dimension of the morphisme of two characters
$begingroup$
I am new to character theory and I have the following question.
In the dual group of a groupe $G$ we can define an inner product. This inner product is equal to :
$$dim Hom_G(a,b) $$
The problem is that I don’t understand what it means. I know that the dimension of a representation $p : G to GL(V)$ is equal to $dim(V)$. So here in the dual group very element has dimension $1$ (since $V = mathbb{C}^*$).
Yet, here what does it mean to take the dimension of a morphisme between two representations ?
Thanks you !
group-theory representation-theory
asked Dec 3 '18 at 0:25
JebfiffkkfnfolzbdJebfiffkkfnfolzbd
642
$endgroup$
add a comment |
$begingroup$
I am new to character theory and I have the following question.
In the dual group of a groupe $G$ we can define an inner product. This inner product is equal to :
$$dim Hom_G(a,b) $$
The problem is that I don’t understand what it means. I know that the dimension of a representation $p : G to GL(V)$ is equal to $dim(V)$. So here in the dual group very element has dimension $1$ (since $V = mathbb{C}^*$).
Yet, here what does it mean to take the dimension of a morphisme between two representations ?
Thanks you !
group-theory representation-theory
asked Dec 3 '18 at 0:25
JebfiffkkfnfolzbdJebfiffkkfnfolzbd
642
$endgroup$
$begingroup$
Do you know the projections formulas (also there) ? $fin Hom_G(a,b)$ is "project $a$ on one of its subrepresentation then embed as a subrepresentation of $b$".
$endgroup$
– reuns
Dec 3 '18 at 1:28
$begingroup$
$operatorname{dim}operatorname{Hom}_{G}(a,b)$ is the vector space dimension of the $mathbb{C}$ vector space of linear maps from the representation $a$ to the representation $b$ that respect the action of $G$. A linear map between two representations is a map of the modules given by the representations.
$endgroup$
– Adam Higgins
Dec 3 '18 at 21:09
add a comment |
$begingroup$
I am new to character theory and I have the following question.
In the dual group of a groupe $G$ we can define an inner product. This inner product is equal to :
$$dim Hom_G(a,b) $$
The problem is that I don’t understand what it means. I know that the dimension of a representation $p : G to GL(V)$ is equal to $dim(V)$. So here in the dual group very element has dimension $1$ (since $V = mathbb{C}^*$).
Yet, here what does it mean to take the dimension of a morphisme between two representations ?
Thanks you !
group-theory representation-theory
asked Dec 3 '18 at 0:25
JebfiffkkfnfolzbdJebfiffkkfnfolzbd
642
$endgroup$
I am new to character theory and I have the following question.
In the dual group of a groupe $G$ we can define an inner product. This inner product is equal to :
$$dim Hom_G(a,b) $$
The problem is that I don’t understand what it means. I know that the dimension of a representation $p : G to GL(V)$ is equal to $dim(V)$. So here in the dual group very element has dimension $1$ (since $V = mathbb{C}^*$).
Yet, here what does it mean to take the dimension of a morphisme between two representations ?
Thanks you !
group-theory representation-theory
group-theory representation-theory
asked Dec 3 '18 at 0:25
JebfiffkkfnfolzbdJebfiffkkfnfolzbd
642
asked Dec 3 '18 at 0:25
JebfiffkkfnfolzbdJebfiffkkfnfolzbd
642
asked Dec 3 '18 at 0:25
JebfiffkkfnfolzbdJebfiffkkfnfolzbd
642
asked Dec 3 '18 at 0:25
JebfiffkkfnfolzbdJebfiffkkfnfolzbd
642
asked Dec 3 '18 at 0:25
JebfiffkkfnfolzbdJebfiffkkfnfolzbd
642
642
$begingroup$
Do you know the projections formulas (also there) ? $fin Hom_G(a,b)$ is "project $a$ on one of its subrepresentation then embed as a subrepresentation of $b$".
$endgroup$
– reuns
Dec 3 '18 at 1:28
$begingroup$
$operatorname{dim}operatorname{Hom}_{G}(a,b)$ is the vector space dimension of the $mathbb{C}$ vector space of linear maps from the representation $a$ to the representation $b$ that respect the action of $G$. A linear map between two representations is a map of the modules given by the representations.
$endgroup$
– Adam Higgins
Dec 3 '18 at 21:09
add a comment |
$begingroup$
Do you know the projections formulas (also there) ? $fin Hom_G(a,b)$ is "project $a$ on one of its subrepresentation then embed as a subrepresentation of $b$".
$endgroup$
– reuns
Dec 3 '18 at 1:28
$begingroup$
$operatorname{dim}operatorname{Hom}_{G}(a,b)$ is the vector space dimension of the $mathbb{C}$ vector space of linear maps from the representation $a$ to the representation $b$ that respect the action of $G$. A linear map between two representations is a map of the modules given by the representations.
$endgroup$
– Adam Higgins
Dec 3 '18 at 21:09
$begingroup$
Do you know the projections formulas (also there) ? $fin Hom_G(a,b)$ is "project $a$ on one of its subrepresentation then embed as a subrepresentation of $b$".
$endgroup$
– reuns
Dec 3 '18 at 1:28
$begingroup$
Do you know the projections formulas (also there) ? $fin Hom_G(a,b)$ is "project $a$ on one of its subrepresentation then embed as a subrepresentation of $b$".
$endgroup$
– reuns
Dec 3 '18 at 1:28
$begingroup$
$operatorname{dim}operatorname{Hom}_{G}(a,b)$ is the vector space dimension of the $mathbb{C}$ vector space of linear maps from the representation $a$ to the representation $b$ that respect the action of $G$. A linear map between two representations is a map of the modules given by the representations.
$endgroup$
– Adam Higgins
Dec 3 '18 at 21:09
$begingroup$
$operatorname{dim}operatorname{Hom}_{G}(a,b)$ is the vector space dimension of the $mathbb{C}$ vector space of linear maps from the representation $a$ to the representation $b$ that respect the action of $G$. A linear map between two representations is a map of the modules given by the representations.
$endgroup$
– Adam Higgins
Dec 3 '18 at 21:09
add a comment |
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$begingroup$
Do you know the projections formulas (also there) ? $fin Hom_G(a,b)$ is "project $a$ on one of its subrepresentation then embed as a subrepresentation of $b$".
$endgroup$
– reuns
Dec 3 '18 at 1:28
$begingroup$
$operatorname{dim}operatorname{Hom}_{G}(a,b)$ is the vector space dimension of the $mathbb{C}$ vector space of linear maps from the representation $a$ to the representation $b$ that respect the action of $G$. A linear map between two representations is a map of the modules given by the representations.
$endgroup$
– Adam Higgins
Dec 3 '18 at 21:09