representation

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## Idea

The notion of _representation_ is closely related to, or even identical, to that of [[action]]: some object $C$ that has a notion of [[composition]] is _represented_ on some object $D$ that has a notion of composition. In this generality _representation_ is just another word for [[functor]] (or potentially $\infty$-[[infinity-functor|functor]]). But in practice the term _representation_ is typically used in the context of [[representation theory]], where we attempt to study $C$ in terms of its representations in $D$, where $D$ is typically rather more familiar.

Most specifically, one studies representations of a [[group]] by linear endomorphisms of a [[vector space]]; that is, $C$ is (the [[delooping]] of) a group and $D$ is [[Vect]].  However, the typical tools in representation theory these days involve vast generalizations of the notion of a linear representation of a group; for instance, one studies [[D-module]]s on [[action groupoid]]s $G //_{Ad}G$ and things like that. This may be thought of as studying representations with values in [[schreiber:∞-vector bundle|∞-vector space]]s.

##Historical Idea (with pedagogic overtones)

###Representing Groups

The notion of a [[group]] of 'operators' was already being used in about 1832 in work by Galois, and others, but there was not a definition of an abstract group until 40 years later when Cayley wrote:

> _A group is defined by the law of composition of its members._

(see the article on [The abstract group concept](http://www-history.mcs.st-and.ac.uk/HistTopics/Abstract_groups.html) in the St Andrews History of Mathematics archive.) Groups as well behaved sets of functions were beginning to be well understood and used, for instance in Klein's work on geometry. Cayley proved that every finite group could be realised as a group of permutations. The theory of representations grew from that.

An abstract group could be studied by mapping it into a group of permutations or of invertible matrices, as then you could bring techniques from one area of mathematics (linear algebra) to the assistance of another, the Theory of Abstract Groups. This was exploited fully by Frobenius, Burnside, Schur and later Bauer.

That is one theme: you take an abstract algebraic thing and study it by mapping it into a similar structure which you think you know more about!

This also uses another basic idea from the start of group theory.  The earlier pioneers thought of groups as groups of 'operators', but that means they have to operate on something. To make things more explicit, let $G$ be a finite group then we know there are homomorphisms from $G$ into $S_n$. (We could take $n$ to be the order of the group and use Cayley's theorem but we do not assume that is the case.) If we look at such a homomorphism we get an [[action]] of $G$ on an $n$-element set.

Similarly if we take a homomorphism from $G$ into a group of invertible matrices $Gl_n(K)$, for $K$ a field, say, then we get a linear action of $G$ on the vector space, $K^n$.

As you would expect, we can generalise and categorify this basic idea in several useful ways.

We can think of $G$ as a groupoid, $\mathbf{B}G$, (the [[delooping]] of $G$), and then a linear representation / action will be a functor from $\mathbf{B}G$ to $Vect$, the category of vector spaces over $K$.  We could replace $G$ by a general groupoid, or a general category, but then a representation of that is the same as a diagram of that 'shape' in $Vect$. We could replace $Vect$ by another more general category, or higher category, but if we are thinking of diagrams as representations, perhaps we should not totally forget that the term 'representation' did mean a process whereby the perhaps abstract 'syntactical' objects of the category gain a 'semantic' meaning, as 'operations' of some type, and which in turn, can be usefully used to gain information on the inherent structure.

## General definition

In a rather general form, we therefore have a __representation__ of a category $C$ in a category $D$ is simply a [[functor]] $F\colon C \to D$.  Similarly, an [[homomorphism]] between representations ("[[intertwiner]]") is simply a [[natural transformation]] between functors when they are being thought of as representations. Thus we have a [[category of representations]].

The term 'representation' is most often used when one or more of the following conditions apply:

*  $D$ is the category $k$-[[Vect]] of vector spaces over some [[field]] $k$; one then has a __$k$-linear representation__.
*  $C$ is the [[delooping]] of a [[group]]; one then has a __group representation__ in $D$.  Such a representation gives us a specific object $V$ of $D$; we say that we have a representation of $G$ on $V$.
*  $C$ is the [[free category]] on a [[quiver]]; one then has a __[[quiver representation]]__.

The classical [[representation theory]] of groups is about representations of (finite, topological, smooth etc.) groups on (topological) vector spaces, that is when the first two conditions apply.

There are also enriched, $k$-linear and other versions, hence one can talk about representations of [[Lie algebras]], [[vertex operator algebras]], etc. See also [[representation theory]].

## Examples

* [[trivial representation]]

* [[linear representation]], [[permutation representation]]

* [[regular representation]], [[alternating representation]]

* [[adjoint representation]], [[coadjoint representation]]

* [[induced representation]], [[coinduced representation]]

* [[Lie algebra representation]]

* [[Galois representation]]

* [[groupoid representation]]

## Related concepts

* [[quiver variety]]

* [[action]], [[∞-action]]

* [[module]], [[∞-module]]

* **representation**, [[∞-representation]]

  * [[faithful representation]]

  * [[virtual representation]]

  * [[2-representation]]

  * [[star-representation]]

    * [[representation of a C-star-algebra]]

  * [[representation sphere]]

* [[character]]

* [[singleton representation]]

* [[equivariant homotopy theory]], [[global equivariant homotopy theory]]

* [[equivariant stable homotopy theory]], [[global equivariant stable homotopy theory]]

[[!include homotopy type representation theory -- table]]

## References

See also the references at _[[representation theory]]_.

[[!redirects representation]]
[[!redirects representations]]

[[!redirects group representation]]
[[!redirects group representations]]
[[!redirects representation of a group]]
[[!redirects representations of a group]]
[[!redirects representations of groups]]


representation of a C-star-algebra

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### Context
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#### Operator algebra
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[[!include AQFT and operator algebra contents]]
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# Contents
* table of contents
{:toc}

## Idea

This page collects a few notions and facts about [[representations]] of [[C-star-algebras]] with an eye towards their use in [[AQFT]].

In [[AQFT]] the observables are given by a [[causal net of algebras]], usually $C^*$-algebras. A concrete physical system corresponds to a [[state]] of the algebra of all observables, which leads, via the [[GNS construction]], to a representation of this algebra on a concrete Hilbert space. In this way the familiar picture of [[quantum mechanics]] reappears. The interpretation of states and their representation as modelling concrete physical systems means that a systematic study of all representation of a given algebra of observables is central to [[AQFT]].

## Definition

+-- {: .num_defn}
###### Definition

A **representation** $\pi$ of a $C^*$-algebra $A$ is a [[star-representation]] on a [[Hilbert space]] $H$, hence a $*$-homomorphism from $A$ to the algebra of [[bounded operators]] on $H$.

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+-- {: .num_remark}
###### Remark

The _[[continuous function|continuity]]_ of the representation is implied by the [[star-representation]]-property.

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###### Definition

A representation is **faithful** if its [[kernel]] is trivial.

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###### Definition

Given two representations $\pi_1$ on $H_1$ and $\pi_2$ on $H_2$, if there is a [[unitary operator]] $U: H_1 \to H_2$ such that $U \pi_1 = \pi_2 U$ then the representations are **unitarily equivalent**. A linear map $U$ (not necessarily unitary) having this property is called an **[[intertwiner]]** or an **intertwining map**. If there is no nontrivial intertwiner the two representations $\pi_1$ and $\pi_2$ are called **disjoint** (or totally / completley different).

=--

In the context of [[AQFT]] the term 'intertwiner' is mostly used in the specific sense defined here.

From the physical viewpoint unitarily equivalent representations describe the same system, so that the classification of not unitarily equivalent representations is an important topic.

+-- {: .num_defn}
###### Definition
If there is a subspace $H_1$ of the Hilbert space $H$ which is invariant under $\pi(A)$, that is $\pi(A)(H_1) \subseteq H_1$, then the restriction of the representation to $H_1$ is again a representation of $A$, it is called a **subrepresentation** of $\pi$.
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+-- {: .num_defn}
###### Definition
Given a family $(\pi_i, H_i)$ of representations, we can form the direct sum $H := \oplus H_i$ of the Hilbert spaces and define a new representation $\pi := \oplus \pi$ via $\pi(A) | H_i := \pi_i$. This is the **direct sum of representations**.
=--

## Related concepts

* [[cyclic vector]]

* [[GNS construction]]

## References

See at _[[operator algebras]]_.

[[!redirects representation of a C-star-algebra]]
[[!redirects representations of a C-star-algebra]]
[[!redirects representations of C-star-algebras]]
[[!redirects representation of a C-star algebra]]
[[!redirects representations of a C-star algebra]]
[[!redirects representations of C-star algebras]]

[[!redirects C-star representation]]
[[!redirects C-star representations]]

[[!redirects C*-representation]]
[[!redirects C*-representations]]

[[!redirects representation of a C*-algebra]]
[[!redirects representations of a C*-algebra]]
[[!redirects representation of C*-algebras]]
[[!redirects representations of C*-algebra]]


representation ring

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## Definition

Any [[group]] $G$ has a [[category]] of finite-dimensional [[complex number|complex]]-linear [[representations]], often denoted $Rep(G)$.  This is a [[symmetric monoidal category|symmetric monoidal]] [[abelian category]] (a "[[tensor category]]") and thus has a [[Grothendieck ring]], which is called the **representation ring** of $G$ and denoted $R(G)$. Elements of the representation ring are hence formal differences (with respect to [[direct sum]]) of ordinary representations: [[virtual representations]].

### In components

More concretely, we get $R(G)$ as follows.  It has a [[basis]] $(e_i)_i$ given by the [[isomorphism classes]] of **[[irreducible representations]]** of $G$: that is, $i$ is an index for an irreducible finite-dimensional complex representation of $G$.  It has a product given by

$$ e_i e_j = \sum_k m_{i j}^k e_k ,$$

where $m_{i j}^k$ is the multiplicity of the $k$th irrep in the
[[tensor product of representations]] of the $i$th and $j$th irreps.

In [[physics]] these coefficients are also known as _[[Clebsch-Gordan coefficients]]_ (specifically for $G$ the [[special orthogonal group]] $SO(3)$), and the relations they satisfy are also known as _[[Fierz identities]]_ (specifically for $G$ a [[spin group]]).

Notice that $R(G)$ is commutative thanks to the [[symmetric monoidal category|symmetry]] of the tensor product.

### As equivariant K-theory of the point
 {#AsEquivariantKTheoryOfThePoint}

Equivalently the representation ring of $G$ over the [[complex numbers]] is the $G$-[[equivariant K-theory]] of the point, or equivalently by the [[Green-Julg theorem]], if $G$ is a [[compact Lie group]], the [[operator K-theory]] of the [[group algebra]] (the [[groupoid convolution algebra]] of the [[delooping]] groupoid of $G$):

$$
  R_{\mathbb{C}}(G) \simeq KU^0_G(\ast) \simeq KK(\mathbb{C}, C(\mathbf{B}G))
  \,.
$$

The first [[isomorphism]] here follows immediately from the elementary definition of equivariant [[topological K-theory]], since a $G$-[[equivariant vector bundle]] over the point is manifestly just a [[linear representation]] of $G$ on a [[complex vector space]].

(e.g. [Greenlees 05, section 3](#Greenlees05), [Wilson 16, example 1.6 p. 3](#Wilson16))

Therefore a similar isomorphism identifies the $G$-representation ring over the [[real numbers]] with the equivariant orthogonal $K$-theory of the point in degree 0:

$$
  R_{\mathbb{R}}(G)
  \;\simeq\;
  KO_G^0(\ast)
  \,.
$$

But beware that equivariant [[KO]], even of the point, is much richer in higher degree ([Wilson 16, remark 3.34](#Wilson16))

In fact, [[equivariant KO-theory]] of the point subsumes the [[representation rings]] over the [[real numbers]], the [[complex numbers]] and the [[quaternions]]:

$$
  KO_G^n(\ast)
  \;\simeq\;
  \left\{
  \array{
    0 &\vert& n = 7
    \\
    R_{\mathbb{H}}(G)/ R_{\mathbb{R}}(G) &\vert& n = 6
    \\
    R_{\mathbb{H}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 5
    \\
    R_{\mathbb{H}}(G) \phantom{/ R_{\mathbb{R}}(G) }  &\vert& n = 4
    \\
    0 &\vert& n = 3
    \\
    R_{\mathbb{C}}(G)/ R_{\mathbb{H}}(G) &\vert& n = 2
    \\
    R_{\mathbb{R}}(G)/ R_{\mathbb{C}}(G) &\vert& n = 1
    \\
    R_{\mathbb{R}}(G) \phantom{/ R_{\mathbb{R}}(G)} &\vert& n =0
  }
  \right.
$$

([Greenlees 05, p. 3](#Greenlees05))

[[!include Segal completion -- table]]

## Properties

### Relation to the character ring

If $G$ is a [[finite group]] and we [[tensor product of modules|tensor]] $R(G)$ with the [[complex numbers]], it becomes [[isomorphism|isomorphic]] to the **[[character ring]]** of $G$: that is, the ring of complex-valued functions on $G$ that are constant on each [[conjugacy class]].  Such functions are called **[[class functions]]**.

Similarly for $G$ a [[compact Lie group]], its complex linear [[representations]] $\rho \colon G \to U(n) \to Aut(\mathbb{C}^n)$ (for all $n \in \mathbb{N}$) are uniquely specified by their [[characters]] $\chi_\rho \coloneqq tr(\rho(-)) \colon G \to \mathbb{C}$. Therefore also here the representation ring is often called the _character ring_ of the group.

### Relation to Schur's lemma

The ([[isomorphism classes]]) of [[finite-dimensional vector space|finite-dimensional]] [[irreducible representations]] $V_i$ form the canonical $\mathbb{Z}$-[[linear basis]] of the representation ring:

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

Moreover, the [[function]] assigning [[dimensions]] of [[hom-object|hom]]-[[vector spaces]] constitutes a canonical bilinear symmetric [[inner product]]:

$$
  \langle -,-\rangle
  \;\colon\;
  R(G) \times R(G)
    \longrightarrow
  \mathbb{Z}
  \,.
$$

In terms of this, _[[Schur's lemma]]_ states that the [[irreducible representations]] generally constitute an _[[orthogonal basis]]_, and even an _[[orthonormal basis]]_ when the [[ground field]] is [[algebraically closed field|algebraically closed]].

For more discussion of this perspective, see

* at _[[Schur's lemma]]_ the section _[In terms of categorical algebra](Schur's+lemma#InterpretationInCategoricalAlgebra)_;

* at _[[Gram-Schmidt process]]_ the section _[Categorified Gram-Schmidt  process](Gram-Schmidt+process#CategorifiedGramSchmidtProcess)_.

### Relation to equivariant K-theory

The representation ring of a [[compact Lie group]] is equivalent to the $G$-[[equivariant K-theory]] of the point.

$$
  Rep(G) \simeq K_G(\ast)
  \,.
$$

The construction of representations by [[index]]-constructions of $G$-equivariant [[Dirac operators]] ([[push-forward in generalized cohomology|push-forward]] in $G$-[[equivariant K-theory]] to the point) is called _[[Dirac induction]]_.

[[!include Segal completion -- table]]

### Lambda-ring structure
 {#LambdaRingStructure}

The [[Adams operations]] equip the representation ring with the structure of a [[Lambda ring]],. (e.g. [tomDieck 79, section 3.5](#tomDieck79), [tom Dieck 09, section 6.2](#tomDieck09), [Meir 17](#Meir17)).

### Relation to the Burnside ring

Let $G$ be a [[finite group]]. Consider

1. the [[Burnside ring]] $A(G)$, which is the [[Grothendieck group]] of the [[monoidal category]] $G Set$ of [[finite set|finite]] [[G-sets]];

1. the [[representation ring]] $R(G)$, which is the [[Grothendieck group]] of the monoidal category $G Rep$ of [[finite dimensional vector space|finite dimensional]] $G$-[[linear representations]].

Then then map that sends a G-set to the corresponding linear [[permutation representation]] is a [[strong monoidal functor]]

$$
  G Set \overset{\mathbb{C}[-]}{\longrightarrow} G Rep
$$

and hence induces a [[ring homomorphism]]

$$
  A(G) \overset{ \mathbb{C}[-] }{\longrightarrow} R(G)
$$

Under the identitification

1. of the [[Burnside ring]] with the [[equivariant stable cohomotopy]] of the point

   $$
     A(G) \;\simeq\; \mathbb{S}_G(\ast)
   $$

   (see [there](Burnside+ring#AsTheEquivariantStableCohomotopyOfThePoint))

1. of the [[representation ring]] with the [[equivariant K-theory]] of the point

   $$
     R(G) \;\simeq\; K_G(\ast)
   $$

   (as [above](#AsEquivariantKTheoryOfThePoint))

this should be image of the initial morphism of [[E-infinity ring spectra]]

$$
  \mathbb{S} \longrightarrow KU
$$

from the [[sphere spectrum]] to [[KU]].

f
### Splitting principle and Brauer induction theorem

The [[Brauer induction theorem]] says that, over the [[complex numbers]], the representation ring is generated already from the [[induced representations]] of 1-dimensional representations. This may be regarded as the [[splitting principle]] for linear representations and for [[characteristic classes of linear representations]] ([Symonds 91](#Symonds91)).

## Examples

\begin{example}\label{RepresentationRingOfUnitaryGroups}
**([[representation ring]] of [[unitary groups]])**
For $n \in \mathbb{N}$, the complex representation ring of [[U(n)]] is

$$
  R_{\mathbb{C}}
  \big(
    U(n)
  \big)
  \;\simeq\;
  \mathbb{Z}[x_1, \cdots, x_{n-1}, x_n, x_n^{-1}]
  \,,
$$

and that of [[SU(n)]] is

$$
  R_{\mathbb{C}}
  \big(
    SU(n)
  \big)
  \;\simeq\;
  \mathbb{Z}[x_1, \cdots, x_{n-1}]
  \,.
$$
\end{example}

(e.g. [Husemöller 66, Ch. 14, Thm. 3.1 (p. 192)](#Husemoeller66))

* The completion of $Rep(Spin(2k))$ is $\mathbb{Z}[ [ e^{\pm x_j} ] ]$ for $1 \leq j \leq k$ (e.g [Brylinski 90, p. 9](#Brylinski90)).

## Related concepts

* [[Clebsch-Gordan coefficients]], [[Fierz identities]]

* [[Deligne's theorem on tensor categories]]

## References

### General

Lecture notes:

* {#tomDieck09} [[Tammo tom Dieck]], Section 4.4. in: _Representation theory_, 2009 ([pdf](http://www.uni-math.gwdg.de/tammo/rep.pdf))

Computations for the [[classical Lie groups]]:

* {#Husemoeller66} [[Dale Husemöller]], *Representation Rings of Classical Groups*, Chapter 14 in: _Fibre bundles_, McGraw-Hill 1966 (300 p.); Springer Graduate Texts in Math. __20__, 2nd ed. 1975 (327 p.), 3rd. ed. 1994 (353 p.) ([pdf](http://www.maths.ed.ac.uk/~aar/papers/husemoller))

Exposition in relation to [[equivariant K-theory]] includes

* [[Akhil Mathew]], _[Equivariant K-theory](https://amathew.wordpress.com/2011/12/03/equivariant-k-theory/)_

* {#Greenlees05} [[John Greenlees]], _Equivariant version of real and complex connective K-theory_, Homology Homotopy Appl. Volume 7, Number 3 (2005), 63-82. ([Euclid:1139839291](http://projecteuclid.org/euclid.hha/1139839291))

* {#Wilson16} [[Dylan Wilson]], _Equivariant K-theory_, 2016 ([pdf](https://www.math.uchicago.edu/~dwilson/notes/equivariant-k-theory-talk.pdf), [[WilsonKTheory16.pdf:file]])

Classical results for [[compact Lie groups]]:

* {#Segal68} [[Graeme Segal]], _The representation ring of a compact Lie group_, Publications Mathématiques de l'Institut des Hautes Études Scientifiques, January 1968, Volume 34, Issue 1, pp 113-128 ([numdam:PMIHES_1968__34__113_0](http://www.numdam.org/item/?id=PMIHES_1968__34__113_0))

* Masaru Tackeuchi, _A remark on the character ring of a compact Lie group_, J. Math. Soc. Japan Volume 23, Number 4 (1971), 555-705 ([Euclid](http://projecteuclid.org/euclid.jmsj/1259849785))

In the generality of [[super Lie groups]]:

* [[Gregory Landweber]], _Representation rings of Lie superalgebras_, K-Theory 36 (2005), no. 1-2, 115-168 ([arXiv:math/0403203](http://arxiv.org/abs/math/0403203))

* [[Gregory Landweber]], _Twisted representation rings and Dirac induction_, J. Pure Appl. Algebra 206 (2006), no. 1-2, 21-54 ([arXiv:math/0403524](http://arxiv.org/abs/math/0403524))

With an eye towards [[loop group representations]]:

* {#Brylinski90} [[Jean-Luc Brylinski]], _Representations of loop groups, Dirac operators on loop space, and modular forms_, Topology, 29(4):461–480, 1990.

Concerning the [[Brauer induction theorem]] and the [[splitting principle]]:

* {#Symonds91} [[Peter Symonds]], _A splitting principle for group representations_, Comment. Math. Helv. (1991) 66: 169 ([dml:140229](https://eudml.org/doc/140229), [doi:10.1007/BF02566643](https://doi.org/10.1007/BF02566643))

### Adams operations

The [[Adams operations]] and [[Lambda-ring]]-structure on representation rings are discussed in

* {#tomDieck79} [[Tammo tom Dieck]], section 3.5 of _[[Transformation Groups and Representation Theory]]_, Lecture Notes in Mathematics 766 Springer 1979

* Robert Boltje, _A characterization of Adams operations on representation rings_, 2001 ([pdf](https://boltje.math.ucsc.edu/publications/p01v.pdf))

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

* [[Michael Boardman]], _Adams operations on Group representations_, 2007 ([pdf](http://www.math.jhu.edu/~jmb/note/adamrept.pdf))

* {#Meir17} Ehud Meir, [[Markus Szymik]], _Adams operations and symmetries of representation categories_ ([arXiv:1704.03389](https://arxiv.org/abs/1704.03389))

[[!redirects representation rings]]


representation sphere

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###Context###
#### Representation theory
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[[!include representation theory - contents]]
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#### Spheres
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[[!include spheres -- contents]]
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

Given a linear [[representation]] of a [[group]] ([[topological group]]) $G$ in a [[vector space]]/[[Cartesian space]] $\mathbb{R}^n$, then the corresponding _representation sphere_ is the [[one-point compactification]] of this $\mathbb{R}^n$ (the $n$-[[sphere]]) regarded as a [[G-space]].

Representation spheres induce the [[looping and delooping]] which is used in
[[RO(G)-grading|RO(G)-graded]] [[equivariant cohomology]] theory, [[Brown representability theorem|represented]] by [[genuine G-spectra]] in [[equivariant stable homotopy theory]].

## Properties

### Equivariant stereographic projection
 {#Construction}

+-- {: .num_prop #RepresentationSpheresAsUnitSpheres}
###### Proposition
**([[representation spheres]] of $V$ are [[unit spheres]] in $\mathbb{R} \oplus V$)**

Let $G$ be a [[finite group]] and $V \in RO(G)$ a [[finite dimensional vector space|finite-dimensional]] [[linear representation]] of $G$.

Consider the [[unit sphere]] $S(\mathbb{R}\oplus V)$ where $\mathbb{R}$ carries the [[trivial representation]]. Then the [[stereographic projection]] homeomorphism

$$
  S(\mathbb{R}\oplus V)\setminus \{(1,\mathbf{0})\}
  \stackrel{\simeq}{\longrightarrow}
  V
$$

is manifestly $G$-equivariant, with its inverse exhibiting $S(\mathbb{R}\oplus V)$ as the [[one-point compactification]] of $V$, hence

$$
  S^V \simeq_G S(\mathbb{R}\oplus V)
  \,.
$$

This also shows that $S^V$ is a [[smooth manifold]] with smooth $G$-[[action]].

=--

(e.g. [Marzantowicz & Prieto 2004, p. 215 (2 of 17)](#MarzantowiczPrieto04))

### Equivariant projective lines

In particular:

+-- {: .num_prop #OneDimensionalRepresentationSpheresAsProjectiveLine}
###### Proposition
**(1-dimensional [[representation spheres]] are [[projective G-spaces]])**

If $\mathbf{1}_V \,\in\, G Representations_k$ is 1-dimensional over the given [[ground field]] $k$, [[stereographic projection]] identifies the [[representation sphere]] of $V$ with the [[projective G-space]] over $k$ of $\mathbf{1}_V \oplus \mathbf{1}$:

$$
  \array{
    V^{cpt}
    &
      \longrightarrow
    &
    k P
   \big(
     \mathbf{1}_V \oplus \mathbf{1}
   \big)
    \\
    v &\mapsto&
    \left\{
    \array{
      [v,1] &\vert& v \in V
      \\
      [1,0] &\vert& v = \infty
    }
    \right.
  }
$$

=--

(e.g. [Atiyah 68, Sec. 4](projective+G-space#Atiyah68), [Greenlees 01, 9.C](projective+G-space#Greenlees01))

Prop. \ref{OneDimensionalRepresentationSpheresAsProjectiveLine} underlies the concept of [[equivariant complex oriented cohomology theory]].

### $G$-CW-Complex structure

+-- {: .num_prop #RepresentationSpheresAreGCWComplexes}
###### Proposition
**([[G-representation spheres are G-CW-complexes]])**

For $G$ a [[compact Lie group]] (e.g. a [[finite group]]) and $V \in RO(G)$ a [[finite dimensional vector space|finite-dimensional]] [[orthogonal group|orthogonal]] $G$-[[linear representation]], the [[representation sphere]] $S^V$ admits the structure of a [[G-CW-complex]].

=--

+-- {: .proof}
###### Proof

Observe that we  have a $G$-equivariant [[homeomorphism]] between the representation sphere of $V$ and the [[unit sphere]] in $\mathbb{R} \oplus V$, where $\mathbb{R}$ is the 1-dimensional [[trivial representation]]  (Prop. \ref{RepresentationSpheresAsUnitSpheres})

\[
  \label{RepresentationSphereAsUnitSphere}
  S^V \;\simeq\; S(\mathbb{R} \oplus V)
  \,.
\]

It is thus sufficient to show that unit spheres in orthogonal representations admit G-CW-complex structure.

This in turn follows as soon as there is a $G$-equivariant [[triangulation]] of $S(\mathbb{R}\oplus V)$, hence a [[triangulation]] with the property that the $G$-[[action]] restricts to a [[bijection]] on its sets of $k$-dimensional cells, for each $k$. Because then if $G/H$ is an [[orbit]] of this $G$-action on the set of $k$-cells, we have a cell $G/H \times D^k$ of an induced [[G-CW-complex]].

Since the unit spheres in (eq:RepresentationSphereAsUnitSphere) are [[smooth manifolds]] with [[smooth function|smooth]] $G$-[[action]], the existence of such $G$-[[equivariant triangulations]] follows for general [[compact Lie groups]] $G$ from the _[[equivariant triangulation theorem]]_ ([Illman 83](#Illman83)).

More explicitly, in the case that $G$ is a [[finite group]] such an equivariant triangulation may be constructed as follows:

Let $\{b_1, b_2, \cdots, b_{n+1}\}$ be an [[orthonormal basis]] of $\mathbb{R} \oplus V$. Take then as [[vertices]] of the triangulation all the distinct points $\pm g(b_i) \in \mathbb{R} \oplus V$, and as [[edges]] the [[geodesics]] (great circle segments) between nearest neighbours of these points, etc.

=--

## References

* {#Blumberg17} [[Andrew Blumberg]], Example 1.1.5 of _Equivariant homotopy theory_, 2017 ([pdf](https://www.ma.utexas.edu/users/a.debray/lecture_notes/m392c_EHT_notes.pdf), [GitHub](https://github.com/adebray/equivariant_homotopy_theory))

* {#MarzantowiczPrieto04} Waclaw Marzantowicz, [[Carlos Prieto]], _The unstable equivariant fixed point index and the equivariant degree_, Jourmal of the London Mathematical Society **69**  1 (2004) 214-230 [[pdf](http://hopf.math.purdue.edu/Marzantowicz-Prieto/Marprieto.pdf), doi:10.1112/S0024610703004721]

[[!redirects representation spheres]]

[[!redirects G-representation sphere]]
[[!redirects G-representation spheres]]


representation stability

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## Idea

The term *representation stability* [[Church & Farb (2013)](#ChuchFarb13)] refers to the phenomenon that for various [[sequences]] of [[group representations]] which arise in [[topology]] --- notably for  ([[symmetric group representations]] on) ([[cohomology groups|co]])[[homology groups]] of  ordered [[configuration spaces of points]] and other [[moduli spaces]] --- certain aspects, such as the multiplicities of [[irrep|irreducible]] sub-representations, eventually stabilize.

Specifically for the case of the [[ordinary homology]] of [[configuration spaces of points]] the phenomenon of representation stability is a refinement of the older notion of *[homological stabilization](configuration+space+of+points#HomologyAndStabilization)*:
Namely, where homological stabilization applies to *un*-ordered configurations of points, and says that the [[ordinary homology]] of these spaces eventually stabilizes (i.e. no longer increases) as the number of points grows, the same is far from true, at face value, for the *ordered* configuration spaces, the rank of whose homology groups instead increases strictly monotonically with the number of points. But the homology of unordered configuration spaces is still equipped with an [[group action|action]] of the [[symmetric group]] $Sym(n)$ on the given number of points, and the statement of *representation stability* is that *as [[representation theory of the symmetric group|symmetric group representations]]* the homology still does stabilize, notably in that the number and multiplicity of [[irreducible representations]] stabilizes.

In particular, the homology of the un-ordered configuration spaces is canonically included into that of ordered configuration spaces as the summand of the [[trivial representation]], and in this way representation stability does subsume and generalize the phenomenon of [homological stabilization](configuration+space+of+points#HomologyAndStabilization).

Moreover, it turns out that the general phenomenon and the details of representation stability for ordered [[configuration spaces of points]] is abstractly all governed by the fact that, as one looks at all numbers of points at once, their homology forms an *[[FI-representation]]* namely a [[functor]] from the [[category]] [[FinSetInj|$FinSet_{inj}$]] of [[finite sets]] with (just) [[injections]] between them.

## References

The observation originates with:

* {#ChurchFarb13} [[Thomas Church]], [[Benson Farb]], *Representation theory and homological stability*, Advances in Mathematics
**245** (2013) 250-314 [[doi:10.1016/j.aim.2013.06.016](https://doi.org/10.1016/j.aim.2013.06.016)]

Lecture notes:

* [[Steven V Sam]], *Representation stability*, lecture notes (2017) [[pdf](https://mathweb.ucsd.edu/~ssam/old/17F-847/notes.pdf)]

Introduction and review:

* [[Benson Farb]], *Representation Stability* [[arXiv:1404.4065](https://arxiv.org/abs/1404.4065)]

* [[Jenny Wilson]], *A brief introduction to representation stability*, Oberwolfach Workshop (Jan 2018) [[pdf](https://dept.math.lsa.umich.edu/~jchw/OberwolfachTalk-IntroductionToRepStability.pdf), [[Wilson-RepresentationStability.pdf:file]]]

* [[Jennifer C. H. Wilson]], *Representation stability and the braid groups*, talk at *[ICERM -- Braids](https://icerm.brown.edu/programs/sp-s22/)* (Feb 2022) [[pdf](https://app.icerm.brown.edu/assets/362/3661/3661_3250_021720221430_Slides.pdf)]

* {#Wilson23} [[Jenny Wilson]], *Stability patterns for braid groups and configuration spaces*, [talk at](Center+for+Quantum+and+Topological+Systems#WilsonApr2023) [[CQTS]] (Apr 2023) [[web](Center+for+Quantum+and+Topological+Systems#WilsonApr2023), slides:[[Wilson-CQTS-Apr-2023.pdf:file]], video:[YT](https://www.youtube.com/watch?v=uO2CFY7h5lo)]

* [[Rita Jimenez Rolland]], [[Jennifer C. H. Wilson]], *Stability properties of moduli spaces*, Notices of the AMS [**69** 4](https://www.ams.org/notices/202204/202204FullIssue.pdf) (April 2022) [[arXiv:2201.04096](https://arxiv.org/abs/2201.04096), [pdf](https://www.ams.org/journals/notices/202204/rnoti-p522.pdf), [web](https://www.ams.org/journals/notices/202204/noti2452/noti2452.html)]

Expressing the [[rational cohomology]] of [[ordered configuration spaces of points]] via [[factorization homology]] and [[Ran spaces]], and relation to higher representation stability:

* [[Quoc P. Ho]], _Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras_ [[arXiv:2004.00252](https://arxiv.org/abs/2004.00252)]

In view of [[FI-representation]]-theory:

* [[Trevor Hyde]], *$FI$-Modules and Representation Stability* (2016) [[pdf](https://math.uchicago.edu/~tghyde/Hyde%20--%20FI-modules%20and%20representation%20stability.pdf)]

* [[Nir Gadish]], *Categories of FI type: a unified approach to generalizing representation stability and character polynomials*, Journal of Algebra **480** (2017) 450-486 [[arXiv:1608.02664](https://arxiv.org/abs/1608.02664), [doi:10.1016/j.jalgebra.2017.03.010](https://doi.org/10.1016/j.jalgebra.2017.03.010)]

* [[Joe Moeller]], *Extensions of representation stable categories* [[arXiv:2209.03879](https://arxiv.org/abs/2209.03879)]

See also:

* [[Christin Bibby]], [[Nir Gadish]], *A generating function approach to new representation stability phenomena in orbit configuration spaces*, Trans. Amer. Math. Soc. Ser. B **10** (2023) 241-287 [[ams:S2330-0000-2023-00130-3](https://www.ams.org/journals/btran/2023-10-09/S2330-0000-2023-00130-3), [arXiv:1911.02125](https://arxiv.org/abs/1911.02125)]

Discussion of [[FI-representations]] in the generality of [[infinity-representations|$\infty$-representations]] in [[stable infinity-categories|stable $\infty$-categories]] and their analysis via [[Goodwillie calculus]]:

* [[Kaya Arro]], *FI-Calculus and Representation Stability* (2023) [[arXiv:2306.13597](https://arxiv.org/abs/2306.13597)]


representation theory

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#### Mathematics
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#Contents#
* table of contents
{:toc}

## Idea

_Representation theory_ is concerned with the study of [[algebra|algebraic]] [[structures]] via their [[representations]]. This concerns notably [[groups]], directly or in their incarnation as [[group algebras]], [[Hopf algebras]] or [[Lie algebras]], and usually concerns [[linear algebra|linear]] [[representations]], hence [[modules]] of these structures. But more generally representation theory also studies [[representations]]/[[modules]]/[[actions]] of generalizations of such structures, such as [[coalgebras]] via their [[comodules]] etc.

See also at _[[geometric representation theory]]_.

## In homotopy type theory
 {#InHomotopyTypeTheory}

The fundamental concepts of representation theory have a particular natural formulation in  [[homotopy theory]] and in fact in [[homotopy type theory]], which also  refines it from the study of [[representations]] of [[groups]] to that of     [[∞-representations]] of [[∞-groups]]. This includes both [[discrete ∞-groups]] as well as [[geometric homotopy types]] such as [[smooth ∞-groups]], the higher analog of [[Lie groups]].

The key observation to this translation is that

1. an [[∞-group]] $G$ is equivalently given by its [[delooping]] $\mathbf{B}G$ regarded with its canonical [[pointed object|point]] (see at [[looping and delooping]]), hence the universal $G$-[[principal ∞-bundle]]

   $$
     \array{
        G &\longrightarrow& \ast
        \\
        && \downarrow
        \\
        && \mathbf{B}G
     }
   $$

1. an [[∞-action]] $\rho$ of $G$ on any [[geometric homotopy type]] $V$ is equivalently given by a [[homotopy fiber sequence]] of the form

   $$
     \array{
        V &\stackrel{}{\longrightarrow}& V//_\rho G
        \\
        && \downarrow
        \\
        && \mathbf{B}G
     }
     \,,
   $$

   hence by a $V$-[[fiber ∞-bundle]] over $\mathbf{B}G$ which is the $\rho$-[[associated ∞-bundle]] to the universal $G$-[[principal ∞-bundle]] (see at _[[∞-action]]_ for more on this).

Under this identification, the representation theory of $G$ is equivalently

* the [[homotopy theory]] in the [[slice (∞,1)-topos]] over $\mathbf{B}G$;

* the [[homotopy type theory]] in the [[context]] of/[[dependent type theory|dependent on]] $\mathbf{B}G$.

More in detail, this yields the following identifications:

[[!include homotopy type representation theory -- table]]

## Examples

* [[representation theory of the symmetric group]]

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

* [[representation theory of the special unitary group]]

* [[Schur-Weyl duality]]

## Related entries

* [[representation]], [[action]], [[module]]

  * [[group]], [[group algebra]], [[groupoid]], [[algebraic group]], [[Lie algebra]]

  * [[vector space]],  [[affine space]], [[symplectic vector space]]

  * [[character]]

  * [[action]], [[module]], [[equivariant object]]

  * [[bimodule]], [[Morita equivalence]], [[induced representation]]

  * [[Hilbert space]], [[Banach space]], [[Fourier transform]], [[functional analysis]]

  * [[weight (in representation theory)]]

  * [[orbit]], [[coadjoint orbit]], [[Killing form]]

  * [[geometric quantization]], [[coherent state]]

  * [[socle]], [[quiver]]

  * [[module algebra]], [[comodule algebra]], [[Hopf action]], [[measuring]]

* [[representation ring]]

* [[irreducible representation]]

* [[Schur's lemma]]

* [[Young diagram]]

* [[Schur index]]

* [[McKay correspondence]], [[ADE classification]]

* [[asymptotic representation theory]]

* [[geometric representation theory]]

  * [[Borel-Weil theorem]], [[Beilinson-Bernstein localization]]

  * [[D-module]], [[perverse sheaf]], [[BBDG decomposition theorem]]

  * [[Kazhdan-Lusztig theory]]

  * [[Dirac induction]]

* [[Verma module]], [[BGG resolution]]

* [[representation stability]]

* [[Grothendieck group]], [[lambda-ring]], [[symmetric function]], [[formal group]]

* [[principal bundle]], [[torsor]], [[vector bundle]], [[Atiyah Lie algebroid]]

* [[character sheaf]], [[Harish Chandra transform]]

* [[geometric function theory]], [[groupoidification]]

* [[Eilenberg-Moore category]], [[algebra over an operad]], [[actegory]], [[crossed module]]

* [[reconstruction theorems]]

## References
 {#References}

Lecture notes:

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

* {#Teleman05} [[Constantin Teleman]], _Representation theory_, lecture notes 2005 ([pdf](https://math.berkeley.edu/~teleman/math/RepThry.pdf))

* Joel Robbin, _Real, Complex and Quaternionic representations_, 2006 ([pdf](http://www.math.wisc.edu/~robbin/angelic/RCH-G.pdf), [[Robbin08RCHRep.pdf:file]])

* [[Igor R. Shafarevich]], [[Alexey O. Remizov]]: §14 in: *Linear Algebra and Geometry* (2012) [[doi:10.1007/978-3-642-30994-6](https://doi.org/10.1007/978-3-642-30994-6), [MAA-review](https://maa.org/press/maa-reviews/linear-algebra-and-geometry)]

Textbook accounts

for [[finite groups]]:

* Charles Curtis, Irving Reiner, _Representation theory of finite groups and associative algebras_, AMS 1962

* {#LuxPahlings10} Klaus Lux, Herbert Pahlings, _Representations of groups -- A computational approach_, Cambridge University Press 2010 ([author page](http://www.math.rwth-aachen.de/~RepresentationsOfGroups/), [publisher page](http://www.cambridge.org/catalogue/catalogue.asp?isbn=9780521768078))

* Caroline Gruson, Vera Serganova, *From Finite Groups to Quivers via Algebras -- A Journey Through Representation Theory*, Springer (2018) [[doi:10.1007/978-3-319-98271-7](https://doi.org/10.1007/978-3-319-98271-7)]

and more generally for [[compact Lie groups]]:

* [[Tammo tom Dieck]], [[Theodor Bröcker]], *Representations of compact Lie groups*, Springer (1985) [[doi:10.1007/978-3-662-12918-0](https://link.springer.com/book/10.1007/978-3-662-12918-0)]

* {#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))

In the context of [[quantum mechanics]]:

* [[Peter Woit]], *Quantum Theory, Groups and Representations: An Introduction*, Springer 2017 [[doi:10.1007/978-3-319-64612-1](https://doi.org/10.1007/978-3-319-64612-1), ISBN:978-3-319-64610-7]

Discussion via [[string diagrams]]/[[Penrose notation]]:

* [[Jeffrey Ellis Mandula]], *Diagrammatic techniques in group theory*, Southampton Univ. Phys. Dept. (1981) ([cds:129911](https://cds.cern.ch/record/129911), [pdf](https://cds.cern.ch/record/129911/files/SHEP%2080-81-7.pdf))

* {#Cvitanovic08} [[Predrag Cvitanović]], _Group Theory: Birdtracks, Lie's, and Exceptional Groups_, Princeton University Press July 2008 ([PUP](https://press.princeton.edu/books/paperback/9780691202983/group-theory), [birdtracks.eu](http://birdtracks.eu/), [pdf](http://www.birdtracks.eu/version9.0/GroupTheory.pdf))

  > (aimed at [[Lie theory]] and [[gauge theory]])

Further references:

* [[Mikhail Khovanov]], _[Resources](http://www.math.columbia.edu/~khovanov/resources)_.

The relation to [[number theory]] and the [[Langlands program]] is discussed in

* [[Robert Langlands]], _Representation theory: Its rise and its role in number theory_ ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.207.3303))

[[!redirects linear representation theory]]

[[!redirects representation theories]]


representation theory - contents

**[[representation theory]]**

**[[geometric representation theory]]**

## Ingredients

* [[linear algebra]], [[algebra]], [[higher algebra]]

## Definitions {#sidebar_definitions}

[[representation]], [[2-representation]], [[∞-representation]]

* [[group]], [[∞-group]]

* [[group algebra]], [[algebraic group]], [[Lie algebra]]

* [[vector space]], [[n-vector space]]

* [[affine space]], [[symplectic vector space]]

* [[action]], [[∞-action]]

* [[module]], [[equivariant object]]

* [[bimodule]], [[Morita equivalence]]

* [[induced representation]], [[Frobenius reciprocity]]

* [[Hilbert space]], [[Banach space]], [[Fourier transform]], [[functional analysis]]

* [[orbit]], [[coadjoint orbit]], [[Killing form]]

* [[unitary representation]]

* [[geometric quantization]], [[coherent state]]

* [[socle]], [[quiver]]

* [[module algebra]], [[comodule algebra]], [[Hopf action]], [[measuring]]

## Geometric representation theory

* [[D-module]], [[perverse sheaf]],

* [[Grothendieck group]], [[lambda-ring]], [[symmetric function]], [[formal group]]

* [[principal bundle]], [[torsor]], [[vector bundle]], [[Atiyah Lie algebroid]]

* [[geometric function theory]], [[groupoidification]]

* [[Eilenberg-Moore category]], [[algebra over an operad]], [[actegory]], [[crossed module]]

* [[reconstruction theorems]]

## Theorems {#sidebar_theorems}

* [[Borel-Weil-Bott theorem]]

* [[Be?linson-Bernstein localization]]

* [[Kazhdan-Lusztig theory]]

* [[BBDG decomposition theorem]]

[Edit this sidebar](/nlab/edit/representation+theory+-+contents)


representation theory of the general linear group

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## Idea

The [[representation theory]] of the [[general linear groups]].

## Properties

### Polynomial irreps

The polynomial [[irreps]] of $GL(n,\mathbb{C})$ are labeled by [[Young diagrams]] $\lambda \in Part(n)$ (e.g. [Fulton 97, Thm. 2 on p. 114](#Fulton97)).

These are equivalently the irreps of the [[special linear group]] (e.g. [Sternberg 94, Sec. 5.8](#Sternberg94)).

And for $rows(\lambda) \leq n-1$ these are also the irreps of the [[special unitary group]] (e.g. [Peluse 14, p. 2](representation+theory+of+the+special+unitary+group#Peluse14)).

### Dimension of irreps and hook-content formula

[[!include hook length and content formulas -- table]]

## Related concepts

* [[Schur-Weyl duality]]

* [[representation theory of the symmetric group]]

* [[representation theory of the special unitary group]]

## References

* {#Sternberg94} [[Shlomo Sternberg]], Section 5 and Appendix C.7 of: *Group Theory and Physics*, Cambridge University Press 1994 ([ISBN:9780521558853](https://www.cambridge.org/gb/academic/subjects/mathematics/algebra/group-theory-and-physics?format=PB&isbn=9780521558853))

* {#Fulton97} [[William Fulton]], Section 8 of: _Young Tableaux, with Applications to Representation Theory and Geometry_, Cambridge U. Press, 1997 ([doi:10.1017/CBO9780511626241](https://doi.org/10.1017/CBO9780511626241))

* Amritanshu Prasad, *Representations of General Linear Groups* ([doi:10.1017/CBO9781139976824.007](https://doi.org/10.1017/CBO9781139976824.007)), Chapter 6 in: *Representation theory -- A Combinatorial Viewpoint*, Cambridge University Press  2014 ([doi:10.1017/CBO9781139976824](https://doi.org/10.1017/CBO9781139976824))

[[!redirects representation theory of general linear groups]]


representation theory of the special unitary group

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## Idea

The [[representation theory]] of the [[special unitary group]].

## Properties

### Irreps

The [[irreps]] of $SU(n)$ are those polynomial [[irreps]] of [[representation theory of the general linear group|GL(n,C)]], hence those irreps of $SL(n,\mathbb{C})$, which are labeled by [[partitions]]/[[Young diagrams]] $\lambda \in Part(n)$ with $rows(\lambda) \leq n - 1$.

(e.g. [Peluse 14, p. 1](#Peluse 14))

### Dimension of irreps and hook/content

[[!include hook length and content formulas -- table]]

## Related concepts

* [[hook-content formula]]

* [[representation theory of the symmetric group]]

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

## References

* [[Shlomo Sternberg]], Section 5 of: *Group Theory and Physics*, Cambridge University Press 1994

* {#Peluse14} [[Sarah Peluse]], *Irreducible representations of $SU(n)$ with prime power degree*, Séminaire Lotharingien de Combinatoire 71 (2014), Article B71d ([pdf](https://www.emis.de/journals/SLC/wpapers/s71peluse.pdf))


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## Idea

The [[representation theory]] of the [[symmetric groups]].

## Properties

### Irreducible representations
 {#IrreducibleRepresentations}

In [[characteristic zero]], the [[irreducible representations]] of the symmetric group are, up to [[isomorphism]], given by the [[Specht modules]] labeled by [[partitions]] $\lambda \in Part(n)$ (e.g. [Sagan 01, Thm. 2.4.6](#Sagan01)).

### Dimension of irreps and hook length

Over the [[complex numbers]]:

The [[dimension]] of the [[irrep]] $S^{(\lambda)}$ ([[Specht module]]) equals the [[number]] of [[standard Young tableau]] of shape $\lambda$:

\[
  \label{DimensionOfSPechtModuleEqualsNumberofStandardYoungDiagrams}
  dim\big(S^{(\lambda)}\big)
  \;=\;
  \left\vert
    sYTableaux_\lambda
  \right\vert
\]

(e.g. [Sagan, Thm. 2.6.5](#Sagan01))

Moreover, the [[number]] of [[standard Young tableaux]] of shape $\lambda$ is given by the [[hook length formula]]

\[
  \label{HookLengthFormulaCountingNumberOfStandardYoungTableau}
  \left\vert
    sYTableaux_\lambda
  \right\vert
    \;=\;
    n!
    \left(
      \prod_{
        { 1 \leq i \leq rows(\lambda) }
        \atop
        { 1 \leq j \leq \lambda_i }
      }
      \ell hook_\lambda(i,j)
    \right)^{-1}
  \,.
\]

This is due to [Frame, Robinson & Thrall 54](hook+length+formula#FrameRobinsonThrall54). Textbook accounts include [Stanley 99, Cor. 7.21.6](hook+length+formula#Stanley99), [Sagan 01 Thm. 3.10.2](#Sagan01).

Combining (eq:DimensionOfSPechtModuleEqualsNumberofStandardYoungDiagrams) with (eq:HookLengthFormulaCountingNumberOfStandardYoungTableau) gives the [[hook length formula]] for the [[dimension]] of the [[Specht modules]]

$$
  dim\big(S^{(\lambda)}\big)
  \;=\;
  n!
  \left(
    \prod_{
      { 1 \leq i \leq rows(\lambda) }
      \atop
      { 1 \leq j \leq \lambda_i }
    }
    \ell hook_\lambda(i,j)
  \right)^{-1}
  \,.
$$

(e.g. [James 78, Thm. 20.1](#James78))

[[!include hook length and content formulas -- table]]

## Related concepts

* [[Specht module]]

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

* [[representation theory of the special unitary group]]

* [[Schur-Weyl duality]]

* [[Jucys-Murphy element]]

* [[FI-representation]]

## References

Textbook accounts:

* {#James78} [[G. D. James]], *The Representation Theory of the Symmetric Groups*,  Lecture Notes in Mathematics, volume 682, Springer 1978 ([doi:10.1007/BFb0067708](https://link.springer.com/book/10.1007/BFb0067708), [pdf](http://www-users.math.umn.edu/~webb/oldteaching/Year2010-11/the-representation-theory-of-the-symmetric-groups-SLN.pdf))

* {#Diaconis88} [[Persi Diaconis]], Chapter 7 of: *Group Representations  in Probability  and  Statistics*, IMS Lecture Notes Monogr. Ser., 11: 198pp. (1988) ([jstor:i397389](https://www.jstor.org/stable/i397389), [ISBN: 0940600145](https://projecteuclid.org/ebooks/institute-of-mathematical-statistics-lecture-notes-monograph-series/Group-representations-in-probability-and-statistics/toc/10.1214/lnms/1215467407), [pdf](https://jdc.math.uwo.ca/M9140a-2012-summer/Diaconis.pdf))

* {#Fulton97} [[William Fulton]], Section 7 of: _Young Tableaux, with Applications to Representation Theory and Geometry_, Cambridge U. Press, 1997 ([doi:10.1017/CBO9780511626241](https://doi.org/10.1017/CBO9780511626241))

* {#Sagan01} [[Bruce Sagan]], _The symmetric group_, Springer 2001 ([doi:10.1007/978-1-4757-6804-6](https://link.springer.com/book/10.1007/978-1-4757-6804-6), [pdf](http://math.sfsu.edu/federico/Clase/RepTh/sagan.pdf))

See also:

* Wikipedia, *[Representation theory of the symmetric group](https://en.wikipedia.org/wiki/Representation_theory_of_the_symmetric_group)*

Notes:

* [[Yufei Zhao]], _Young Tableaux and the Representations of the Symmetric Group_ ([pdf](https://yufeizhao.com/research/youngtab-hcmr.pdf), [[ZhaoYoungTableaux.pdf:file]])

Discussion of [[character of a linear representation|characters]] for the [[symmetric group]] that depend only on [[Cayley distance]] from the [[neutral element]] ("block character"):

* {#GnedinGorinKerov11} [[Alexander Gnedin]], [[Vadim Gorin]], [[Sergei Kerov]], *Block characters of the symmetric groups*, Journal of Algebraic Combinatorics, 38, no. 1 (2013), 79-101 ([arXiv:1108.5044](https://arxiv.org/abs/1108.5044), [doi:10.1007/s10801-012-0394-9](https://doi.org/10.1007/s10801-012-0394-9))

From the perspective of the [[seminormal representation]]:

On the [[representation theory of the symmetric group]] via the [[seminormal representation]]:

* {#VershikOkounkov04} [[Anatoly Vershik]], [[Andrei Okounkov]], *A New Approach to the Representation Theory of the Symmetric Groups*,
Part I: Selecta Mathematica, New Series **2**, 581-605 ([arXiv:math/0503040](https://arxiv.org/abs/math/0503040), [doi:10.1007/BF02433451](https://doi.org/10.1007/BF02433451));
Part II (incorporates Part I in revised and improved form):
Russian version: [Записки научных семинаров ПОМИ 307 (2004), 57–98](ftp://ftp.pdmi.ras.ru/pub/publicat/znsl/v307/p057.ps.gz) (Zapiski nauchnyh seminarov POMI 307 (2004), 57–98);
English version: Journal of Mathematical Sciences 131 (2005), 5471–5494 ([doi:10.1007/s10958-005-0421-7](https://doi.org/10.1007/s10958-005-0421-7)).

In relation to [[quantum information theory]]:

* [[Alonso Botero]], *Quantum Information and the Representation Theory of the Symmetric Group*, Rev. colomb. mat. vol.50 no. 2 Bogotá July/Dec. 2016 ([doi:10.15446/recolma.v50n2.62210](https://doi.org/10.15446/recolma.v50n2.62210), [pdf](http://www.scielo.org.co/pdf/rcm/v50n2/v50n2a05.pdf))

[[!redirects representation theory of the symmetric groups]]
[[!redirects representation theory of symmetric groups]]

[[!redirects symmetric group representation]]
[[!redirects symmetric group representations]]


representative function

## Definition

Let either

* $G$ be a Hausdorff [[topological group]] and $F$ either the field of real or complex numbers or

* $G$ be a linear algebraic group over an infinite field $F$ .

Then an $F$-valued representative function on $G$ is $F$-valued function which arises in the form $t\circ\rho$ where $\rho:G\to End V$ is a representation of $G$ on a _finite dimensional_ $F$-vector space and $t:End V\to F$ a linear functional.

In the (Hausdorff) topological group case the representative function is automatically continuous and in the algebraic case it is automatically a regular function.

On the other hand if we assume that a function $f:G\to F$ is continuous or respectively regular then it is a representative function iff

* all left translates $L_g f : h\mapsto f(h g)$ where $g\in G$ form a finite dimensional $F$-vector space

what is true iff

* all right translates $R_g f : h\mapsto f(g h)$ where $g\in G$ form a finite dimensional $F$-vector space

The set of all representative functions on $G$ is a Hopf $F$-algebra.

[Hochschild1971](#Hochschild1971) If $G$ is an arbitrary monoid with multiplication $m:G\times G\to G$ then $m$ induces a map $m^*:Fun(G,k)\to Fun(G\times G,k)$, $m^*(f):f\mapsto f\circ m$. We say that $f\in Fun(G,k)$ is representative if $m^*(f)$ is in the image of the canonical map $Fun(G,k)\otimes Fun(G,k)\hookrightarrow Fun(G\times G,k)$. Equivalently, $f$ is representative if the span of all functions $g\cdot f : h\mapsto f(h\cdot g)$ is finite dimensional. It follows then that $m^*(f)$ is in fact in (the image of) $R(G)\otimes R(G)$ where $R(G)$ is the space of all representative functions on $G$.

## Relations to other spaces of functions on a group

Let $G$ be an algebraic subgroup of $GL(d,k)$. The Hopf algebra of regular ("coordinate") functions $\mathcal{O}(G)$ is a finitely generated subHopf algebra of the Hopf algebra of continuous representative functions on $G$, [CartierHopf](#CartierHopf).

[[Peter-Weyl theorem]] says that the representative functions on a compact topological group form a dense subspace of the space of all continuous functions.

## Literature

Related entries include [[Tannaka duality]], [[locally compact topological group]], [[Hopf algebra]], [[harmonic analysis]], [[representation theory]], [[linear algebraic group]], [[distribution on an affine algebraic group]]

* G. Hochschild,  G. D. Mostow, _Representations and representative functions of Lie groups_, Annals of Mathematics, Second ser. 66 (1957) 495--542 [jstor](https://www.jstor.org/stable/1969906) [MR0098796](https://www.ams.org/mathscinet-getitem?mr=0098796)

* {#CartierHopf} [[Pierre Cartier]], _A primer of Hopf algebras_, Frontiers in number theory, physics, and geometry II, 537--615, Springer 2007.; preprint IH\'ES 2006/40 [pdf](http://preprints.ihes.fr/2006/M/M-06-40.pdf)

* J. C. Jantzen, Representations of algebraic groups, Acad. Press 1987 (Pure and Appl. Math. vol 131); 2nd edition AMS Math. Surveys and Monog. 107 (2003; reprinted 2007)

* A. Derighetti, _Representative functions on topological groups_, Comm. Math. Helv. __44__:1 (1969) 476--483

* {#Hochschild1971} G. Hochschild, _Introduction to algebraic group schemes_, 1971

[[!redirects representative functions]]


reproducing kernel Hilbert space

#Contents#
* table of contents
{:toc}

## Idea

A __reproducing kernel Hilbert space__ is a [[Hilbert space]] of functions in which point evaluation is a continuous linear functional. Using spectral measures one makes connection to specific kind of integral kernels.

## Related pages

* [[projection measure]], [[Bergman kernel]], [[Bargmann-Segal transform]], [[coherent state]]. Using reproducing kernels in the context of [[machine learning]] is known as the [[kernel method]]. In probability theory, the analogues are [[Markov kernel]]s (related to [[Chapman-Kolmogorov formula]] for conditional probabilities), see
[[monads of probability, measures, and valuations]]. The analogue for quantum amplitudes is used in derivation of Feynman integral approach, including the formal coherent state path integrals.

## Literature

* wikipedia: [reproducing kernel Hilbert space](https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space), [Bergman kernel](https://en.wikipedia.org/wiki/Bergman_kernel)
* N. Aronszajn, _Theory of reproducing kernels_, Trans. Amer. Math. Soc. 68 (1950) 337--404
* V. Bargmann, _On a Hilbert space of analytic functions and an associated integral transform_, Communications on Pure and Applied Mathematics __14__ (1961) 187--214 [MR0157250](https://www.ams.org/mathscinet-getitem?mr=157250) [doi](https://doi.org/10.1002/cpa.3160140303)
* J. H. Rawnsley, _Coherent states and Kähler manifolds_, Quart. J. Math. Oxford (2), 28 (1977) 403--415
* V. V. Kisil, _Integral representations and coherent states_, Bulletin of the Belgian Mathematical Society, v. 2 (1995), No 5, pp. 529-540.
* Daniel Beltiţă, José E. Galé, _Universal objects in categories of reproducing kernels_, Rev. Mat. Iberoamericana __27__:1 (2011) 123--179 [arXiv:0912.0091](https://arxiv.org/abs/0912.0091) [MR2815734](https://www.ams.org/mathscinet-getitem?mr=2815734) [euclid](http://projecteuclid.org/euclid.rmi/1296828831)
* Daniel Beltiţă, [[Tudor S. Ratiu]], _Geometric representation theory for unitary groups of operator algebras_, Advances in Mathematics __208__:1 (2007) 299--317 [doi](https://doi.org/10.1016/j.aim.2006.02.009) arXiv:[math.RT/0501057](https://arxiv.org/abs/math/0501057)

[[!redirects reproducing kernel]]


Republic dialogue

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### Context
#### Philosophy
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**The Republic** is a long dialogue written by [[Plato]], which covers a wide range of philosophical topics. The question of the best political organization of the _polis_, or city state, is the central theme of the book. In describing the role of mathematical training in the formation of the city-state's future leaders, or Guardians, Plato gives expression to his views on [[mathematics]] through the words he gives to [[Socrates]]. In particular, Plato divides mathematics into five branches: [[arithmetic]], [[plane geometry]], solid geometry (see also [[Platonic solids]]), astronomy, and harmonics. This syllabus was taken up in medieval university as the [quadrivium](https://en.wikipedia.org/wiki/Quadrivium).

Here the study of astronomy and of harmonics is claimed to be related (by what later came to be called the _[[The Music of the Spheres]]_):

> As the eyes, said I, seem formed for studying astronomy, so do the ears seem formed for harmonious motions: and these seem to be twin sciences to one another, as also the Pythagoreans say. (Republic VII.XII)

The Guardians are to study mathematics for many years to 'awaken the eye of the soul', the faculty by which they can appreciate also the Forms, especially those of Goodness and Beauty. On this theme, the dialog is also famous for the allegory of 'The Cave', where the best education is likened to releasing a prisoner condemned to see only shadows projected on a wall and eventually to emerge from the cave into the light (see at _[[doctrine of ideas]]_).

## Related entries

* [[Music of the Spheres]]

* [[Parmenides dialogue]]

* [[Timaeus dialogue]]

## References

* Wikipedia _[Republic (Plato)](https://en.wikipedia.org/wiki/Republic_%28Plato%29)_

[[!redirects Republic]]


request for help > history


Reshetikhin-Turaev construction

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#Contents#
* table of contents
{:toc}

## Idea

The Reshetikhin-Turaev construction is the [[FQFT]] construction of a 3d [[TQFT]] from the data of  a [[modular tensor category]] $\mathcal{C}$. It is something like the "square root" of the [[Turaev-Viro model]] on $\mathcal{C}$.

In the case that $C$ is a category of
[[positive energy representation]]s of a [[loop group]] $\Omega G$ of a [[Lie group]] $G$, then this algebraically defined QFT is thought to be the result of [[quantization]] of [[Chern-Simons theory]] over the group $G$.

## Properties

### As a boundary of the Crane-Yetter model

The Reshetikhin-Turaev model is a [[boundary field theory]] of the [[4d TQFT]] [[Crane-Yetter model]] ([Barrett&Garci-Islas&Martins 04, theorem 2](#BarrettGarciIslasMartins04)) Related discussion is in [[Freed]] "[[4-3-2 8-7-6]]".

### Relation to Chern-Simons theory

The RT-construction for group $G$ is expected to be the [[FQFT]] of $G$-[[Chern-Simons theory]], though a fully explicit proof of this via [[quantization]] is currently not in the literature.

See at _[[quantization of Chern-Simons theory]]_ for more on this.

### Relation to conformal field theory

The [[CFT|Fuchs-Runkel-Schweigert-construction]] builds from the RT-construction explicitly the rational 2-dimensional [[2d CFT]] boundary theory (see at _[[holographic principle]]_).

## References

Original articles:

* {#ReshetikhinTuraev91} [[Nikolai Reshetikhin]], [[Vladimir Turaev]], _Invariants of 3-manifolds via link polynomials and quantum groups_. Invent. Math. 103 (1991), no. 3, 547–597. ([doi:10.1007/BF01239527](https://doi.org/10.1007/BF01239527), [pdf](http://mathlab.snu.ac.kr/~top/quantum/article/Reshetikhin01.pdf))

* [[Vladimir Turaev]], *Modular categories and 3-manifold invariants*,  International Journal of Modern Physics B Vol. 06, No. 11n12, pp. 1807-1824 (1992) ([doi:10.1142/S0217979292000876](https://doi.org/10.1142/S0217979292000876))

  > (introducing [[modular tensor categories]])

Textbook accounts:

* [[Vladimir Turaev]], _Quantum invariants of knots and 3-manifolds_, de Gruyter Studies in Mathematics **18** Walter de Gruyter & Co., Berlin, 1994 ([doi:10.1515/9783110435221](https://doi.org/10.1515/9783110435221))

* [[Bojko Bakalov]], [[Alexandre Kirillov]], _Lectures on tensor categories and modular functors_ AMS, University Lecture Series, (2000) ([ISBN:978-1-4704-2168-7](https://bookstore.ams.org/ulect-21), [webpage](http://www.math.sunysb.edu/~kirillov/tensor/tensor.html)).

Review:

* Wikipedia, *[Reshetikhin-Turaev invariant](https://en.wikipedia.org/wiki/Reshetikhin%E2%80%93Turaev_invariant)*

See also:

* [[Jørgen Ellegaard Andersen]], William Petersen, *Construction of Modular Functors from Modular Tensor Categories* ([arXiv:1602.04907](https://arxiv.org/abs/1602.04907))

* [[Jørgen Ellegaard Andersen]], *Construction of the Witten-Reshetikhin-Turaev TQFT from conformal field theory* ([arXiv:1110.5027](https://arxiv.org/abs/1110.5027))

* [[Kevin Walker]], _On Witten's 3-Manifold Invariants_ ([old version](http://canyon23.net/math/1991TQFTNotes.pdf) [draft of new version](http://canyon23.net/math/tc.pdf))

Discussion that relates the [[geometric quantization]] of $G$-[[Chern-Simons theory]] to the Reshetikhin-Turaev construction of a 3d-[[TQFT]] from the [[modular tensor category]] induced by $G$ is in

* [[Jørgen Andersen]], _A geometric formula for the Witten-Reshetikhin-Turaev Quantum Invariants and some applications_ ([arXiv:1206.2785](http://arxiv.org/abs/1206.2785))
 {#Andersen}

and references cited there.

* Alain Bruguières, Alexis Virelizier, _Hopf diagrams and quantum invariants_, [math.QA/0505119](http://arxiv.org/abs/math/0505119); _Categorical centers and Reshetikhin-Turaev invariants_, [arxiv/0812.2426](http://arxiv.org/abs/0812.2426)

Relation to the [[Crane-Yetter model]]:

* {#BarrettGarciIslasMartins04} [[John Barrett]], J. Garcia-Islas, [[João Faria Martins]], _Observables in the Turaev-Viro and Crane-Yetter models_, J. Math. Phys. 48:093508, 2007 ([arXiv:math/0411281](http://arxiv.org/abs/math/0411281))

[[!redirects Reshetikhin-Turaev construction]]
[[!redirects Reshetikhin--Turaev construction]]

[[!redirects Reshetikin-Turaev model]]
[[!redirects Reshetikhin-Turaev model]]
[[!redirects Reshetikhin-Turaev model]]
[[!redirects Reshetikhin--Turaev model]]

[[!redirects Reshetikin-Turaev theory]]
[[!redirects Reshetikhin-Turaev theory]]

[[!redirects Turaev-Reshetikhin TQFT]]
[[!redirects Turaev-Reshetikhin TQFTs]]

[[!redirects Reshetikhin-Turaev invariant]]
[[!redirects Reshetikhin-Turaev invariants]]

[[!redirects RT-invariant]]
[[!redirects RT-invariants]]


residual

> This entry as written remains a partial duplicate of *[[internal hom]]*. See there for more. But [[lattices]] with internal homs are also known as *[[residuated lattice|residuated lattices]]*.

#Contents#
* table of contents
{:toc}

## Idea

Residuals, in the sense here, are "[[exponential object|exponential objects]] for a [[monoidal category]]": they are compatible with the monoidal product $\otimes$. Because $\otimes$ is (in general) not symmetric, we distinguish a *left residual* and a *right residual*. In a [[symmetric monoidal category]], these are the same.

Recall the definition of an exponential object for a cartesian category:

Let $A,C$ be two objects of the cartesian category. An _exponential object_ is an object $C^A$ with data $\mathit{ev},\varepsilon$ such that for every object $B$, there is a one-to-one mapping between morphisms

$$ f : A \times B \to C $$

and morphisms

$$ \varepsilon[f] : B \to C^A \; . $$

The backwards direction is given by $\mathit{ev} : A \times C^A \to A$, namely $f = \mathit{ev} \circ (A \times \varepsilon[f])$. Going back and forth is also required to be the identity: for $g : B \to C^A$, we must have $g = \varepsilon[ev \circ (A \times g)]$.

## Definition

Let $A,C$ be two objects of a monoidal category.

A _left residual_ of $C$ by $A$ is an object $A{\backslash}C$ together with an evaluation map $\mathit{\lev} : A \otimes (A{\backslash}C) \to C$ and for objects $B$ a transformation from morphisms

$$ f : A \otimes B \to C $$

to morphisms

$$ \lambda[f] : B \to A{\backslash}C $$

such that $f = \mathit{\lev} \circ (A \otimes \lambda[f])$, and $g = \lambda[\mathit{\lev} \circ (A \otimes g)]$ for every morphism $g : B \to A{\backslash}C$.

A _right residual_ of $C$ by $A$ is an object $C{/}A$ together with an evaluation map $\mathit{rev} : (C{/}A) \otimes A \to C$ and for objects $B$ a transformation from morphisms

$$ f : B \otimes A \to C $$

to morphisms

$$ \rho[f] : B \to C{/}A $$

such that $f = \mathit{rev} \circ (\rho[f] \otimes A)$, and $g = \rho[\mathit{rev} \circ (g \otimes A)]$ for every morphism $g : B \to C{/}A$.

Mnemonic for the notation: $C{/}A$ looks like dividing $C$ by $A$ on the right, and $A{\backslash}C$ looks like dividing $C$ by $A$ on the left.

## Properties

Left and right residuals are unique up to isomosphism.

## Examples

* In every cartesian category, the exponential objects are left and right residuals.
* Any [[monoidal closed category]] has all right residuals.

## Related concepts

* [[exponential object]]
* [[internal hom]]
* [[monoidal closed category]]

[[!redirects left residual]]
[[!redirects right residual]]

## References

The term "residual" for left/right [[internal homs]] is (non-standard and) used in

* [[Paul-André Melliès]] and [[Noam Zeilberger]], _Functors Are Type Refinement Systems_, in _Proceedings of the 42nd Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages_, POPL '15, 3–16. New York, NY, USA, 2015. ACM. doi:[10.1145/2676726.2676970](http://doi.acm.org/10.1145/2676726.2676970).


residuated idempotent semiring

##Idea

In general, in the [[idempotent semiring]] $(max,+)$, an equation of form $Ax=b$ has no solution, but the inequality $Ax\leq b$ does by taking $x=\mathbb{0}$.  (Here recall that in any idmepotent semiring, there is a natural partial order on the elements.)  It is natural to relax equality in the search for somultions, and study instead the set of its 'subsolutions'.  One way forward in this approach is to use the notion of [[residuated mapping]] from the theory of [[posets]].

## Definition

An idempotent semiring, $S$ is residuated if the right and left multiplication maps

$$\lambda_a:x\mapsto ax$$

and

 $$\rho_a:x\mapsto xa$$

from $S$ to itself are [[residuated mapping|residuated]].

## Example

Any complete idempotent semiring is automatically residuated.  We set

$a \backslash b:= \lambda^\#_a (b) = max \{x \mid ax \leq b\}$

and

$b / a:= \rho_a^\# (b) = max \{ x \mid xa \leq b\}.$

In the completed $(max,+)$ semiring, $\overline{\mathbb{R}}_{max}$, $a\backslash b$ and $b/a$ are equal and both equal $b-a$, provided that $a\neq \mathbb{0}$, in which case they equal $+\infty$.

##References

* [[Stéphane Gaubert]], _Methods and Applications of $(max,+)$ Linear Algebra,_ [Report 3088](https://hal.inria.fr/inria-00073603/document), January 1997, INRIA.


residuated lattice

##Definition

A _residuated lattice_ is an algebra, $L=(L,\wedge,\vee, \cdot, \backslash, /,1)$ such that

* $(L,\wedge,\vee)$ is a lattice;

* $(L,\cdot,1)$ is a monoid;

* for all $a,b,c\in L$,

$$(ab\le c)\Leftrightarrow (b\le a \backslash c)\Leftrightarrow (a\le c / b).$$

##Examples

###Powerset of a monoid.
 Let $(M,\cdot, e)$ be a monoid and consider $L=\mathcal{P}(M)$.

If $X, Y\subseteq M$, define

$$X\cdot Y=\{x\cdot y\mid x\in X, y\in Y\};$$

$$X\backslash Y = \{z\in M\mid X\cdot\{z\}\subseteq Y\};$$

$$Y/ X= \{z\in M\mid \{z\}\cdot X\subseteq Y\}$$

and then $L$ with this structure is a residuated lattice.

* A special instance of this is with the quantale of ideals in a ring, $R$; see [[ideals in a monoid]] and [[quantale]], especially the section on examples.

###Relation algebra on a set

 For any set, $X$, the relation algebra, $Rel(X)=\mathcal{P}(X\times X)$ of binary relations on $X$ is a residuated lattice.

Writing $R^-$ for the complement of a relation, $R$, and $R;S$ for the relational composition of two relations, $R$ and $S$, we define

$R \backslash S = (R;S^-)^- ,$

$S / R = (S^-;R)^- $

and

 $R\to S= (R\cap S^-)^- = R^-\cup S.$

We have

* $(Rel(X), \cap,\cup,\to, \emptyset, X^2)$ is a Boolean algebra,

*  $(Rel(X), ; \Delta)$ is a monad, where $\Delta$ is the diagonal / equality relation;

* the residuation relation is satisfied:

$$((R;S)\subseteq T)\Leftrightarrow (S\subseteq (R\backslash T))\Leftrightarrow (R\subseteq T / S)).$$

###Lattice-ordered group

Any [[lattice ordered group]] gives a residuated lattice. This is described in the entry on lattice ordered groups.  It is noteworthy that in this context, the residuals are quotients and, in fact, the inverses in the group are residuals.

##Categorical interpretations of residuation

Looking at the poset structure of $L$ as a category, one rewrites $a\le b$ as $a\to b$ and  $L(a,b)$ for the singleton set consisting just of the morphism $a\to b$.

We have for each element $a$ in $L$, two mappings given by, respectively, left and right multiplication by $a$:

$$\lambda_a:x\mapsto a\cdot x$$
$$\rho_a:x\mapsto x\cdot a.$$

These mappings are order preserving, so give endofunctors on the _category_ $L$.  For instance, suppose $x\le y$ and we have $a\cdot y\leq b$.  We then have $y\leq a/b$, and hence $x\le a/b$, i.e. $a\cdot x\leq b$.  Now take $b= a\cdot y$.  We have thus that $a\cdot x\leq a\cdot y$, as required.

In this interpretation, both these functors have right adjoints, namely the two residuals. We have, for instance,
$$L(\lambda_a(x),c) \cong L(x,a\backslash c).$$
The residual functor, $a\backslash -$, acts like a left exponential object for the left multiplication functor. Likewise $-/a$ acts like a right exponential object functor.

This point of view is explored further in the entry on [[residual]]s.

##Related entries

* [[residual]]

* [[quantale]]

* [[idempotent semiring]]

* [[residuated morphism]]

##References

A classical reference is

* M. Ward and R. P. Dilworth, _Residuated Lattices_,  Transactions of the American Mathematical Society, Vol. 45, No. 3 (May, 1939), pp. 335-354.[doi.org/10.2307/1990008 ](https://doi.org/10.2307/1990008 )

A set of slides from a modern treatment is

* Nikolaos Galatos, _Residuated lattices_, [slides](https://www2.karlin.mff.cuni.cz/~ssaos/2008/ResiduatedLatticesTutorial.pdf)


residuated morphism

##Idea

Coming originally from the theory of  [[poset|(partially) ordered sets]], the notion of residuated morphism or residuated mapping is, from a categorical viewpoint, just the condition that a map of posets when considered as a functor between the corresponding categories, has a left adjoint.

The notion is used in the theory of [[idempotent semirings]] to give a form of `pseudo-solution' to equations which fail to have actual solutions.

## Definition
 {#Definition}

Given [[posets]] $(E,\le)$ and $(F,\le)$, a monotone map, $f:E\to F$ is said to be _residuated_ if, and only if, for each $y\in F$, the set $\{x\in E\mid f(x)\le y\}$ has a maximal element, which we denote $f^\#(y)$.

##Translation into categorical terms

Each poset gives a small category, and each monotone map gives a functor.  From the categorical viewpoint, the condition that $f$ be residuated interprets as saying it has a left adjoint.

More exactly, the assignment $y\mapsto f^#(y)$ defines a monotone mapping $f^\#:F\to E$ called the _residual mapping_.  We have

$$f\circ f^\# \le id_F$$

and

$$f^\#\circ f \ge id_F.$$

##Standard example

The [[ceiling]] function $\lceil{-}\rceil:\mathbb{R}\to \mathbb{Z}$ is residuated. (The details are given in the discussion at [[floor]].)

##Related entries

* [[residuated lattice]]

* [[residuated idempotent semiring]]

* [[Galois connection]]

##References.

One of the standard references for the poset viewpoint is

* T. S. Blyth and M. F. Janowitz, _Residuation Theory_, Pergamon Press, 1972.

The Wikipedia entry is

* [residuated mapping](https://en.wikipedia.org/wiki/Residuated_mapping)

[[!redirects residuated mapping]]


residue

#Contents#
* table of contents
{:toc}

## Idea

The _residue_ of a [[meromorphic function]] $f$ at some point $\zeta \in \mathbb{Z}$ is the [[coefficient]] of $(z - \zeta)^{-1}$ (the first order [[pole]]) of its [[Laurent series]] expansion around $\zeta$. Up to a prefactor, this is what is picked up by the [[contour integral]] around that point, accrding to the [[Cauchy integral formula]].

## Related entries

* [[pole]]

* [[Cauchy integral formula]]

## References

* Wikipedia, _Residue (complex analysis)_

[[!redirects residues]]


residue field

> This entry is about the notion of a residue field in [[algebraic geometry]]. There is another (related) notion of a residue field in [[constructive mathematics]]; see [[weak Heyting field]].

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## Idea

Given a [[local ring]] $R$, by the definition there is a [[maximal ideal]] $\mathfrak{m}\subset R$. The quotient $R/\mathfrak{m}$ is therefore a [[division ring]], and in commutative case, therefore a [[field]], called the residue field. In algebraic geometry, the __residue field__ at a point $x$ of a scheme $X$ is the residue field of the corresponding stalk $\mathcal{O}_{X,x}$ of the structure sheaf, which is by the definition a local ring.

[[!redirects residue field]]
[[!redirects residue fields]]


resizing axiom

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## Idea

In [[formal logic]] and specifically in [[type theory]], a _resizing rule_ is an [[introduction rule]] which allows, under suitable conditions, to find a [[type]] that is in some [[type universe]] $U_2$  also in a smaller [[type universe]] $U_1$.

## References

* [[Vladimir Voevodsky]], _Resizing Rules - their use and semantic
justification_, 2011 ([pdf](http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2011_Bergen.pdf))

[[!redirects resizing axioms]]

[[!redirects resizing rule]]
[[!redirects resizing rules]]


resolution

> This entry is about resolutions in the sense of [[homotopy theory]]. For resolutions of [[singularities]] see at _[[resolution of singularities]]_.

***

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## Idea

In the context of [[homotopy theory]] modeled by [[homotopical categories]], where [[equality]] is relaxed to a weaker concept of ([[weak homotopy equivalence|weak]]) [[homotopy equivalence]] or generally to _[[weak equivalences]]_, a _resolution_ of any [[object]] $X$ is a choice of weak equivalence to or from another object $X_{res}$, such that $X_{res}$ has some prescribed good properties which $X$ itself may be lacking.

This general concept may be formalized by [[homotopical categories]] that carry extra information on what counts as a "good property" of an object, called _[[fibrant objects]]_ or _[[cofibrant objects]]_. This includes [[fibration categories]]/[[cofibration categories]] and notably [[model categories]].

A common cause for need of resolutions occurs when in a [[category]] of nice objects certain [[universal constructions]] -- such as [[quotients]] or [[intersections]] -- fail to exist. Homotopical resolution may allow to embed the nice objects in categories of _systems_ of nice objects (such as [[simplicial objects]] or [[chain complexes]]) inside which homotopical resolutions may be found (such as [[simplicial resolutions]] or [[homological resolutions]], respectively).

For example the [[quotient]] of a [[scheme]] or a [[smooth manifold]] by the [[action]] of an [[algebraic group]] or [[Lie group]] may fail to be a scheme or smooth manifold again, respectively, but the corresponding _[[action groupoid]]_ ([[orbifold]]) serves as a resolution of the quotient in a suitable [[homotopical category]] of [[simplicial objects]] in schemes/manifolds (presenting the "[[quotient stack]]", see at _[[higher geometry]]_ for more).

Dually the [[intersection]] of two subschemes/[[submanifolds]] may fail to be a scheme/manifold itself, but the corresponding [[derived intersection]] may provide a resolution in a homotopical category of [[cosimplicial objects]] in schemes/manifolds, respectively (see at _[[derived geometry]]_ for more).

## In a model category

If $C$ is a [[model category]] then the most important resolutions are _cofibrant resolutions_ and _fibrant resolutions_.

A **fibrant resolution** (or *fibrant approximation*) of $X$ is a [[fibrant object]] $\hat X$ equipped with a weak equivalence into it

$$
  X \stackrel{\simeq}{\to} \hat X \to *
  \,.
$$

If the weak equivalence is also a [[cofibration]], the fibrant resolution is a *good fibrant resolution.*

A **cofibrant resolution** (or *cofibrant approximation*) of $X$ is a [[cofibrant object]] $\hat X$ equipped with a weak equivalence out of it

$$
  \emptyset \hookrightarrow \hat X \stackrel{\simeq}{\to} X
  \,.
$$

If the weak equivalence is also a [[fibration]] the cofibrant resolution is a *good cofibrant resolution.*

Notice that the factorization axioms of a [[model category]] ensure that such resolutions always exist.

Of course for the notion of fibrant resolution to make sense, also the ambient structure of a [[category of fibrant objects]] works. For cofibrant resolutions a [[Waldhausen category]] does the job, etc.

In the context of [[cofibration categories]], the  term used is *fibrant model*. (One also finds the term *fibrant replacement* used.)

## Examples

### In chain complexes

We consider the case of _[[homological resolutions]]_ in one of the standard [[model structures on chain complexes]].

If $C$ is a category of [[chain complex]]es in a suitable (possibly structured) [[abelian category]] or [[semiabelian category]] $A$ then one can in particular consider resolutions of ordinary objects of $A$ -- regarded as a chain complex concentrated in degree 0 - by chain complexes of $A$.

A _resolution_ is an acyclic nonpositive complex $P_\cdot$ which coaugments $M$ or an acyclic nonnegative complex $I^\cdot$ which augments $M$, i.e. it is equipped with a map of complexes $P_\cdot \to M$ or a map of complexes $M\to I^\cdot$.

If each object $P_n$ is a [[projective object]] then  $P_\cdot \to M$ is a **[[projective resolution]]** , and if each $I^n$ is an [[injective object]] then $M\to I^\cdot$ is an **[[injective resolution]]** . These are fibrant and cofibrant resolutions in the suitable [[model structure on chain complexes]].

There are further generalizations like unbounded resolutions etc.

### In simplicial objects

See at _[[simplicial resolution]]_.

## Related concepts

* [[fibrant object]]

* [[canonical resolution]], [[bar construction]]

* [[homological resolution]]

* [[minimal fibration]]

[[!redirects resolutions]]

[[!redirects fibrant resolution]]
[[!redirects cofibrant resolution]]
[[!redirects fibrant resolutions]]
[[!redirects cofibrant resolutions]]

[[!redirects cofibrant replacement functor]]
[[!redirects fibrant replacement functor]]

[[!redirects cosimplicial resolution functor]]
[[!redirects simplicial resolution functor]]


resolution of singularities

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## Idea

Given a [[space]] $X$ of sorts, in particular a [[scheme]] or [[variety]], with [[singularities]], i.e. with a subspace $S \subset X$ at which the [[geometry]] is non-regular (specifically: not [[smooth scheme|smooth]]), a _resolution_ of the singularity is a suitable regular (non-singular) space
$\widehat X$ equipped with a [[morphism]] back to the original space

$$
  p \;\colon\; \widehat X \longrightarrow X
$$

which is an [[isomorphism]] away from the singular locus.

Typical resolution of singularities is by "blow-up" of the singularity where the singular point is replaced by an [[n-sphere]]/[[projective space]] (and its neighbourhood by a [[tautological line bundle]]), then called the "exceptional [[divisor (algebraic geometry)|divisor]]" of the blow-up.

(quick review of the basic details includes [Berghoff 14, section 4.1](#Berghoff14))

## Examples

* the [[Fulton-MacPherson compactification]] of [[configuration spaces of points]] may be regarded as resolution of the [[fat diagonal]] inside an iterated [[Cartesian product]];

* this generalizes to  "[[wonderful compactifications]]" resolving singularities given by more general subspace arrangements

* [[Bridgeland stability]] over resolution of [[ADE-singularities]]: see [there](Bridgeland+stability+condition#OverResolutionsOfADESingularities)

## Related concept

* [[ADE singularity]]

  * [[M-theory lift of gauge enhancement on D6-branes]]

  * [[triple membrane junction]]

* [[tautological line bundle]]

  [[tautological equivariant line bundle]]

* [[derived blow-up]]

## References

The existence of resolutions of singularities by "blow-up" was established, for [[ground fields]] of [[characteristic zero]], in some generality in

* {#Hironaka64} [[Heisuke Hironaka]], _Resolution of Singularities of an Algebraic Variety Over a Field of Characteristic Zero: I_, Annals of Mathematics Second Series, Vol. 79, No. 1 (Jan., 1964), pp. 109-203 (95 pages) ([jstor:1970486](https://www.jstor.org/stable/1970486))

Basic review:

* {#Berghoff14} [[Marko Berghoff]], section 4 of: _Wonderful renormalization_, 2014 ([pdf](http://www2.mathematik.hu-berlin.de/%7Ekreimer/wp-content/uploads/Berghoff-Marko.pdf), [doi:10.18452/17160](https://doi.org/10.18452/17160))

* [[Loring Tu]], Section 29.2 of: *Introductory Lectures on Equivariant Cohomology*, Annals of Mathematics Studies **204**, AMS 2020 ([ISBN:9780691191744](https://press.princeton.edu/books/hardcover/9780691191744/introductory-lectures-on-equivariant-cohomology))

The theorem of [Hironaka 64](#Hironaka64) was used to discuss singular [[distributions]] (in the sense of [[generalized functions]]) in

* {#Atiyah70} [[Michael Atiyah]], _Resolution of singularities and Division of Distributions_, Communications in Pure and Applied Mathematics, vol. XXIII, 145-150, 1970 ([doi:10.1002/cpa.3160230202](https://doi.org/10.1002/cpa.3160230202))

This method is closely related to the resolution of singularities of [[propagators]]/[[Feynman amplitudes]] by passage to [[Fulton-MacPherson compactification|compactified]] [[configuration spaces of points]], as disucussed at _[[Feynman amplitudes on compactified configuration spaces of points]]_.

See also

* Wikipedia, _[Resolution of singularities](https://en.wikipedia.org/wiki/Resolution_of_singularities)_

* Wikipedia, _[Blowing up](https://en.wikipedia.org/wiki/Blowing_up)_

* Wikipedia, _[Exceptional divisor](https://en.wikipedia.org/wiki/Exceptional_divisor)_

[[!redirects resolutions of singularities]]

[[!redirects crepant resolution]]
[[!redirects crepant resolutions]]

[[!redirects blowing up]]
[[!redirects blow up]]
[[!redirects blow-up]]
[[!redirects blowup]]

[[!redirects exceptional divisor]]
[[!redirects exceptional divisors]]


resolvent

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## Idea

(...)

## Definition

(...)

## Related concepts

* [[operator spectrum]]

## References

See also

* Wikipedia, *[Resolvent formalism](https://en.wikipedia.org/wiki/Resolvent_formalism)*

[[!redirects resolvents]]


rest energy

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## Idea

In [[special relativity]], the [[energy]] of a [[particle]] at rest is

$$
  E_0 \;=\; m c^2
$$

where

* $m$ is the [[mass]] of the particle;

* $c$ is the [[speed of light]].

## Related concepts

* [[Compton wavelength]]

* [[Schwinger limit]]

* [[Planck mass]]

## References

See also

* WIkipedia, _[Rest energy](https://en.wikipedia.org/wiki/Invariant_mass#Rest_energy)_

[[!redirects rest energies]]


restricted formal power series

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## Definition

Let $A$ be a commutative [[topological ring]].

A [[formal power series]]
$$
\sum c_{n_1,\ldots,n_p}X_1^{n_1}\cdots X_p^{n_p} \in A[[X_1,\ldots,X_p]]
$$
is called **restricted** if for every neigbourhood $V$ of 0 in $A$, there is only a finite number of coefficients $c_{n_1,\ldots,n_p}$ not belonging to $V$. One can view this as the coefficients "converging to $0\in A$".

If $A$ is linearly topologised (that is, it has a fundamental system of [[neighbourhoods]] of $0$ consisting of [[ideals]]) then restricted formal power series form a _subring_ $A\{X_1,\ldots,X_p\} \subset A[[X_1,\ldots,X_p]]$.

Every derivative (in the formal sense) of a restricted formal power series is also restricted.

If $A$ is discrete, then a $A\{X_1,\ldots,X_p\} = A[X_1,\ldots,A_p]$, the ring of [[polynomials]].


restricted Lie algebra

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## Idea

A _restricted Lie algebra_ is a [[Lie algebra]] in [[positive characteristic]] equipped with a compatible $p$-power operation ([[Frobenius morphism]]).

## Related concepts

* [[Milnor-Moore theorem]]

## References

* {#MilnorMoore65} [[John Milnor]], [[John Moore]], section 6 of _On the structure of Hopf algebras_, Annals of Math. __81__ (1965), 211-264. ([pdf](http://www.uio.no/studier/emner/matnat/math/MAT9580/v12/undervisningsmateriale/milnor-moore-ann-math-1965.pdf))

* Wikipedia, _[Restricted Lie algebra](https://en.wikipedia.org/wiki/Restricted_Lie_algebra)_

[[!redirects restricted Lie algebras]]


restricted product

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## Idea

Given a [[set]] $S$ and a collection $\{K_s \stackrel{\iota_s}{\hookrightarrow} X_s\}_{s \in S}$ of [[morphisms]] in some [[category]] (typically [[monomorphisms]]), then the _restricted product_ $\underset{s\in S}{\prod}^\prime X_s$ is vaguely the like the actual [[product]] $\underset{s\in S}{\prod} X_s$ of all the $X_s$, but subject to the restriction that for each [[element]] $(x_s)_{s \in S}$ all but a [[finite number]] of components $x_s$ are in the [[image]] of $\iota_s$.

## Definition

Assume that the ambient [[category]] $\mathcal{C}$ has ordinary [[products]]. Write $\mathcal{P}_{fin} S$ for the [[poset]] of [[finite set|finite]] [[subsets]] of $S$ and consider the [[functor]]

$$
  X^K \colon \mathcal{P}_{fin}\longrightarrow \mathcal{C}
$$

which is given on a [[finite set|finite]] [[subset]] $U \subset S$ by the [[products]]

$$
  U \mapsto X^K_U \coloneqq \left(\underset{u \in U}{\prod} X_u\right)
   \times
  \left(
     \underset{s \in S-U}{\prod} K_s
  \right)
$$

and which sends an inclusion $U_1 \hookrightarrow U_2$ of subsets to the evident [[morphism]] between these products whose components $f_s$ are the [[identity]] for $s \in S-(U_2-U_1)$ and are $\iota_s$ for $s \in U_2-U_1$.

Then the restricted product is the ([[filtered colimit|filtered]]) [[colimit]] over this functor

$$
  \underset{s\in S}{\prod}^\prime X_s
  \coloneqq
  \underset{\underset{U \in \mathcal{P}_{fin}S}{\longrightarrow}}{\lim}
  X^K_U
  \,.
$$

## Examples

* A [[weak direct product]] is a restricted product whose trivial subobjects are specified [[global elements]], hence where each $K_s = \ast$ are [[terminal objects]].

* The [[ring of adeles]] of a [[global field]] $k$ is the restricted product of the [[formal completions]] $k_v$ at all its [[places]] $v$, with the restriction being that only finitely many components have [[norm]] greater than unity.

  Equivalently this is the restricted product for the inclusions $\iota_v \colon \mathcal{O}_v \hookrightarrow k_v$ of the [[ring of integers]] of $k_v$ into $k_v$.

* Under the [[function field analogy]] the above example is a special case of the following more general example: for $\Sigma$ an [[arithmetic curve]] then the restricted product of all the algebras of functions on all its punctured [[formal disks]], with the restrictions being along the inclusions of the functions on the un-punctured formal disks, appears in the description of [[Cech cohomology|Cech cocycles]] with respect to [[covers]] of the curve by the complement of any finite points and the formal disks around these points (e.g [Frenkel 05, section 3.2](#Frenkel05)). This is discussed in detail at _[moduli stack of bundles -- over curves](moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence)_.

## References

* Wikipedia _[Restricted product](http://en.wikipedia.org/wiki/Restricted_product)_

* {#Frenkel05} [[Edward Frenkel]], _Lectures on the Langlands Program and Conformal Field Theory_, in _Frontiers in number theory, physics, and geometry II_, Springer Berlin Heidelberg, 2007. 387-533. ([arXiv:hep-th/0512172](http://arxiv.org/abs/hep-th/0512172))

* [MO/96137/categorical-description-of-the-restricted-product-adeles](http://mathoverflow.net/questions/96137/categorical-description-of-the-restricted-product-adeles)

[[!redirects restricted product]]
[[!redirects restricted products]]
[[!redirects restricted direct product]]
[[!redirects restricted direct products]]


restricted representation

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## Idea

Given a [[group]] $G$ and a [[representation]] $V$ of $G$, then any [[subgroup]] inclusion $H\hookrightarrow G$ makes also $H$ [[action|act]] on $V$, this is the _restricted representation_.

Given an [[irreducible representation]] of $G$, then its decomposition as a [[direct sum]] of irreducible representations of $H$, after restricting the respesentation to $H$, is often called the "branching rule" of the restriction.

More generally, for any group [[homomorphism]] $H \to G$ representations of $G$ "pull back" to representations of $H$.

The [[left adjoint]] to this construction is the operation of forming _[[induced representations]]_, the [[right adjoint]] that of _[[coinduced representations]]_.

## Examples

* [[supermultiplet]]

## References

* Wikipedia, _[Restricted representation](https://en.wikipedia.org/wiki/Restricted_representation)_

[[!redirects restricted representations]]

[[!redirects branching rule]]
[[!redirects branching rules]]

[[!redirects restricted action]]
[[!redirects restricted actions]]


restricted Yoneda embedding

# Restricted Yoneda embedding

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## Definition

For any [[functor]] $i:A\to C$ (often the inclusion of a [[full subcategory]]), the **restricted Yoneda embedding** is the composite

$$ C \hookrightarrow [C^{op},Set] \xrightarrow{i^\ast} [A^{op},Set] $$

of the ordinary [[Yoneda embedding]] of $C$ with the restriction functor $i^\ast$ along $i$.

When $C$ is cocomplete, the restricted Yoneda embedding has a left adjoint: the [[nerve and realization|realization]].

One important example of a restricted Yoneda embedding is that of the fully faithful inclusion $i : \Delta \hookrightarrow Cat$, where $\Delta$ is the [[simplex category]]. This is known as the [[nerve functor]].

## Related pages

* [[Yoneda embedding]]
* [[dense subcategory]]
* [[nerve]]


restriction

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## Definition

The __restriction__ of a [[function]] $f: X \to Y$ to a [[subset]] $U$ of $X$ is simply the [[composite]] of $f$ and the [[inclusion function]] of $U$:

$$ U \stackrel{i_U}\to X \stackrel{f}\to Y .$$

It is often written as $f_{|U}: U \to Y$, or a variation on that theme.

## Related concepts

* [[extension]]

* [[lift]]

* [[Cauchy principal value]]

[[!redirects restrictions]]


restriction and extension of sheaves

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## Idea

Recall that for [[presheaf|presheaves]] on a [[site]] $X$ (with underlying category $S_X$) with values in a category $A$ that admits small [[limit]]s and small colimits (so in particular for $A =$  [[Set]]), $PSh(X, A) = [S_X^{op}, A]$, every functor $f^t : S_Y \to S_X$ induces three functors of presheaf catgeories:

| Notation                                 | | Definition                    |
| ---------------------------------------- |-| ----------------------------- |
| $(f^t)_* : PSh(X,A) \to PSh(Y,A)$        | | direct image                  |
| $(f^t)^\dagger : PSh(Y,A) \to PSh(X,A)$  | | left adjoint to direct image  |
| $(f^t)^\ddagger : PSh(Y,A) \to PSh(X,A)$ | | right adjoint to direct image |

Recall moreover that for $f : X \to Y$ any [[site|morphism of sites]], the  left adjoint to [[direct image]] followed by [[sheafification]] $\bar{(-)}$ is the [[inverse image]] map of sheaves:

$$
  f^{-1} : Sh(Y,A) \to Sh(X,A)
  \,.
$$

Now, if the morphism of sites $f$ happens to be restriction to a sub-site  $f : X \to U$ with $U \in PSh(X,A)$ with $U$ carrying the induced [[Grothendieck topology|topology]], then

* the direct image is called **restriction** of sheaves;

* the right adjoint takes sheaves to sheaves and is called **extension** of sheaves.

## Definition

Given a [[site]] $X$ with underlying [[category]] $S_X$ and given a [[presheaf]] $U \in PSh(X)$ with the induced sub-site $j_{U \to X} : X \to U$
corresponding to the [[stuff, structure, property|forgetful]] functor
$j^t_{U \to X} : (Y_{S_X}/U) \to S_X$
from the [[comma category]] $S_U = (Y_{S_X}/U) \to S_X$ underlying the [[site]] $U$
(as discussed at [[site]]) the right [[adjoint functor]]

$$
  j^{\ddagger}_{U \to X} : PSh(U) \to PSh(X)
$$

to the [[direct image]] or, in this case, **restriction** functor

$$
  (j_{U \to X})_* : Sh(X) \to Sh(U)
$$
whose action may suggestively be denoted
$$
  (j_{U \to X})_* : F \mapsto F|_U
$$

happens to take sheaves to sheaves
(when $U$ is equipped with the canonical induced topology as described at [[site]]):

one calls
$$
  j^{\ddagger}_{U \to X} : Sh(U) \to Sh(X)
$$

the **extension** of sheaves on $U$ to sheaves on $X$.

To summarize notation and terminology:

| Terminology         | | Notation                                                 | | Definition                                              |
| ------------------- |-| -------------------------------------------------------- |-| ------------------------------------------------------- |
| morphism of sites   | | $j_{U \to X} : X \to U $                                 | |                                                         |
| underlying functor  | | $j^t_{U \to X} : (Y_{S_X}/U) \to S_X $                   | |                                                         |
| sheaf restriction   | | $(j_{U \to X})_* : Sh(X) \to Sh(U)$                      | | direct image                                            |
| sheaf extension     | | $j^{\ddagger}_{U \to X} : Sh(U) \to Sh(X)$               | | right adjoint to direct image                           |
| sheaf inverse image | | $\overline{(j^t)^{\dagger}_{U \to X}} : Sh(U) \to Sh(X)$ | | left adjoint to direct image followed by sheafification |

## Remarks

Notice the difference to the [[inverse image]] operation
$$
  j^{-1}_{U \to X} : Sh(U) \to Sh(X)
  \,.
$$

## References
 {#References}

For instance section 17.6 of

* Kashiwara, Schapira, _[[Categories and Sheaves]]_

* [[Stacks Project]], [Tag 00VC](http://stacks.math.columbia.edu/tag/00VC) and [Tag 00XF](http://stacks.math.columbia.edu/tag/00XF).