Schur functor

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### Context
#### Representation theory
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[[!include representation theory - contents]]
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#Contents#
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## Idea and definition ##

Classically, a 'Schur functor' is a specific sort of [[functor]]

$$
  S \,\colon\, FinDimVect \longrightarrow FinDimVect
$$

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 '[[natural transformation|naturally]]' --- or more precisely, '[[pseudonatural transformation|pseudonaturally]]' defined on all [[symmetric monoidal category|symmetric monoidal]] [[linear category|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 representation]]s of $S_n$ correspond to $n$-box [[Young diagram|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:

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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]]:
$$ (p^A_\lambda)^2 = p^A_\lambda \, , \; \; (p^S_\lambda)^2 = p^S_\lambda  $$
Now, it is easy to see that the product of commuting idempotents is idempotent.   The elements $p^S_\lambda$ and $p^A_\lambda$ do not commute, but amazingly, their product
$$ p_\lambda = p^A_\lambda p^S_\lambda $$
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**:
$$  S_\lambda \colon FinDimVect \to FinDimVect $$
defined on any finite-dimensional vector space $X$ as follows:
$$  S_\lambda(X) = p_\lambda X^{\otimes n} $$
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
$$   V_\lambda = p_\lambda \mathbb{C}[S_n] \subseteq \mathbb{C}[S_n] $$
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
$$   X^{\otimes n} \cong \mathbb{C}[S_n] \otimes_{\mathbb{C}[S_n]} X^{\otimes n} $$
It follows that
$$ \array{ S_\lambda(X) &=& p_\lambda X^{\otimes n} \\
&\cong& p_\lambda \mathbb{C}[S_n] \otimes_{\mathbb{C}[S_n]} X^{\otimes n} \\
&\cong& V_\lambda \otimes_{\mathbb{C}[S_n]} X^{\otimes n}
}
$$
These isomorphisms here are natural, so there is no harm in _defining_ the Schur functor $S_\lambda$ by
$$   S_\lambda(-) = V_\lambda \otimes_{\mathbb{C}[S_n]} (-)^{\otimes n} $$
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
$$  S_R \colon FinDimVect \to FinDimVect $$
for _any_ representation $R$ of $S_n$, as follows:
$$   S_R(-) = R \otimes_{\mathbb{C}[S_n]} (-)^{\otimes n} \, . $$
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**
$$   S_R(-) = \bigoplus_{n \ge 0} R_n \otimes_{\mathbb{C}[S_n]} (-)^{\otimes n} \, . $$

* 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]]:
$$ \mathbb{P} = \bigsqcup_{n \ge 0} S_n \, , $$
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
$$R : \mathbb{P} \to Vect $$
But conceptually, the importance of $\mathbb{P}$ is that it is a [[skeleton]] of the groupoid of finite sets and bijections.  So,
$$ \mathbb{P} \simeq core(FinSet) \, .$$

As a result, any Schur functor gives a functor
$$    R : core(FinSet) \to Vect $$
Following [[Andre Joyal|Joyal]]'s work on combinatorics, such functors are known as $Vect$-valued [[species]], or $Vect$-valued [[structure type|structure types]].  The idea here is that just as an ordinary Set-valued species
$$    T : core(FinSet) \to Set $$
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
$$F: Set \to Vect \, ,$$
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]]:

* {#AMbook} [[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
$$   S_R(X) = \bigoplus_{n \ge 0} R_n \otimes_{\mathbb{C}[S_n]} X^{\otimes n} $$
gives rise to a formal power series called its **generating function**:
$$      \sum_{n \ge 0} \frac{dim(R_n) \, x^n}{n!}  \, , $$
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 category|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

$$ (F \oplus G)(V) = F(V) \oplus G(V) $$

and

$$ (F \otimes G)(V) = F(V) \otimes G(V) $$

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](#AMbook) 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 category|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 category|symmetric monoidal]] [[Cauchy complete category|Cauchy complete]] [[linear category]] on one object.

## Schur functors on more general categories ##

We have described Schur functors as special functors

$$  F \,\colon\, FinDimVect \to FinDimVect  $$

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 category|symmetric monoidal]] [[Cauchy complete category|Cauchy complete]] [[linear category]] ("[[tensor category]]").  Here by **linear category** we mean a category [[enriched 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

* [[idempotent|idempotents]] [[split idempotent|split]].

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 space|graded vector spaces]] or [[chain complexes]] over $k$

* for $k = \mathbb{R}$ or $\mathbb{C}$, the category of finite-dimensional real or complex [[vector bundle|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

$$Mat \to C$$

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

$$Mat \to FinDimVect,$$

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

$$i \colon FinDimVect \to C$$

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

$$I^m \otimes I^n \cong I^{m n},$$

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

In summary, we have the following proposition.

+-- {: .num_prop}
######Proposition
There is exactly one symmetric monoidal linear functor
$i \colon FinDimVect \to C$, up to symmetric monoidal linear isomorphism.
=--

### 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

$$\bigoplus_{\lambda} hom(V_\lambda, V_\lambda)$$

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]]:

$$ p_\lambda^2 = p_\lambda \, . $$

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

$$   k[S_n] = \bigoplus_{\sigma \in S_n}  k $$

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

\[  k[S_n] \cong \bigoplus_{\sigma \in S_n}  I  \label{sum} \]

There is a general concept of what it means for a monoid in a monoidal category to [[action|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

$$ \sigma : X^{\otimes n} \to X^{\otimes n} $$

Putting these together with the help of \eqref{sum} we obtain a morphism

$$k[S_n] \otimes X^{\otimes n} \to X^{\otimes n}$$

which is the desired action.

Finally, we would like to describe how each Young symmetrizer $p_\lambda \in k[S_n]$ 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

$$ x : I \to k[S_n]$$

This then yields an endomorphism

$$ \widetilde{x}: X^{\otimes n} \to X^{\otimes n }$$

given as the composite

$$ X^{\otimes n} \stackrel{\cong}{\longrightarrow} I \otimes X^{\otimes n} \stackrel{x \otimes 1}{\longrightarrow} k[S_n] \otimes X^{\otimes n} \longrightarrow X^{\otimes n} $$

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

$$ \widetilde{x y} = \widetilde{x} \widetilde{y} $$

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

$$ \widetilde{p}_\lambda : X^{\otimes n} \to X^{\otimes n} $$

is idempotent, because $p_\lambda$ is.

### Constructing Schur functors ###

By construction, the morphisms

$$ \widetilde{p}_\lambda : X^{\otimes n} \to X^{\otimes n} $$

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

$$ X^{\otimes n} \stackrel{\widetilde{p}_\lambda}{\underset{1}{\rightrightarrows}} X^{\otimes n} $$

+-- {: .num_defn}
######Definition
For any Young diagram $\lambda$, the **Schur functor** $S_\lambda: C \to C$ is defined as follows.  Given an object $X$ of $C$, let $S_\lambda(X)$ be the cokernel of $(1 - \widetilde{p}_\lambda) : X^{\otimes n} \to X^{\otimes n}$.  Given a morphism $f: X \to Y$ in $C$, let $S_\lambda(f)$ be the unique map $S_\lambda(X) \to S_\lambda(Y)$ such that
$$\array{
X^{\otimes n} & \to & S_\lambda(X) \\
f^{\otimes n} \downarrow \; \; & & \downarrow S_\lambda(f) \\
Y^{\otimes n} & \to & S_\nu(Y)
}$$
commutes, where the horizontal arrows are the cokernel maps.
=--

More generally we can define a **Schur functor**

$$  S_R : C \to C  $$

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

$$  R = \bigoplus_i V_{\lambda_i}  $$

and then define

$$  S_R(-) = \bigoplus_i S_{\lambda_i}(-) \, .$$

### 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

$$\array{
S_{\lambda, D} (G X) = V_{\lambda, D} \otimes_{S_n} (G X)^{\otimes n} & \cong & V_{\lambda, D} \otimes_{S_n} G(X^{\otimes n}) \\
 & \cong & G(V_{\lambda, C}) \otimes_{S_n} G(X^{\otimes n}) \\
 & \cong & G(V_{\lambda, C} \otimes_{S_n} X^{\otimes n}) \\
 & \cong & G(S_{\lambda, C}(X))
}$$

where the first isomorphism uses the symmetric monoidal structure of $G$; 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

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

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

$$S_R: C \to C$$

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

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

commutes up to a canonical isomorphism $\phi_G: S_{R, D} \circ G \cong G \circ S_{R, C}$, 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

$$U: SymMonLinCauch \to Cat$$

where:

+-- {: .num_defn}
######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 $U$ 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:

+-- {: .num_defn}
######Definition
Given two 2-functors $U, V: S \stackrel{\to}{\to} C$ between 2-categories, a **pseudonatural transformation** $\phi: U \to V$ is a rule that assigns to each 0-cell $s$ of $S$ a 1-cell $\phi(s): U(s) \to V(s)$ of $C$, and to each 1-cell $f: r \to s$ of $S$ an invertible 2-cell $\phi(f)$ of $C$:
$$\array{
U(r) & \stackrel{U(f)}{\to} & U(s) \\
\phi(r) \downarrow & \phi(f) \swArrow & \downarrow \phi(s) \\
V(r) & \underset{V(f)}{\to} & V(s)
}$$
such that the following pasting diagram equalities hold:
$$\array{
U(r) & \stackrel{U(f)}{\to} & U(s) & \stackrel{U(g)}{\to} & U(t) & & U(r) & \stackrel{U(g f)}{\to} & U(t) & & \\
\phi(r) \downarrow & \phi(f) \swArrow & \downarrow \phi(s) & \phi(g) \swArrow & \downarrow \phi(t) & = & \phi(r) \downarrow & \phi(g f) \swArrow & \downarrow \phi(t) & & & \\
V(r) & \underset{V(f)}{\to} & V(s) & \underset{V(g)}{\to} & V(t) & & V(r) & \underset{V(g f)}{\to} & V(t) & & &
}$$
and
$$\array{
 U(r) & \stackrel{U(1_r)}{\to} & U(r) & & \\
\phi(r) \downarrow & \phi(1_r) \swArrow & \downarrow \phi(r) & = & 1_{\phi(r)} \\
V(r) & \underset{V(1_r)}{\to} & V(r) & &
}$$

There is also a requirement that the assignation f \mapsto \phi(f) should be natural as f varies along a $2$-cell \alpha: f \Rightarrow g. That is, \phi(g).U\alpha =V\alpha.\phi(f).
=--

+-- {: .num_defn}
######Definition
With notation as above, let $\phi, \psi: U \to V$ be two pseudonatural transformations. A **modification** $x: \phi \to \psi$ is a rule which associates to each 0-cell $s$ of $S$ a 2-cell $x(s): \phi(s) \to \psi(t)$ of $C$, such that the following compatibility condition holds:
=--

+-- {: style="text-align:center"}
[[tincan.png:pic]]
=--

We now propose our conceptual definition of Schur functor:

+-- {: .num_defn}
######Definition
An (abstract) **Schur functor** is a pseudonatural transformation $S: U \to U$, where
$$U: SymMonLinCauch \to Cat$$
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 $Schur$ 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

$$ U : CommRing \to Set \, . $$

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

$$ P_R : x \mapsto P(x) $$

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

$$ U : CommRing \to Set  $$

has a left adjoint, the 'free commutative ring' functor

$$ F : Set \to CommRing \, . $$

The free commutative ring on a 1-element set is

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

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

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

So, we say $F(1)$ [[representable functor|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 $U(F(1))$:

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

In the first step here we use the representability $U \cong hom(F(1), -)$; 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

$$U: SymMonLinCauch \to Cat$$

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

+-- {: .query}

[[John Baez]]: I added explanatory remarks above.  Okay?

=--

## 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:

+-- {: .num_theorem}
######Theorem
The underlying 2-functor
$$U: SymMonLinCauch \to Cat$$
is represented by $\widebar{k \mathbb{P}}$.  In other words:

$$   U(-) \simeq hom(-, \widebar{k \mathbb{P}}) $$

=--

+-- {: .proof}
######Proof (Sketch)

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

$$U_0: SymMonCat \to Cat \, .$$

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

$$k \cdot - : Set \to Vect$$

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

$$(k(-): SymMonCat \to SymMonLin) \dashv (U_1: SymMonLin \to SymMonCat)$$

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

$$(\widebar{(-)}: SymMonLin \to SymMonLinCauch) \dashv (U_2: SymMonLinCauch \to SymMonLin)$$

For this, the key fact is that if $A \otimes B$ denotes the tensor product of two $V$-enriched categories, then there is a canonical enriched functor  $\overline{A} \otimes \overline{B} \simeq \overline{A \otimes B}$ making Cauchy completion into a lax 2-monoidal functor on $V$-$Cat$.  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 $U: SymMonLinCauch \to Cat$ is the evident composite

$$SymMonLinCauch \stackrel{U_2}{\to} SymMonLin \stackrel{U_1}{\to} SymMonCat \stackrel{U_0}{\to} Cat$$

and therefore we have pseudonatural equivalences

$$\array{
SymMonLinCauch(\widebar{k(\mathbb{P})}, -) & \cong & SymMonLin(k(\mathbb{P}), U_2 -) \\
 & \cong & SymMonCat(\mathbb{P}, U_1 U_2 -) \\
 & \cong & U_0 U_1 U_2 \\
 & \cong & U
}$$

so that $\widebar{k \mathbb{P}}$ is the representing object.
=--

+-- {: .query}

[[John Baez]]: I took remarks from the query box here and used them to improve the proof above.  However, it could still use more improvement.  Could you polish it up a bit, Todd?

=--

### 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

$$F \colon \mathbb{P}^{op} \to FinDimVect$$

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:

+-- {: .num_defn}
######Definition
A **polynomial species** is a functor $F: \mathbb{P}^{op} \to FinDimVect$ where $F(n) = 0$ for all sufficiently large $n$.  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

$$S_m \times S_n \to S_{m+n}$$

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

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

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

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

+-- {: .query}

[[John Baez]]: Todd had written $S_n$ where I have put $k[S_n]$ here.  Okay?

=--

or, in other notation,

$$(F G)(n) = \sum_{j+k = n} Ind_{S_j \times S_k}^{S_n} F(j) \otimes G(k)$$

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

$$F(x) = \sum_{0 \leq j \leq M} \frac{f_j x^j}{j!} \qquad G(x) = \sum_{0 \leq k \leq N} \frac{g_k x^k}{k!}$$

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

$$\sum_{j+k = n} \frac{n!}{j! k!} f_j g_k$$

So in summary:

+-- {: .num_theorem}
######Theorem
$\widebar{k \mathbb{P}}$ is equivalent to the symmetric monoidal category of polynomial species.
=--

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

$$[U, U] \cong SymMonLinCauch(\widebar{k\mathbb{P}}, \widebar{k\mathbb{P}}) \cong U(\widebar{k\mathbb{P}})$$

In other words,

+-- {: .num_theorem}
######Theorem
The category $Schur$ is equivalent to the category of polynomial species $\mathbb{P}^{op} \to FinDimVect$.
=--

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

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

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

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

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

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

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 $\overline{k\mathbb{P}}$.)

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,

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

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

$$(G \boxtimes F)(n) = \sum_{k \geq 0} G(k) \otimes_{S_k} F^{\otimes k}(n)$$

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** $F, G: \mathbb{P}^{op} \to Vect$.

+-- {: .num_prop}
######Proposition
The product $\boxtimes$ makes the category of polynomial species into a monoidal category. The unit for this product is polynomial species $X$ given by the representable $\mathbb{P}(-, 1): \mathbb{P}^{op} \to FinDimVect$.
=--

+-- {: .proof}
######Proof (Sketch)
The following proof is adapted from a similar argument due to Max Kelly [ref]: we exhibit an associativity isomorphism $\alpha: (- \boxtimes F) \boxtimes G \to - \boxtimes (F \boxtimes G)$ on the basis of universal properties. The point is that by the universal property of $\overline{k\mathbb{P}}$, the category of functors
$$F: 1 \to \overline{k\mathbb{P}}$$
is equivalent to the category of symmetric monoidal linear functors
$$H: \overline{k\mathbb{P}} \to \overline{k\mathbb{P}}$$
The correspondence in one direction takes $F$ to the symmetric monoidal functor $H = - \boxtimes F$, and in the other direction takes $H$ to $F = H(X)$. By the equivalence, we have a unit isomorphism $X \boxtimes F \cong F$. Also by this equivalence, symmetric monoidal linear transformations between symmetric monoidal linear functors of the form
$$(- \boxtimes F) \boxtimes G \to - \boxtimes (F \boxtimes G)$$
are in natural bijection with morphisms $(X \boxtimes F) \boxtimes G \to X \boxtimes (F \boxtimes G)$, which by the unit isomorphism reduce to morphisms $F \boxtimes G \to F \boxtimes G$. Thus, corresponding to the identity on $F \boxtimes G$ we obtain an associativity map $\alpha: (- \boxtimes F) \boxtimes G \to - \boxtimes (F \boxtimes G)$. By similar arguments that appeal to the universal property of $\overline{k \mathbb{P}}$, we get all the required axioms: the invertibility of $\alpha$, the pentagon, etc.
=--

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

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

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](http://www.thehcmr.org/issue2_2/tableaux.pdf))

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](https://arxiv.org/abs/2106.00190))

[[species|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](http://www.math.tamu.edu/~maguiar/a.pdf))
* 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](https://arxiv.org/abs/1610.02180))

## Old Stuff ##

[[John Baez]]: this stuff should go somewhere, perhaps:

+-- {: .query}

The first thing that should be understood from the beginning is that a general Schur functor $F$ is _nonlinear_: the action on [[hom-set|hom-sets]]

$$hom(V, W) \to hom(F(V), F(W))$$

is not assumed to respect the linear structure. In fact, linear Schur functors are rather uninteresting: because every finite-dimensional space is a finite [[direct sum]] of copies of the $1$-dimensional space $\mathbb{C}$, and because [[linear functor|linear functors]] preserve finite direct sums (that is, [[biproduct|biproducts]], it turns out that every linear Schur functor $F$ is [[representable functor|representable]] as $hom(X, -)$ where $X = F(\mathbb{C})$.  So, the category of linear Schur functors is equivalent to $FinDimVect$.

=--

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

+-- {: .query}

### Modules over a bimonoid ###

Next we exploit the fact that, just like any [[group algebra]], $k[S_n]$ is a [[bialgebra]] --- or in fancier language, a  [[bimonoid]] in the [[symmetric monoidal category]] [[FinDimVect]]${}_k$.  Since $i \colon FinDimVect_k \to C$ is a [[symmetric monoidal functor]], this means that $i$ carries $k[S_n]$ to a  bimonoid in $C$.  As noted above, we call this bimonoid by the same name, $k[S_n]$.

The [[category of modules]] over a [[bimonoid]] is a [[monoidal category]]. More explicitly, in the case of the bimonoid $k[S_n]$ in $C$ with comultiplication

$$\delta: k[S_n] \to k[S_n] \otimes k[S_n] \,,$$

the tensor product $V \otimes W$ of two $k[S_n]$-modules in $C$ carries a module structure where the action is defined by

$$k[S_n] \otimes V \otimes W \stackrel{\delta \otimes 1_{V \otimes W}}{\to} k[S_n] \otimes k[S_n] \otimes V \otimes W \stackrel{1 \otimes \sigma \otimes 1}{\to} k[S_n] \otimes V \otimes k[S_n] \otimes W \stackrel{\alpha_V \otimes \alpha_W}{\to} V \otimes W$$

where $\sigma$ is a symmetry isomorphism and $\alpha_V$, $\alpha_W$ are the actions on $V$ and $W$.

Now we consider a particular case of tensor product representations. If $X$ is an object of $C$, the symmetric group $S_n$ has a representation on $X^{\otimes n}$. (Indeed, for each $\sigma \in S_n$, there is a corresponding symmetry isomorphism $X^{\otimes n} \to X^{\otimes n}$. From this one may construct an action

$$k[S_n] \otimes X^{\otimes n} \cong \bigoplus_{\sigma \in S_n} I \otimes X^{\otimes n} \to X^{\otimes n}$$

which is the required representation.) So, if $V_\nu$ is a Young tableau representation of $k[S_n]$ in $C$, we obtain a tensor product representation

$$V_\nu \otimes X^{\otimes n}$$

of $k[S_n]$ in $C$.

Consider next the **averaging operator** $e = \frac1{n!} \sum_{\sigma \in S_n} \sigma$:

$$e: V_\nu \otimes X^{\otimes n} \to V_\nu \otimes X^{\otimes n}$$

This operator makes sense since $k$ has characteristic zero, and crucially, this operator is _idempotent_ (because $e = \sigma e$ for all $\sigma \in S_n$). Because we assume idempotents split in $C$, we have a (split) coequalizer

$$V_\nu \otimes X^{\otimes n} \stackrel{\overset{e}{\to}}{\underset{1}{\to}} V_\nu \otimes X^{\otimes n} \to V_\nu \otimes_{S_n} X^{\otimes n}$$

This [[coequalizer]] is indeed the object of $S_n$-[[coinvariants]] of $V_\nu \otimes X^{\otimes n}$, i.e., the joint coequalizer of the diagram consisting of all arrows

$$V_\nu \otimes X^{\otimes n} \stackrel{\sigma \cdot-}{\to} V_\nu \otimes X^{\otimes n}$$

ranging over $\sigma \in S_n$ (it is the joint coequalizer because $e = \sigma e$ for all $\sigma$).

We may now define the Schur functor $S_\nu$ on $C$ attached to a Young tableau $\nu$.

=--

[[!redirects Schur functors]]


Schur index

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[representation theory]], a _Schur_ index is a measure for how much an [[irreducible representation]] $V$ over some [[field]] $k$ becomes the [[direct sum]] of several irreducible representations as one passes from the [[ground field]] $k$ to some [[field extension]] $\widehat k$ of $k$.

There are two different-looking but equivalent perspectives on Schur indices, one via [[Galois theory]], the other via the [[lambda-ring]]-structure on [[representation rings]] $R_{\widehat k}(G)$.

## Definition

Throughout, $G$ is a [[finite group]] with [[order of a group|order]] denoted ${\vert G\vert} \in \mathbb{N}$.

Given a [[field]] $k$, we write $R_K(G)$ for the [[representation ring]] of $G$ over $k$.

### Galois action and Adams operations

+-- {: .num_remark #RepresentationRingIsLambdaRing}
###### Remark
**([[representation ring]] is a [[lambda-ring]])**

Let $k$ be a [[field]] of [[characteristic zero]].

The [[representation ring]] $R_k(G)$ is canonically a [[lambda-ring]].

Hence, in particular, for each $n \in \mathbb{N}$, there exists the corresponding [[Adams operation]]

$$
  \psi^n
    \;\colon\;
  R_k(G)
    \longrightarrow
  R_k(G)
  \,.
$$

Notice that for $k = \mathbb{C}$ the [[complex numbers]], the [[representation ring]] is identified with the $G$-[[equivariant K-theory]] of the point

$$
  R_{\mathbb{C}}(G)
  \;\simeq\;
  KU^0_G(\ast)
$$

and under this identification the [[Adams operations]] here are those familiar from K-theory.

=--

([tom Dieck 79, section 3.5](#tomDieck79))

+-- {: .num_prop #AdamsOperationActionOnCharacters}
###### Proposition
**([[Adams operation]]-[[action]] on [[characters]])**

Let $k$ be a [[field]] of [[characteristic zero]].

Then the $n$th Adams operation on the [[representation ring]] (Remark \ref{RepresentationRingIsLambdaRing})

$$
  \psi^n
    \;\colon\;
  R_k(G)
    \longrightarrow
  R_k(G)
  \,.
$$

has the following simple explicit description in terms of the [[characters]] $\chi_V$ of representations $V \in R_{k}(G)$:

$$
  \chi_{\psi^n V}
  =
  \chi_{V}\left( (-)^n \right)
$$

hence for all $g \in G$

$$
  \chi_{\psi^n(V)}(g)
  \;=\;
  \chi_V( g^n )
$$

=--

[tom Dieck 79, Prop. 3.5.1](#tomDieck79)

+-- {: .num_prop #AdamsOperationsRepresentGaloisAction}
###### Proposition
**([[Adams operations]] represent [[Galois group|Galois]] [[action]])**

Now let $\widehat k$ be a [[splitting field]] for $G$, for instance the [[complex numbers]] $\mathbb{C}$, but containing at least the [[cyclotomic field]] $\mathbb{Q}\left[ \exp(2 \pi i/e(G) \right]$, where $e(G) \in \mathbb{N}$ is the [[exponent of a group|exponent]] of $G$.

Then the [[action]] of the [[Adams operations]] (Remark \ref{RepresentationRingIsLambdaRing})

$$
  \psi^n
    \;\colon\;
  R_{\widehat k}(G)
    \longrightarrow
  R_{\widehat k}(G)
  \,.
$$

for $n$ [[coprime integer|coprime]] to the [[order of a group|order]] of the group,

$$
  (n, {\vert G \vert}) = 1
  \,,
$$

equals the canonical [[action]] of the [[Galois group]] of $\mathbb{Q}\left[ e^{2 \pi i/e(G)} \right]$ over $\mathbb{Q}$ on $R_{\widehat k}(G)$ by [[field]] [[automorphisms]], which in turns is [[isomorphism|isomorphic]] to the [[multiplicative group of integers modulo n|multiplicative group of integers modulo e(G)]]:

$$
  Gal\left(
    \mathbb{Q}\left[ e^{2 \pi i/e(G)} \right]
    \;:\;
    \mathbb{Q}
  \right)
  \;\simeq\;
  \left(
    \mathbb{Z}/{e(G)}
  \right)^{\times}
  \;\simeq\;
  \big\{
    \psi^{n} \;\vert\; (n,{\vert G\vert}) = 1
  \big\}
$$

One hence also says that $\psi^n V$ is a _Galois translate_ of $V$.

Moreover, if $V \in R_{\widehat k}(G)$ is an [[irreducible representation]], then also its Galois translate $\psi^n V$ is an irreducible representation, for $n$ coprime to ${\vert G \vert}$.

=--

([tom Dieck 79, Prop. 3.5.2](#tomDieck79))

### The Schur index

+-- {: .num_defn #GaloisGroupAveraging}
###### Definition
**([[Galois group|Galois]] [[group averaging]] on [[representations]])**

For $V \in R_{\widehat k}(G)$ an [[irreducible representation]], say that its _[[group averaging]]_ with respect to the [[Galois group]]/[[Adams operations]] from Prop. \ref{AdamsOperationsRepresentGaloisAction} is the smallest representation $V_{avg}$ containing $V$ as a [[subrepresentation]] such that

$$
  \Psi^n\left(V_{avg}\right) - V_{avg}
  \;=\;
  0
  \;\;\;\;\;
  \in
  R_{\widehat{k}}(G)
$$

for all $n$ [[coprime integer|coprime]] to ${\vert G\vert}$. (See also at _[[Adams conjecture]]_.)

By Prop. \ref{AdamsOperationsRepresentGaloisAction} this is equivalently the [[direct sum]]

$$
  V + \Psi^{n_1}\left(V\right) + \psi^{n_2}\left(V\right) + \cdots
$$

of $V$ with all of its _distinct_ Galois translates.

=--

+-- {: .num_prop #SchurIndexForComplexToRational}
###### Proposition
**(Schur index for complex-to-rational)**

Let $k = \mathbb{Q}$ be the [[rational numbers]] and let $\widehat k$ be the [[complex numbers]] or at least the [[cyclotomic field]] $\mathbb{Q}\left[e^{2 \pi i/e(G)}\right]$ for $e(G) \in \mathbb{Z}$ the [[exponent of a group|exponent]] of $G$.

Then for

$$
  V \in R_{\widehat k}(G)
$$

a $\widehat k$-linear [[representation]], its _Schur index_

$$
  s_V \in \mathbb{N}
$$

is the smallest [[natural number]] such that there exists a rational [[irreducible representation]] $W \in R_{\mathbb{Q}}(G)$ whose [[extension of scalars]] (e.g. [[complexification]]) is $s$ times the Galois group averaging $V_{avg}$ (Def. \ref{GaloisGroupAveraging}) of $V$:

$$
  W \otimes_{\mathbb{Q}} \mathbb{C}
  \;\simeq\;
  s_V \cdot V_{avg}
$$

Such irreducible $W$ exists uniquely.

=--

(e.g. [tom Dieck 09, theorem 6.2.1](#tomDieck09))

+-- {: .num_remark #EquivariantJHomomorphism}
###### Remark
**([[equivariant J-homomorphism]]?)**

The uniqueness of the rational irrep $W$ in Prop. \ref{SchurIndexForComplexToRational} means that the Schur index construction there  provides a [[linear map]] (not necessarily a [[ring homomorphism]])

$$
  \array{
    R_{\widehat k}(G)
      \;
      &\overset{J_{\mathbb{Q}}}{\longrightarrow}&
      \;
    R_{\mathbb{Q}}(G)
    \\
    V &\mapsto& s_V \cdot \big(V + \Psi^{n_1}(V) + \cdots \big)
  }
  \,.
$$

For some [[finite groups]] $G$ the [[permutation representation]]-assigning map

$$
  A(G) \overset{\beta}{\longrightarrow} R_{\mathbb{Q}}(G)
$$

from the [[Burnside ring]] to the rational [[representation ring]] is [[surjective map|surjective]] ([this Prop.](permutation+representation#WhenAllVirtualLinearRepsAreVirtualPermutationReps)) and admits a canonical [[section]]; for $G = C_n$ a [[cyclic group]] it is actually an [[isomorphism]]. Hence in these cases we may regard the Schur index construction of Prop. \ref{SchurIndexForComplexToRational} as a [[linear map]] of the form

$$
  J
    \;\colon\;
  R_{\mathbb{C}}(G)
    \longrightarrow
  A(G)
$$

hence going from the [[equivariant K-theory]] of the point to [[equivariant stable cohomotopy]] of the point

$$
  J
    \;\colon\;
  KU^0_G(\ast)
    \longrightarrow
  \mathbb{S}_G^0(\ast)
  \,.
$$

> Question:

> Is this the $G$-[[equivariant J-homomorphism]] over the point?

Notice that, by the construction in Prop. \ref{SchurIndexForComplexToRational} this map satisfies

$$
  J \big( \Psi^n(V) - V \big)
  \;=\;
  0
$$

for all $n$ [[coprime integer|coprime]] to ${\vert G \vert}$.

> Question:

> Is this the $G$-[[equivariant Adams conjecture]]-statement over the point?

=--

## References

Lecture notes include

* {#tomDieck09} [[Tammo tom Dieck]], section 6.2 of _Representation theory_, 2009 ([pdf](http://www.uni-math.gwdg.de/tammo/rep.pdf))

Textbook accounts include

* {#tomDieck79} [[Tammo tom Dieck]], _[[Transformation Groups and Representation Theory]]_, Lecture Notes in Mathematics 766, Springer 1979

* Bertram Huppert, chapter 38 of _Character theory of finite groups_, de Gruyter 1998

[[!redirects Schur indices]]


Schur multiplier

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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

##Idea

The **Schur multiplier** of a group, $G$, is $H^2(G;\mathbb{C}^{\times}) \cong H^3(G;\mathbb{Z})$. It is also the second homology group, $H_2(G; \mathbb{Z})$ with coefficients in the trivial $G$-module $\mathbb{Z}$.

A group is called _perfect_ if its abelianization is trivial, i.e., its first homology group, $H_1(G; \mathbb{Z})$, vanishes, and a perfect group is called _superperfect_ if its Schur multiplier also vanishes.

Let $G$ be a perfect discrete group. Then $G$ possesses a Schur cover, whose central subgroup is the Schur multiplier $A_{uni} = H_2(G; \mathbb{Z})$.

##Related concepts

* [[group cohomology]]

##References

* Gregory Karpilovsky 1987 _The Schur Multiplier_, Oxford University Press.


Schur orthogonality relation

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###Context###
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given a [[finite group]] $G$ of [[cardinality]] ${\vert G \vert}$, then the [[complex number|complex]] valued [[class functions]]  $\chi \colon G \to \mathbb{C}$ have an [[inner product]] -- the _Schur inner product_ -- given by

\[
  \langle \alpha, \beta\rangle
  \coloneqq
  \frac{1}{{\vert G \vert}}
  \underset{g \in G}{\sum} \alpha(g) \overline{\beta(g)}
  \,.
\]

The _[[Schur orthogonality relation]]_ is the statement that the [[complex numbers|complex]] [[irreducible representation|irreducible]] [[group characters]] $\{\chi^{(a)}\}_a$ form an [[orthonormal basis]] of the space of [[class functions]] under this [[inner product]]:

\[
  \label{FirstSchurOrthogonalityRelation}
  \langle \chi^{(a)}, \chi^{(b)} \rangle
  =
  \left\{
    \array{
       1 & if \; a = b
       \\
       0 & otherwise
    }
  \right.
\]

(e.g. [Fulton & Harris 91, Thm. 2.12](#FultonHarris91), [tom Dieck 09, Prop. 2.2.1 & (2.3.3)](#tomDieck09))

In fact, this holds also at the level of complex [[irreducible representations]] $\rho^{(a)}$ ("grand Schur orthogonality"), regarded for each $g \in G$ as [[unitary matrices]] $\rho^{(a)}(g) \in U(\chi^{(a)}(e))$ with matrix entries $\big(\rho^{(a)}(g)_{i j}\big)_{i j}$:

\[
  \label{SchurOrthogonalityForIrreps}
  \underset{g \in G}{\sum}
    \rho^{(a)}(g)_{i_1 j_1}
    \cdot
    \overline{\rho^{(b)}(g)_{i_2 j_2}}
  \;=\;
  \delta^{a b} \delta_{i_1 i_2} \delta_{j_1 j_2}
  \cdot
  \frac{
    \left\vert G \right\vert
  }{
    dim(\rho^{(a)})
  }
  \,.
\]

(e.g. [tom Dieck 09 (2.2.7)](#tomDieck09))

Conversely, the sum over all ([[isomorphism classes]]) of [[irreducible characters]] $\chi_i$ yields:

\[
  \underset{\chi^{(a)}}{\sum}
  \chi^{(a)}(g)
  \cdot
  \overline{
    \chi^{(a)}(h)
  }
  \;=\;
  \left\{
  \array{
    \left\vert Central_G(g) \right\vert
    &
    \text{if} \; h,g \;\text{are conjugate}
    \\
    0 & \text{otherwise}
  }
  \right.
\]

(e.g. [Fulton & Harris 91, Ex. 2.21](#FultonHarris91), [tom Dieck 09, Prop. 2.2.1 & (2.3.3)](#tomDieck09))

## Related concepts

* [[Schur orthogonality relation]]

* [[1d Dijkgraaf-Witten theory]]

## References

Textbook accounts:

* {#FultonHarris91} [[William Fulton]], [[Joe Harris]], _Representation Theory: a First Course_, Springer, Berlin, 1991 ([doi:10.1007/978-1-4612-0979-9](https://link.springer.com/book/10.1007/978-1-4612-0979-9))

Lecture notes:

* {#tomDieck09} [[Tammo tom Dieck]], _Representation theory_, 2009 ([pdf](http://www.uni-math.gwdg.de/tammo/rep.pdf), [[tomDieckRepresentationTheory.pdf:file]])

* Andrei Yafaev, _Characters of finite groups_ ([pdf](http://www.ucl.ac.uk/~ucahaya/Characters.pdf))

See also:

* Wikipedia, _[Schur orthogonality relation](http://en.wikipedia.org/wiki/Schur_orthogonality_relations)_

[[!redirects Schur orthogonality relations]]
[[!redirects Schur orthogonality]]

[[!redirects grand Schur orthogonality relation]]
[[!redirects grand Schur orthogonality relations]]
[[!redirects grand Schur orthogonality]]

[[!redirects Schur inner product]]


Schur positivity

#Contents#
* table of contents
{:toc}

## Definition

A [[symmetric function]] $f$ with [[coefficients]] in the [[ground field]] is *Schur positive* if, when expanded in the [[linear basis]] of [[Schur polynomials]] $f = \sum_\mu a_\mu s_\mu$, all [[coefficients ]] $a_\mu$ are [[positive numbers]].

## References

* Rebecca Patrias, *What is Schur positivity and how common is it?* ([arXiv:1809.04448](https://arxiv.org/abs/1809.04448))

* Thomas Lam, Alexander Postnikov, Pavlo Pylyavskyy, _Schur positivity and Schur log-concavity_, [math.CO/0502446](http://arxiv.org/abs/math/0502446)

* James M. Borger, _Witt vectors, semirings, and total positivity_, [arxiv/1310.3013](http://arxiv.org/abs/1310.3013)

> We extend the big and p-typical Witt vector functors from commutative rings to commutative semirings. In the case of the big Witt vectors, this is a repackaging of some standard facts about monomial and Schur positivity in the combinatorics of symmetric functions. In the p-typical case, it uses positivity with respect to an apparently new basis of the p-typical symmetric functions.


Schur's lemma

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### Context
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Schur's lemma is one of the fundamental facts of [[representation theory]]. It concerns basic properties of the [[hom-sets]] between [[irreducible representations|irreducible]] [[linear representations]] of [[groups]].

The lemma consists of two parts that depend on different assumptions (a distinction often not highlighted in the literature):

1. The first statement applies over _every_ [[ground field]]:

   It says that there are no non-[[zero morphism|zero]] [[homomorphisms]] between distinct (i.e. non-[[isomorphism|isomorphic]]) [[irreducible representations]]

1. The second statement applies only in the special case that the [[ground field]] is an [[algebraically closed field]] (such as the [[complex numbers]]) and that the [[representations]] are [[finite dimensional vector space|finite-dimensional]]:

   It says that, in this case, moreover the only non-trivial [[endomorphisms]] of an [[irreducible representation]] are multiples of the [[identity morphism]].

## Statement

Let $G$ be a [[group]]. In the following:

* *representation* means _[[linear representation]] of $G$_, linear over some [[ground field]].

* *finite dimensional representation* means that the underlying [[vector space]] is a [[finite-dimensional vector space]].

* an *[[irreducible representation]]* is one whose only $G$-[[invariant]] [[subspaces]] ([[fixed point spaces]]) are the trivial degenerate cases: the [[zero object|zero]]-subspace and the full space itself.

\begin{prop}\label{SchurLemma}
**(Schur's lemma)**
\linebreak

1. {#FirstItem} A [[homomorphism]] $\phi \;\colon\; V\to W$ between [[irreducible representations]], is either the [[zero morphism]] or an [[isomorphism]].

   It follows that the [[endomorphism ring]] of an [[irreducible representation]] is a [[division ring]].

2. {#SecondItem} In the case that the [[ground field]] is an [[algebraically closed field]]; [[endomorphisms]] $\phi \;\colon\; V \to V$ of a [[finite-dimensional vector space|finite dimensional]] [[irreducible representations]] $V$ are a multiple $c \cdot id$ of the [[identity morphism|identity operator]].

   In other words, nontrivial automorphisms of irreducible representations, _a priori_ possible by [(1)](#FirstItem), are ruled out over algebraically closed fields.

\end{prop}

## Proof

As it goes with very fundamental lemmas, the [[proof]] of Schur's lemma follows by elementary inspection.

\begin{proof}(of Prop. \ref{SchurLemma})
\linebreak
For the first statement:

It is immediate to see that both the [[kernel]] as well as [[image]] of a [[homomorphism]]

$$
  V \overset{f}{\longrightarrow} W
$$

of any $G$-[[representations]] are $G$-[[invariant]] [[subspaces]] ([[fixed point spaces]]). But by the very definition of [[irreducible representation|irreducibility]], the only such subspaces of $V$ and $W$ are the degenerate ones: their zero subspaces and the full spaces themselves.

Now if the [[kernel]] is all of $V$ or the [[image]] is [[zero object|zero]], then $f$ is the [[zero morphism]]. The only case left is that the [[kernel]] is [[zero object|zero]] _and_ the [[image]] is all of $W$, but this means that $f$ is [[injective map|injective]] and [[surjective map|surjective]] and is hence an [[isomorphism]].

For the second statement:

Now we use that over an [[algebraically closed field]] $k$ of every [[linear map|linear]] [[endomorphism]] of a [[finite dimensional vector space]] has an [[eigenvalue]] $c \in k$ (which is, ultimately, due to the [[fundamental theorem of algebra]] for [[algebraically closed fields]]). Now, with $f$ also the linear combination

$$
  (f - c \cdot \mathrm{id}) \;\colon\; V \longrightarrow V
$$

is a [[homomorphism]] of $G$-[[representations]]. But then, by the first part, this must be an [[isomorphism]] or [[zero morphism|zero]]. But it is not an [[isomorphism]], by construction, since now the [[eigenvectors]] with [[eigenvalue]] $c$ are in the [[kernel]]. Therefore, all of the linear combination must be [[zero morphism|zero]]

$$
  f - c \cdot id = 0
$$

and hence

$$
  f = c \cdot id
$$

is a multiple of the identity.
\end{proof}

## Interpretation in categorical algebra
 {#InterpretationInCategoricalAlgebra}

The statement of _Schur's lemma_ is particularly suggestive in the language of [[categorical algebra]].

Here it says that [[irreducible representations]] form a [[categorification|categorified]] _[[orthogonal basis]]_ for the [[2-Hilbert space]] of [[finite-dimensional vector space|finite-dimensional]] [[representations]], and even an _[[orthonormal basis]]_ if the [[ground field]] is [[algebraically closed field|algebraically closed]].

After [[decategorification]] this becomes equivalently the statement that the [[isomorphism classes]] of [[irreducible representations]] form an _[[orthogonal basis]]_ for the [[representation ring]], and even an _[[orthonormal basis]]_ if the [[ground field]] is [[algebraically closed field|algebraically closed]].

For more on this perspective see also at _[[Gram-Schmidt process]]_ the section _[Categorified Gram-Schmidt  process](Gram-Schmidt+process#CategorifiedGramSchmidtProcess)_.

$\,$

We now explain this perspective of in more detail:

$\,$

Notice that the [[hom-sets]] in a [[category of representations]] $G Rep$ are canonically [[vector spaces]]: given any two [[homomorphisms]] $f,g \;\colon\; V \to W$ of $G$-[[representations]], also the (value-wise) [[linear combination]] $c_1 f + c_2 g \;\colon\; V \to W$ is a $G$-homomorphism. ($G Rep$ is canonically [[enriched category|enriched]] over [[Vect]].)

Now it makes sense to regard this [[vector space]]-valued [[hom-functor]] on $G Rep$ as analogous

$$
  hom_G(-,-) \;\colon\; G Rep^{op} \times G Rep \longrightarrow Vect
$$

as a [[categorification|categorified]] [[inner product]] (see at _[[2-Hilbert space]]_ for more on this).

This is a useful perspective even after [[decategorification]]:

For $V, W \in G Rep^{fin}$ two [[finite-dimensional vector space|finite-dimensional]] [[representations]], write

$$
  \langle V,W\rangle
  \;\coloneqq\;
  dim\big( hom_G(V,W) \big)
  \;\in\;
  \mathbb{N}
$$

for the [[dimension]] of the vector space of homomorphism between them (e.g. [tom Dieck 09, p. 29](#tomDieck09)).

This construction only depends on the [[isomorphism classes]] of $V$ and $W$, and hence descends to a [[function]]

$$  \langle -,-\rangle
  \;\colon\;
  G Rep^{fin}_{/\sim} \times G Rep^{fin}_{/\sim}
  \longrightarrow
  \mathbb{N}
$$

on [[sets]] of [[isomorphism classes]]. In fact, under [[direct sum]] and [[tensor product of representations]], $G Rep^{fin}_{/\sim}$ is a [[rig]], and this pairing is [[linear map|linear]] with respect to the underlying [[commutative monoid|additive monoid]] structure:

$$
  \left\langle
    V_1 \oplus V_2
    \,,\,
    W
  \right\rangle
  \;=\;
  \left\langle
    V_1
    \,,\,
    W
  \right\rangle
  +
  \left\langle
    V_2
    \,,\,
    W
  \right\rangle
$$

and

$$
  \left\langle
    V
    \,,\,
    W_1 \oplus W_w
  \right\rangle
  \;=\;
  \left\langle
    V
    \,,\,
    W_1
  \right\rangle
  +
  \left\langle
    V
    \,,\,
    W_2
  \right\rangle
$$

(This is, ultimately, due to the [[universal property]] of [[direct sum]] as a [[biproduct]].)

To further strengthen the emerging picture, we may consider the [[group completion]] of the [[commutative monoid]] $G Rep^{fin}_{/\sim}$ by passing to its [[Grothendieck group]], in fact its [[Grothendieck ring]] if we remember also the [[tensor product of representations]]. This [[commutative ring]] is called the [[representation ring]]

$$
  R(G) \;\coloneqq\; K\left(  G Rep^{fin}_{/\sim} \right)
$$

of $G$. By the evident $\mathbb{Z}$-linear extension, the above pairing gives an actual symmetric $\mathbb{Z}$-[[bilinear map|bilinear]] [[inner product]] on $R(G)$:

\[
  \label{DecategorifiedInnterProduct}
  \langle
   -,-
  \rangle
  \;\colon\;
  R(G) \times R(G)
  \longrightarrow
  \mathbb{Z}
\]

Now  the underlying [[abelian group]] of $R(G)$ is a [[free abelian group]] whose canonical [[generators and relations|generators]] are nothing but the [[isomorphism classes]] $V_i$ of the [[finite-dimensional vector space|finite-dimensional]] [[irreducible representations]]:

$$
  R(G) \;\simeq_{\mathbb{Z}}\;  \mathbb{Z}\big[ \{V_i\}_i \big]
  \,.
$$

In summary, in the language of [[linear algebra]], the [[irreducible representations]] $[V_i]$ constitute a canonical _[[linear basis]]_ of the [[representation ring]].

In terms of this language, **Schur's lemma** becomes the following statement:

The canonical [[linear basis]] of the [[representation ring]] given by the [[irreducible representations]] is

1. generally: an _[[orthogonal basis|ortho-gonal basis]]_;

1. even an _[[orthonormal basis|ortho-normal basis]]_ if the [[ground field]] is [[algebraically closed field|algebraically closed field]]

with respect to the canonical [[decategorification|decategorified]] [[inner product]] from (eq:DecategorifiedInnterProduct).

## Generalizations and variants
 {#GeneralizationsAndVariants}

### For simple modules

Part [(1)](#FirstItem) of Schur's lemma is essentially [[category theory|category-theoretic]] and can be generalized in many ways, for example, by replacing the [[group]] $G$ by some $k$-[[associative algebra|algebra]] and taking the representations compatible with the action of $k$.

### For simple objects in an abelian category

More generally, part [(1)](#FirstItem) of  Schur's lemma applies to [[simple objects]] in any [[abelian category]], and the  [[endomorphism ring]] of such a [[simple object]] is a [[division ring]], as proved [here](simple+object#in_an_abelian_category).

For [(2)](#SecondItem), if the [[endomorphism rings]] of all objects in an [[abelian category]] are [[finite-dimensional vector space|finite-dimensional]] over an [[algebraically closed field]] $k$, then the endomorphism ring of a simple object is $k$ itself, as proved [here](simple+object#in_an_abelian_category).  This is the case for the category of finite-dimensional complex representations of a group.

### For Bridgeland stable objects

The statement of Schur's lemma applies also to objects which are stable with respect to a [[Bridgeland stability condition]], see [there](Bridgeland+stability+condition#SchurLemma).

## References

Named after *[[Issai Schur]]*.

Lecture notes include:

* {#tomDieck09} [[Tammo tom Dieck]], §1.1.2 in: _Representation theory_ (2009) [[pdf](http://www.uni-math.gwdg.de/tammo/rep.pdf), [[tomDieckRepresentationTheory.pdf:file]]]

See also

* Wikipedia, _[Schur's lemma](https://en.wikipedia.org/wiki/Schur%27s_lemma)_

[[!redirects Schur's lemmas]]

[[!redirects Schur lemma]]
[[!redirects Schur lemmas]]


Schur-Weyl duality

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### Context
#### Representation theory
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[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Schur-Weyl duality is a relation between the [[finite-dimensional vector space|finite-dimensional]] [[irreducible representations]] of the [[general linear groups]] $GL(k)$ and of the [[symmetric group]] $Sym(n)$.

Concretely, for $k, n \in \mathbb{N}$ consider the $k \times n$-dimensional [[complex vector space]]; regarded as a [[tensor product of vector spaces|tensor product]] of $n$ copies of $\mathbb{C}^n$:

$$
  \underset{
    n \; tensor \; factors
  }{
    \underbrace{
      \mathbb{C}^k \otimes \cdots \otimes \mathbb{C}^k
    }
  }
  \;\;\;
  \in
  \;
  \mathbb{C}VectorSpaces
$$

As such, this carries

* a canonical [[linear representation|linear]] [[group action]] by $GL(k) \coloneqq GL(k,\mathbb{C})$, namely the $n$-fold [[diagonal action]] of its defining action on $\mathbb{C}^k$ (its [[fundamental representation]]);

* a canonical [[linear representation|linear]] [[group action]] by $Sym(n)$ given by [[permutation]] of the $n$ tensor factors.

The statement of Schur-Weyl duality is that the decomposition of this [[linear representation]] of the [[direct product group]] $GL(k) \times Sym(n)$ decomposes as a [[direct sum]] of [[tensor product of vector spaces|tensor products]] of [[irreducible representations]] for either group, as follows:

$$
  \underset{
    n \; tensor \; factors
  }{
    \underbrace{
      \mathbb{C}^k \otimes \cdots \otimes \mathbb{C}^k
    }
  }
  \;\simeq\;
  \underset{
    {
    (\lambda_1 \geq \cdots \geq \lambda_k)
    }
    \atop
    {
    \underset{i}{\sum} \lambda_i = n
    }
  }{\oplus}
  D^{(\lambda)}
  \otimes
  S^{(\lambda)}
$$

Here the [[direct sum]] is indexed by [[partitions]] $\lambda$ of $n$ by $k$ [[positive number|positive]] summands, and $D^{(\lambda)}$ and $S^{(\lambda)}$ denote the [[irreducible representations]] of the [[general linear group]] and of the [[symmetric group]] ([[Specht modules]]), respectively, which are indexed by these according to the [[representation theory of the general linear group]] and the [[representation theory of the symmetric group]], respectively.

## Related concepts

* [[representation theory of the general linear group]]

* [[representation theory of the symmetric group]]

* [[Schur-Weyl measure]]

## References

Original article:

* [[Hermann Weyl]],  *The Classical Groups:  Their Invariants and Representations*,  Princeton University Press, Princeton, N.J., 1939 ([jstor:j.ctv3hh48t](https://www.jstor.org/stable/j.ctv3hh48t))

Review:

Textbook accounts:

* {#FultonHarris91} [[William Fulton]], [[Joe Harris]], Section 6.1 of: _Representation Theory: a First Course_, Springer, Berlin, 1991 ([doi:10.1007/978-1-4612-0979-9](https://link.springer.com/book/10.1007/978-1-4612-0979-9))

* Ambar N. Sengupta, Section 10.2 of: *Representing Finite Groups -- A Semisimple Introduction*, Springer 2012 ([doi:10.1007/978-1-4614-1231-1](https://link.springer.com/book/10.1007/978-1-4614-1231-1))

In a context of [[quantum information theory]]:

* Jordi Tura i Brugués, *Schur-Weyl duality* ([pdf](https://link.springer.com/content/pdf/bbm%3A978-3-319-49571-2%2F1.pdf)), Appendix A in: *Characterizing Entanglement and Quantum Correlations Constrained by Symmetry*,  Springer Theses (2017) ([doi:10.1007/978-3-319-49571-2](https://link.springer.com/book/10.1007/978-3-319-49571-2))

See also

* Wikipedia, *[Schur-Weyl duality](https://en.wikipedia.org/wiki/Schur%E2%80%93Weyl_duality)*

In the [[AdS-CFT correspondence]]:

* [[Sanjaye Ramgoolam]], *Schur-Weyl duality as an instrument of Gauge-String duality*, AIPConf. Proc. 1031:255-265, 2008 ([arXiv:0804.2764](https://arxiv.org/abs/0804.2764))

[[!redirects Schur Weyl duality]]


Schur-Weyl measure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Measure and probability theory
+-- {: .hide}
[[!include measure theory - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

 The *Schur-Weyl measure* is [[probability distribution]] on the [[set]] of  [[Young diagrams]] of given number of boxes, closely related to expressions appearing in [[Schur-Weyl duality]].

## Definition

For $n \in \mathbb{N}_+$, write:

* $YDiagrams_n$ for the [[set]] of [[partitions]] of $n$, hence of [[Young diagrams]] with $n$ boxes;

* for $\lambda \in YDiagrams_n$ write:

  * $sYTableaux_\lambda$ for the set of [[standard Young tableaux]] of shape $\lambda$;

  * for $N \in \mathbb{N}_+$ write

    * $ssYTableaux_\lambda(N)$ for the set of [[semistandard Young tableaux]] of shape $\lambda$ and with labels  $\leq N$.

Then the *Schur-Weyl measure* for $n,N \in \mathbb{N}$ is

$$
  \array{
    YDiagrams_N
    &
    \overset{ p^{SW} }{\longrightarrow}
    &
    [0,1]
    \\
    \lambda
    &\mapsto&
    \frac
      {
        \left\vert
          sYTableaux_\lambda
        \right\vert
        \cdot
        \left\vert
          ssYTableaux_\lambda(N)
        \right\vert
      }
      {N^n}
  }
$$

(e.g. [Petrov 19, slide 76](#Petrov19))

In the special case that $N = n$, and writing

* $S^{(\lambda)} \in Rep(Sym(n))$ for the [[complex numbers|complex]] [[irreducible representation]] of the [[symmetric group]] which is labeled by $\lambda$ (the [[Specht module]], see the *[[representation theory of the symmetric group]]*);

* $V^{(\lambda)} \in Rep(Sym(n))$ for the [[complex numbers|complex]] [[irreducible representation]] of the ([[special linear group|special]]) [[general linear group]] which is labeled by $\lambda$ (see the *[[representation theory of the general linear group]]*),

this is equal to (see at *[[hook length formula]]*):

$$
  p^{SW}(\lambda)
  \;=\;
  \frac
    {
      dim\big( S^{(\lambda)} \big)
      \cdot
      dim\big( V^{(\lambda)} \big)
    }
    {N^n}
  \,.
$$

## Properties

### Entropy
 {#Entropy}

With respect to the Schur-Weyl measure on $Part(n)$ and in the [[limit of a sequence|limit]] of large $n = c N^2 \to \infty$, the [[logarithm]] of the Schur-Weyl probability is [[almost surely]] approximately constant (i.e. independent of $\lambda$) on the value

$$
  -
  ln
  p^{SW}
  \;\sim\;
  \sqrt{n}\cdot H_c
$$

for some $H_c \in \mathbb{R}$, in that for all $\epsilon \in \mathbb{R}_+$ we have

$$
 \underset{n = c N^2 \to \infty}{\lim}
 p^{SW}
 \left(
   \left\{
     \lambda \in Part(n)
     \;\left\vert\;
       \tfrac{-1}{\sqrt{n}}
       ln
       p^{SW}(\lambda)
       -
       H_c
       \;\lt\;
       \epsilon
     \right.
   \right\}
 \right)
 \;=\;
 1
  \,.
$$

([Mkrtchyan 14, Thm. 1.1](#Mkrtchyan14))

### Relation to Cayley distance kernel

The Schur-Weyl measure is the [[pushforward measure]] of the probability distribution on [[pure states]] encoded by the "Cayley state" ([here](Cayley+distance+kernel#TheCayleyState)): the [[quantum state]] that is given by the [[Cayley distance kernel]] -- see [there](Cayley+distance+kernel#TheCayleyStateProbabilityDistribution) for details.

## Related concepts

* [[Plancherel measure]]

* [[asymptotic representation theory]]

* [[Robinson-Schensted-Knuth algorithm]]

## References

Review:

* {#Petrov19} [[Leonid Petrov]], slides 73-76 in: *Random Matrices*, lecture notes 2019 ([pdf slides](https://rmt-fall2019.s3.amazonaws.com/rmt-fall2019.pdf), [[PetrovRandomMatrices.pdf:file]], [webpage](https://lpetrov.cc/2019/07/rmt-announce))

  > (in the context of [[random matrix theory]])

See also:

* {#Mkrtchyan14} [[Sevak Mkrtchyan]], *Entropy of Schur-Weyl Measures*, Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 678-713.  ([arXiv:1107.1541](https://arxiv.org/abs/1107.1541), [numdam:AIHPB_2014__50_2_678_0](http://www.numdam.org/item/AIHPB_2014__50_2_678_0))

* [[Pierre-Loïc Méliot]], *Kerov’s central limit theorem for Schur-Weyl and Gelfand measures*, [DMTCS proc.](https://www.dmtcs.org/dmtcs-ojs/index.php/proceedings/) AO, 2011, 669–680 ([pdf](https://www.imo.universite-paris-saclay.fr/~meliot/files/kerovclt.pdf))

* {#Meliot10a} [[Pierre-Loïc Méliot]], _Kerov's central limit theorem for Schur-Weyl measures of parameter 1/2_ ([arXiv:1009.4034](https://arxiv.org/abs/1009.4034))

* M. S. Boyko and N. I. Nessonov, *Entropy of the Shift on $II_1$-representations of the Group $S(\infty)$* ([pdf](http://iamm.su/upload/iblock/9e9/t2-n1-15-37.pdf))

[[!redirects Schur-Weyl measures]]


Schwartz kernel

+-- {: .rightHandSide}
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### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

Let $X_1, X_2$ be two [[open subsets]] of [[Cartesian spaces]]. Then a [[smooth function]] $K \in C^\infty(X_1 \times X_2)$ on the [[Cartesian product]] of these two manifolds defines a [[linear function]]

$$
  \array{
    C^\infty_{cp}(X_2) &\overset{}{\longrightarrow}& C^\infty(X_1)
    \\
    \phi &\mapsto& \mathcal{K}(\phi)
  }
  \,,
$$

by the "[[integral transform]]":

$$
  \mathcal{K}(\phi) \;\colon\; x_1 \mapsto \int_{X_2} K(x_1, x_2) \phi(x_2) \, dvol(x_2)
$$

More generally, this expression makes sense for

$$
  K
  \;\in\;
  \mathcal{D}'(X_1 \times X_2)
$$

a [[distribution]] on $X_1 \times X_2$ (a "distribution of two variables"), which makes the result itself in general be a distribution

$$
  \array{
    C^\infty_{cp}(X_2) &\overset{}{\longrightarrow}& \mathcal{D}'(X_1)
    \\
    \phi &\mapsto& \mathcal{K}(\phi)
  }
  \,.
$$

Here $K$ is called the _[[integral kernel]]_ and $\mathcal{K}(\phi)$ the corresponding _[[integral transform]]_.

## Properties

### Schwartz kernel theorem

The _Schwarz kernel theorem_ states that this construction constitutes a [[linear isomorphism]] between Schwartz [[integral kernels]] and "distribution-valued distributions"

$$
  \array{
    \mathcal{D}'(X_1 \times X_2)
    &\overset{\simeq}{\longrightarrow}&
    \mathcal{D}'( X_2, \mathcal{D}'(X_1) )
    \\
    K &\mapsto& \mathcal{K}
  }
$$

$\,$

+-- {: .num_prop #PartialProductOfDistributionsOfSeveralVariables}
###### Proposition
**(partial [[product of distributions|product of]] [[distributions of several variables]])**

Let

$$
  K_1 \in \mathcal{D}'(X \times Y)
  \phantom{AAA}
  K_2 \in \mathcal{D}'(Y \times Z)
$$

be two [[distributions of two variables]]. For their [[product of distributions]] to be defined over $Y$, [[Hörmander's criterion]] on the [[pair]] of [[wave front sets]] $WF(K_1), WF(K_2)$  needs to hold for the [[wave front set|wave front]] [[wave vectors]] along $X$ and $Y$ taken to be zero.

If this is satisfied, then composition of [[integral kernels]] (if it exists)

$$
  (K_1 \circ K_2)(-,-)
  \;\coloneqq\;
  \underset{Y}{\int}
    K_1(-,y) K_2(y,-)
  dvol_Y(y)
  \;\in\;
  \mathcal{D}'(X \times Z)
$$

has [[wave front set]] constrained by

$$
  WF(K_1 \circ K_2)
  \;\subset\;
  WF(K_1) \circ WF(K_2)
  \;\cup\;
  (X \times \{0\}) \times WF(K_2)
  \;\cup\;
  WF(K_1) \times (Z \times \{0\})
  \,,
$$

where on the left the composition symbol means composition of [[relations]] of [[wave vectors]] over points in $Y$.

=--

([Hörmander 90, theorem 8.2.14](product+of+distributions#Hoermander90))

## Related concepts

* [[tensor product of distributions]]

* [[polynomial observable]], [[microcausal observable]]

* [[correlators as differential forms on configuration spaces of points]]

## References

* {#Hoermander90} [[Lars Hörmander]], section 5.2 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990 ([pdf](http://www.ctr.maths.lu.se/media/MATP11/2014vt2014/distho.pdf))

See also

* Wikipedia, _[Schwartz kernel theorem](http://en.wikipedia.org/wiki/Schwartz_kernel_theorem)_

[[!redirects Schwartz kernels]]

[[!redirects Schwartz kernel theorem]]
[[!redirects Schwartz kernel theorems]]

[[!redirects distribution of two variables]]
[[!redirects distributions of two variables]]

[[!redirects distribution in two variables]]
[[!redirects distributions in two variables]]

[[!redirects distribution of several variables]]
[[!redirects distributions of several variables]]


Schwartz space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Functional analysis
+--{: .hide}
[[!include functional analysis - contents]]
=--
#### Harmonic analysis
+-- {: .hide}
[[!include harmonic analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[functional analysis]],

1. _a_ [[Schwartz space]] ([Terzioglu 69](#Terzioglu69), [Kriegl-Michor 97, below 52.24](#KrieglMichor97)) is a [[locally convex topological vector space]] $E$ with the property that whenever $U$ is an [[absolutely convex]] neighbourhood of $0$ then it contains another, say $V$, such that $U$ maps to a [[precompact set]] in the [[normed vector space]] $E_V$.

2. _the_ Schwartz space of an [[open subset]] of [[Euclidean space]] is the space of [[functions with rapidly decreasing partial derivatives]] (def. \ref{RapidlyDecreasingFunction} below). On this space the operation of [[Fourier transform]] is a linear automorphism (prop. \ref{FourierTransformIsIsomorphismOnSchwartzSpace} below). The [[continuous linear functionals]] on this space are the [[tempered distributions]].

The Schwartz spaces in the second sense are examples of the Schwartz spaces in the first sense.

## Definition

+-- {: .num_defn #RapidlyDecreasingFunction}
###### Definition
**([[functions with rapidly decreasing partial derivatives]])**

For $n \in \mathbb{N}$, a [[smooth function]] $f \colon \mathbb{R}^n \to \mathbb{R}$ on the [[Euclidean space]] $\mathbb{R}^n$ has _[[rapidly decreasing function|rapidly decreasing]] [[partial derivatives]]_ if the [[absolute value]] of the product of any [[partial derivative]] $\partial_\beta f$ of the function with any [[polynomial]] function is a [[bounded function]]:

$$
  \underset{\alpha, \beta \in \mathbb{N}^n}{\forall}
  \left(
  \underset{x \in \mathbb{R}^n}{sup} {\Vert x^\alpha \partial_{\beta} f(x)  \Vert} \lt K_{\alpha, \beta}
  \right)
$$

for some choices of positive constants $K_{\alpha, \beta}$.
=--

(e.g. [Hörmander 90, def. 7.1.2](#Hoermander90))

+-- {: .num_defn #SchwartzSpaceOfFunctionsWithRapidlyDecreasingDerivatives}
###### Definition
**(Schwartz space of [[functions with rapidly decreasing partial derivatives]])**

For $n \in \mathbb{N}$ the _Schwartz space_ $\mathcal{S}(\mathbb{R}^n)$ is the  [[topological vector space]] whose

* underlying [[real vector space]] is the subspace of the space $C^\infty(\mathbb{R}^n)$ of [[smooth functions]] (with pointwise addition and scalar multiplication) on the [[functions with rapidly decreasing partial derivatives]] (def. \ref{RapidlyDecreasingFunction});

* whose [[topological space|topology]] is that induced by the [[semi-norms]] given by $p_{\alpha,\beta}(f) \coloneqq {\Vert x^\alpha \partial_{\beta} (f)(x)  \Vert}$ is called the _Schwartz space_ $\mathcal{S}(\mathbb{R}^n)$.

=--

(e.g. [Hörmander 90, def. 7.1.2](#Hoermander90))

+-- {: .num_prop #SchwartSpaceOfFunctionsIsFrechetSpace}
###### Proposition
**(the Schwartz space is a [[Fréchet space]])**

The Schwartz space $\mathcal{S}$ of [[functions with rapidly decreasing partial derivatives]] (def. \ref{SchwartzSpaceOfFunctionsWithRapidlyDecreasingDerivatives})
is a [[Fréchet space]].

=--

(e.g. p. 2 here: [pdf](https://www.math.ucdavis.edu/~hunter/m218a_09/ch5A.pdf))

+-- {: .num_defn #TemperedDistribution}
###### Definition
**([[tempered distributions]])**

A _[[tempered distribution]]_ on $\mathbb{R}^n$ is a [[continuous linear functional]] on the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ (def. \ref{SchwartzSpaceOfFunctionsWithRapidlyDecreasingDerivatives}).

=--

(e.g. [Hörmander 90, def. 7.1.7](#Hoermander90))

## Properties

+-- {: .num_prop #FourierTransformIsIsomorphismOnSchwartzSpace}
###### Proposition
**([[Fourier transform]] is linear automorphism of Schwartz space)**

For $n \in \mathbb{N}$ the operation of [[Fourier transform]] $f \mapsto \hat f$ is well defined on all smooth functions on $\mathbb{R}^n$ with rapidly decreasing derivatives (def. \ref{RapidlyDecreasingFunction}) and indeed constitutes a [[linear isomorphism]] from the Schwartz space (def. \ref{SchwartzSpaceOfFunctionsWithRapidlyDecreasingDerivatives}) to itself:

$$
  \widehat {(-)}
  \;\colon\;
  \mathcal{S}(\mathbb{R}^n)
    \longrightarrow
  \mathcal{S}(\mathbb{R}^n)
$$

=--

(e.g. [Hörmander 90, lemma 7.1.3](#Hoermander90), [Melrose 03, theorem 1.3](#Melrose03))

## Related entries

* [[Schwartz-Bruhat function]]

## References

Named by [[Alexander Grothendieck]] after [[Laurent Schwartz]] (according to [Terzioglu 69](#Terzioglu69)).

The general concept of Schwartz spaces appears in

* Horvath, 1966, p277

* {#Terzioglu69} T. Terzioglu, _On Schwartz spaces_,  Mathematische Annalen September 1969, Volume 182, Issue 3, pp 236–242  ([web](https://link.springer.com/article/10.1007/BF01350326))

* Jarchow, 1981, 10.4.3, p202

* {#KrieglMichor97} [[Andreas Kriegl]], [[Peter Michor]], on p. 585 after Result 52.24 of _[[The Convenient Setting of Global Analysis]]_, 1997

Specifically the Schwartz spaces of [[functions with rapidly decreasing partial derivatives]] are discussed for instance in

* {#Hoermander90} [[Lars Hörmander]], section 7.1 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

* {#Melrose03} [[Richard Melrose]], chapter 1 of _Introduction to microlocal analysis_, 2003 ([pdf](http://www-math.mit.edu/~rbm/iml90.pdf))

See also

* Wikipedia, _[Schwartz space](https://en.wikipedia.org/wiki/Schwartz_space)_

[[!redirects smooth function with rapidly decreasing derivatives]]
[[!redirects smooth functions with rapidly decreasing derivatives]]

[[!redirects function with rapidly decreasing derivatives]]
[[!redirects functions with rapidly decreasing derivatives]]
[[!redirects rapidly decreasing derivatives]]

category: functional analysis