What does it mean if a representation “factors through a group”?

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I encountered this in this paper:

Suppose $rho$ factors through a finite cyclic group $G = operatornameGal(F/K) simeq C_n$.

I know a little bit about representation theory, but unfortunately, I have never heard of factoring through a group before, so I do not know what exactly this is supposed to mean. Could you please explain that to me? Thank you!

abstract-algebra group-theory notation representation-theory galois-theory

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edited Jul 23 at 13:51

asked Jul 23 at 13:41

Diglett

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favorite

I encountered this in this paper:

Suppose $rho$ factors through a finite cyclic group $G = operatornameGal(F/K) simeq C_n$.

I know a little bit about representation theory, but unfortunately, I have never heard of factoring through a group before, so I do not know what exactly this is supposed to mean. Could you please explain that to me? Thank you!

abstract-algebra group-theory notation representation-theory galois-theory

share|cite|improve this question

edited Jul 23 at 13:51

asked Jul 23 at 13:41

Diglett

502311

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

I encountered this in this paper:

Suppose $rho$ factors through a finite cyclic group $G = operatornameGal(F/K) simeq C_n$.

I know a little bit about representation theory, but unfortunately, I have never heard of factoring through a group before, so I do not know what exactly this is supposed to mean. Could you please explain that to me? Thank you!

abstract-algebra group-theory notation representation-theory galois-theory

share|cite|improve this question

edited Jul 23 at 13:51

asked Jul 23 at 13:41

Diglett

502311

I encountered this in this paper:

Suppose $rho$ factors through a finite cyclic group $G = operatornameGal(F/K) simeq C_n$.

I know a little bit about representation theory, but unfortunately, I have never heard of factoring through a group before, so I do not know what exactly this is supposed to mean. Could you please explain that to me? Thank you!

abstract-algebra group-theory notation representation-theory galois-theory

share|cite|improve this question

edited Jul 23 at 13:51

asked Jul 23 at 13:41

Diglett

502311

share|cite|improve this question

share|cite|improve this question

edited Jul 23 at 13:51

edited Jul 23 at 13:51

edited Jul 23 at 13:51

asked Jul 23 at 13:41

Diglett

502311

asked Jul 23 at 13:41

Diglett

502311

asked Jul 23 at 13:41

Diglett

502311

502311

add a comment | 

add a comment | 

2 Answers
2

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oldest

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up vote
2
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accepted

Generally speaking, you can see (or even define) a representation $rho$ of a group $G$ as a group homomorphism $Gto GL(V)$ for some vector space $V$. Then saying that the representation factors through a group $H$ means that this group homomorphism factors through $H$, in the sense that there exist group homomorphisms $varphi: Gto H$ and $psi: H to GL(V)$ such that $rho=psicirc varphi$.

If you prefer to see a representation as a certain function $Gtimes Vto V:(g,v) mapsto gast_rho v$ satisfying certain conditions, then the above tells you that $gast_rho v=varphi (g)ast_psi v$ for all $g,v$.

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edited Jul 23 at 14:07

answered Jul 23 at 14:00

Arnaud D.

14.7k52141

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1
down vote

The reason for this "factoring" terminology is that homomorphisms between groups behave, to a certain extent, like multiplication: there is a binary operation called "composition"; and this operation satisfies the associative law. For example, we teach our students to factor functions as compositions in order to apply the chain rule: $f(x)=cos(x^2)$ factors as $f=g circ h$ where $h(x)=x^2$ and $g(x)=cos(x)$.

However, one way this differs from multiplication (besides the failure of commutativity) is that the domains and ranges need not all be the same. The point of "factoring through $G$" is that one can factor the representation so that the group in the middle is $G$. Denoting your given group representation as $rho : K to Gamma$ (where $Gamma$ might be the general linear group of some vector space, or a Lie group, or any other kind of group), this means that there are two group homomorphisms
$$K xrightarrowsigma G xrightarrowtau Gamma
$$
such that $rho = tau circ sigma$.

share|cite|improve this answer

edited Jul 23 at 14:17

answered Jul 23 at 14:10

Lee Mosher

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2 Answers
2

active

oldest

votes

2 Answers
2

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
2
down vote



accepted

Generally speaking, you can see (or even define) a representation $rho$ of a group $G$ as a group homomorphism $Gto GL(V)$ for some vector space $V$. Then saying that the representation factors through a group $H$ means that this group homomorphism factors through $H$, in the sense that there exist group homomorphisms $varphi: Gto H$ and $psi: H to GL(V)$ such that $rho=psicirc varphi$.

If you prefer to see a representation as a certain function $Gtimes Vto V:(g,v) mapsto gast_rho v$ satisfying certain conditions, then the above tells you that $gast_rho v=varphi (g)ast_psi v$ for all $g,v$.

share|cite|improve this answer

edited Jul 23 at 14:07

answered Jul 23 at 14:00

Arnaud D.

14.7k52141

add a comment | 

up vote
2
down vote



accepted

Generally speaking, you can see (or even define) a representation $rho$ of a group $G$ as a group homomorphism $Gto GL(V)$ for some vector space $V$. Then saying that the representation factors through a group $H$ means that this group homomorphism factors through $H$, in the sense that there exist group homomorphisms $varphi: Gto H$ and $psi: H to GL(V)$ such that $rho=psicirc varphi$.

If you prefer to see a representation as a certain function $Gtimes Vto V:(g,v) mapsto gast_rho v$ satisfying certain conditions, then the above tells you that $gast_rho v=varphi (g)ast_psi v$ for all $g,v$.

share|cite|improve this answer

edited Jul 23 at 14:07

answered Jul 23 at 14:00

Arnaud D.

14.7k52141

add a comment | 

up vote
2
down vote



accepted

up vote
2
down vote



accepted

Generally speaking, you can see (or even define) a representation $rho$ of a group $G$ as a group homomorphism $Gto GL(V)$ for some vector space $V$. Then saying that the representation factors through a group $H$ means that this group homomorphism factors through $H$, in the sense that there exist group homomorphisms $varphi: Gto H$ and $psi: H to GL(V)$ such that $rho=psicirc varphi$.

If you prefer to see a representation as a certain function $Gtimes Vto V:(g,v) mapsto gast_rho v$ satisfying certain conditions, then the above tells you that $gast_rho v=varphi (g)ast_psi v$ for all $g,v$.

share|cite|improve this answer

edited Jul 23 at 14:07

answered Jul 23 at 14:00

Arnaud D.

14.7k52141

Generally speaking, you can see (or even define) a representation $rho$ of a group $G$ as a group homomorphism $Gto GL(V)$ for some vector space $V$. Then saying that the representation factors through a group $H$ means that this group homomorphism factors through $H$, in the sense that there exist group homomorphisms $varphi: Gto H$ and $psi: H to GL(V)$ such that $rho=psicirc varphi$.

If you prefer to see a representation as a certain function $Gtimes Vto V:(g,v) mapsto gast_rho v$ satisfying certain conditions, then the above tells you that $gast_rho v=varphi (g)ast_psi v$ for all $g,v$.

share|cite|improve this answer

edited Jul 23 at 14:07

answered Jul 23 at 14:00

Arnaud D.

14.7k52141

share|cite|improve this answer

share|cite|improve this answer

edited Jul 23 at 14:07

edited Jul 23 at 14:07

edited Jul 23 at 14:07

answered Jul 23 at 14:00

Arnaud D.

14.7k52141

answered Jul 23 at 14:00

Arnaud D.

14.7k52141

answered Jul 23 at 14:00

Arnaud D.

14.7k52141

14.7k52141

add a comment | 

add a comment | 

up vote
1
down vote

The reason for this "factoring" terminology is that homomorphisms between groups behave, to a certain extent, like multiplication: there is a binary operation called "composition"; and this operation satisfies the associative law. For example, we teach our students to factor functions as compositions in order to apply the chain rule: $f(x)=cos(x^2)$ factors as $f=g circ h$ where $h(x)=x^2$ and $g(x)=cos(x)$.

However, one way this differs from multiplication (besides the failure of commutativity) is that the domains and ranges need not all be the same. The point of "factoring through $G$" is that one can factor the representation so that the group in the middle is $G$. Denoting your given group representation as $rho : K to Gamma$ (where $Gamma$ might be the general linear group of some vector space, or a Lie group, or any other kind of group), this means that there are two group homomorphisms
$$K xrightarrowsigma G xrightarrowtau Gamma
$$
such that $rho = tau circ sigma$.

share|cite|improve this answer

edited Jul 23 at 14:17

answered Jul 23 at 14:10

Lee Mosher

45.6k33478

add a comment | 

up vote
1
down vote

The reason for this "factoring" terminology is that homomorphisms between groups behave, to a certain extent, like multiplication: there is a binary operation called "composition"; and this operation satisfies the associative law. For example, we teach our students to factor functions as compositions in order to apply the chain rule: $f(x)=cos(x^2)$ factors as $f=g circ h$ where $h(x)=x^2$ and $g(x)=cos(x)$.

However, one way this differs from multiplication (besides the failure of commutativity) is that the domains and ranges need not all be the same. The point of "factoring through $G$" is that one can factor the representation so that the group in the middle is $G$. Denoting your given group representation as $rho : K to Gamma$ (where $Gamma$ might be the general linear group of some vector space, or a Lie group, or any other kind of group), this means that there are two group homomorphisms
$$K xrightarrowsigma G xrightarrowtau Gamma
$$
such that $rho = tau circ sigma$.

share|cite|improve this answer

edited Jul 23 at 14:17

answered Jul 23 at 14:10

Lee Mosher

45.6k33478

add a comment | 

up vote
1
down vote

up vote
1
down vote

The reason for this "factoring" terminology is that homomorphisms between groups behave, to a certain extent, like multiplication: there is a binary operation called "composition"; and this operation satisfies the associative law. For example, we teach our students to factor functions as compositions in order to apply the chain rule: $f(x)=cos(x^2)$ factors as $f=g circ h$ where $h(x)=x^2$ and $g(x)=cos(x)$.

However, one way this differs from multiplication (besides the failure of commutativity) is that the domains and ranges need not all be the same. The point of "factoring through $G$" is that one can factor the representation so that the group in the middle is $G$. Denoting your given group representation as $rho : K to Gamma$ (where $Gamma$ might be the general linear group of some vector space, or a Lie group, or any other kind of group), this means that there are two group homomorphisms
$$K xrightarrowsigma G xrightarrowtau Gamma
$$
such that $rho = tau circ sigma$.

share|cite|improve this answer

edited Jul 23 at 14:17

answered Jul 23 at 14:10

Lee Mosher

45.6k33478

The reason for this "factoring" terminology is that homomorphisms between groups behave, to a certain extent, like multiplication: there is a binary operation called "composition"; and this operation satisfies the associative law. For example, we teach our students to factor functions as compositions in order to apply the chain rule: $f(x)=cos(x^2)$ factors as $f=g circ h$ where $h(x)=x^2$ and $g(x)=cos(x)$.

However, one way this differs from multiplication (besides the failure of commutativity) is that the domains and ranges need not all be the same. The point of "factoring through $G$" is that one can factor the representation so that the group in the middle is $G$. Denoting your given group representation as $rho : K to Gamma$ (where $Gamma$ might be the general linear group of some vector space, or a Lie group, or any other kind of group), this means that there are two group homomorphisms
$$K xrightarrowsigma G xrightarrowtau Gamma
$$
such that $rho = tau circ sigma$.

share|cite|improve this answer

edited Jul 23 at 14:17

answered Jul 23 at 14:10

Lee Mosher

45.6k33478

share|cite|improve this answer

share|cite|improve this answer

edited Jul 23 at 14:17

edited Jul 23 at 14:17

edited Jul 23 at 14:17

answered Jul 23 at 14:10

Lee Mosher

45.6k33478

answered Jul 23 at 14:10

Lee Mosher

45.6k33478

answered Jul 23 at 14:10

Lee Mosher

45.6k33478

45.6k33478

add a comment | 

add a comment | 

 

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