxtimes F)(n) = \sum_{k \geq 0} G(k) \otimes_{S_k} F^{\otimes k}(n)

It should be noted that (GF)(n)(G \boxtimes F)(n) is indeed 00 for n>(degG)(degF)n \gt (deg G)(deg F), so that GFG \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: opVectF, G: \mathbb{P}^{op} \to Vect.

Proposition

The product \boxtimes makes the category of polynomial species into a monoidal category. The unit for this product is polynomial species XX given by the representable (,1): opFinVect\mathbb{P}(-, 1): \mathbb{P}^{op} \to \Fin\Vect.

Proof (Sketch)

The following proof is adapted from a similar argument due to Max Kelly [ref]: we exhibit an associativity isomorphism α:(F)G(FG)\alpha: (- \boxtimes F) \boxtimes G \to - \boxtimes (F \boxtimes G) on the basis of universal properties. The point is that by the universal property of k¯\overline{k\mathbb{P}}, the category of functors

F:1k¯F: 1 \to \overline{k\mathbb{P}}

is equivalent to the category of symmetric monoidal linear functors

H:k¯k¯H: \overline{k\mathbb{P}} \to \overline{k\mathbb{P}}

The correspondence in one direction takes FF to the symmetric monoidal functor H=FH = - \boxtimes F, and in the other direction takes HH to F=H(X)F = H(X). By the equivalence, we have a unit isomorphism XFFX \boxtimes F \cong F. Also by this equivalence, symmetric monoidal linear transformations between symmetric monoidal linear functors of the form

(F)G(FG)(- \boxtimes F) \boxtimes G \to - \boxtimes (F \boxtimes G)

are in natural bijection with morphisms (XF)GX(FG)(X \boxtimes F) \boxtimes G \to X \boxtimes (F \boxtimes G), which by the unit isomorphism reduce to morphisms FGFGF \boxtimes G \to F \boxtimes G. Thus, corresponding to the identity on FGF \boxtimes G we obtain an associativity map α:(F)G(FG)\alpha: (- \boxtimes F) \boxtimes G \to - \boxtimes (F \boxtimes G). By similar arguments that appeal to the universal property of k¯\overline{k \mathbb{P}}, we get all the required axioms: the invertibility of α\alpha, the pentagon, etc.

To summarize: we have equivalences between

  • The category of pseudonatural transformations UUU \to U;

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

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

The equivalence SchurSymMonLinCauch(k¯,k¯)Schur \to \Sym\Mon\Lin\Cauch(\overline{k \mathbb{P}}, \overline{k \mathbb{P}}) takes a polynomial species FF to F- \boxtimes F. Moreover, the associativity isomorphism is precisely a structure

(G)(F)(FG)(- \boxtimes G) \circ (- \boxtimes F) \to - \boxtimes (F \boxtimes G)

of strong monoidal equivalence from (Schur,)(Schur, \boxtimes) to the monoidal category SymMonLinCauch(k¯,k¯)\Sym\Mon\Lin\Cauch(\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 SchurSchur goes by another name: it is the plethystic tensor product.

References

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

  • 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.

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

  • Marcelo Aguiar and Swapneel Mahajan, Monoidal functors, species and Hopf algebras, available online.

Old Stuff

There is also some old material that may still be useful.

Revised on July 31, 2013 16:51:06 by John Baez (137.132.250.13)