Charater of induced representation
$begingroup$
Suppose we have an induced representation of $theta: H to GL(W)$, we define the space
$$
V := mathbb{C}[G] otimes_{mathbb{C}[H]} W,
$$
$V$ has as a basis ${e_r otimes_{mathbb{C}[H]} w}_{r in mathcal{R}, w in B}$ with $mathcal{R}$ a complete set of representatives of $G$ and $B$ a basis for $W$. The representations is defined as $g cdot (e_r otimes_{mathbb{C}[H]} w) = (e_{gr} otimes_{mathbb{C}[H]} w)$.
Then, in order to compute the character, we define the subspaces:
$$
V_g = e_g otimes_{mathbb{C}[H]} W subseteq V
$$
(notice that this only depends on the coset).
My question is: how does it follow that this is a subspace? In order to show it, we need to check it is closed under addition and scalar multiplication, for example is
$$
e_g otimes_{mathbb{C}[H]} w + e_g otimes_{mathbb{C}[H]} w' in V_g?
$$
And scalar multiplication as well. I realized that I don't even know how these two (scalar multiplication and addition) are defined in this highly complex $V$ (I don't even know over which field we should use scalar multiplication @_@, $mathbb{C}$? $mathbb{C}[H]$?), since everything is constructed so abstractly and indirectly. Therefore I have no idea how to check that $V_g$ is indeed closed under addition and scalar multiplication. Can someone enlight me on this case?
vector-spaces representation-theory tensor-products characters
asked Nov 27 '18 at 14:45
SigurdSigurd
480211
$endgroup$
add a comment |
$begingroup$
Suppose we have an induced representation of $theta: H to GL(W)$, we define the space
$$
V := mathbb{C}[G] otimes_{mathbb{C}[H]} W,
$$
$V$ has as a basis ${e_r otimes_{mathbb{C}[H]} w}_{r in mathcal{R}, w in B}$ with $mathcal{R}$ a complete set of representatives of $G$ and $B$ a basis for $W$. The representations is defined as $g cdot (e_r otimes_{mathbb{C}[H]} w) = (e_{gr} otimes_{mathbb{C}[H]} w)$.
Then, in order to compute the character, we define the subspaces:
$$
V_g = e_g otimes_{mathbb{C}[H]} W subseteq V
$$
(notice that this only depends on the coset).
My question is: how does it follow that this is a subspace? In order to show it, we need to check it is closed under addition and scalar multiplication, for example is
$$
e_g otimes_{mathbb{C}[H]} w + e_g otimes_{mathbb{C}[H]} w' in V_g?
$$
And scalar multiplication as well. I realized that I don't even know how these two (scalar multiplication and addition) are defined in this highly complex $V$ (I don't even know over which field we should use scalar multiplication @_@, $mathbb{C}$? $mathbb{C}[H]$?), since everything is constructed so abstractly and indirectly. Therefore I have no idea how to check that $V_g$ is indeed closed under addition and scalar multiplication. Can someone enlight me on this case?
vector-spaces representation-theory tensor-products characters
asked Nov 27 '18 at 14:45
SigurdSigurd
480211
$endgroup$
add a comment |
$begingroup$
Suppose we have an induced representation of $theta: H to GL(W)$, we define the space
$$
V := mathbb{C}[G] otimes_{mathbb{C}[H]} W,
$$
$V$ has as a basis ${e_r otimes_{mathbb{C}[H]} w}_{r in mathcal{R}, w in B}$ with $mathcal{R}$ a complete set of representatives of $G$ and $B$ a basis for $W$. The representations is defined as $g cdot (e_r otimes_{mathbb{C}[H]} w) = (e_{gr} otimes_{mathbb{C}[H]} w)$.
Then, in order to compute the character, we define the subspaces:
$$
V_g = e_g otimes_{mathbb{C}[H]} W subseteq V
$$
(notice that this only depends on the coset).
My question is: how does it follow that this is a subspace? In order to show it, we need to check it is closed under addition and scalar multiplication, for example is
$$
e_g otimes_{mathbb{C}[H]} w + e_g otimes_{mathbb{C}[H]} w' in V_g?
$$
And scalar multiplication as well. I realized that I don't even know how these two (scalar multiplication and addition) are defined in this highly complex $V$ (I don't even know over which field we should use scalar multiplication @_@, $mathbb{C}$? $mathbb{C}[H]$?), since everything is constructed so abstractly and indirectly. Therefore I have no idea how to check that $V_g$ is indeed closed under addition and scalar multiplication. Can someone enlight me on this case?
vector-spaces representation-theory tensor-products characters
asked Nov 27 '18 at 14:45
SigurdSigurd
480211
$endgroup$
Suppose we have an induced representation of $theta: H to GL(W)$, we define the space
$$
V := mathbb{C}[G] otimes_{mathbb{C}[H]} W,
$$
$V$ has as a basis ${e_r otimes_{mathbb{C}[H]} w}_{r in mathcal{R}, w in B}$ with $mathcal{R}$ a complete set of representatives of $G$ and $B$ a basis for $W$. The representations is defined as $g cdot (e_r otimes_{mathbb{C}[H]} w) = (e_{gr} otimes_{mathbb{C}[H]} w)$.
Then, in order to compute the character, we define the subspaces:
$$
V_g = e_g otimes_{mathbb{C}[H]} W subseteq V
$$
(notice that this only depends on the coset).
My question is: how does it follow that this is a subspace? In order to show it, we need to check it is closed under addition and scalar multiplication, for example is
$$
e_g otimes_{mathbb{C}[H]} w + e_g otimes_{mathbb{C}[H]} w' in V_g?
$$
And scalar multiplication as well. I realized that I don't even know how these two (scalar multiplication and addition) are defined in this highly complex $V$ (I don't even know over which field we should use scalar multiplication @_@, $mathbb{C}$? $mathbb{C}[H]$?), since everything is constructed so abstractly and indirectly. Therefore I have no idea how to check that $V_g$ is indeed closed under addition and scalar multiplication. Can someone enlight me on this case?
vector-spaces representation-theory tensor-products characters
vector-spaces representation-theory tensor-products characters
asked Nov 27 '18 at 14:45
SigurdSigurd
480211
asked Nov 27 '18 at 14:45
SigurdSigurd
480211
asked Nov 27 '18 at 14:45
SigurdSigurd
480211
asked Nov 27 '18 at 14:45
SigurdSigurd
480211
asked Nov 27 '18 at 14:45
SigurdSigurd
480211
480211
add a comment |
add a comment |
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