Induction representation of the center is not irreducible
Suppose $varphi: Z(G) to GL(V)$ is an irreducible representation of the center of a non-abelian group $G$. I want to show that $Ind^G_{Z(G)} varphi$ is not irreducible. Any hints?
So far, I have consider that $Z(G)$ is a normal subgroup. Moreover, the character is given by
$$
chi_{Ind^G_{Z(G)}} varphi(g) = Ind^G_{Z(G)}(g) = frac{1}{|Z(G)|} sum_{x in G} dot{chi}(x^{-1}gx).
$$
representation-theory characters
asked Nov 20 at 17:52
Sigurd
480211
add a comment |
Suppose $varphi: Z(G) to GL(V)$ is an irreducible representation of the center of a non-abelian group $G$. I want to show that $Ind^G_{Z(G)} varphi$ is not irreducible. Any hints?
So far, I have consider that $Z(G)$ is a normal subgroup. Moreover, the character is given by
$$
chi_{Ind^G_{Z(G)}} varphi(g) = Ind^G_{Z(G)}(g) = frac{1}{|Z(G)|} sum_{x in G} dot{chi}(x^{-1}gx).
$$
representation-theory characters
asked Nov 20 at 17:52
Sigurd
480211
It might be easiest to calculate the inner product of this character with itself. First, you should note that $x^{-1}gxin Z(G)$ implies $gin Z(G)$, which should simplify the calculation.
– Tobias Kildetoft
Nov 20 at 18:48
Thanks for the hint, that's quite helpful! I'm working on the computations right now.
– Sigurd
Nov 20 at 19:27
add a comment |
Suppose $varphi: Z(G) to GL(V)$ is an irreducible representation of the center of a non-abelian group $G$. I want to show that $Ind^G_{Z(G)} varphi$ is not irreducible. Any hints?
So far, I have consider that $Z(G)$ is a normal subgroup. Moreover, the character is given by
$$
chi_{Ind^G_{Z(G)}} varphi(g) = Ind^G_{Z(G)}(g) = frac{1}{|Z(G)|} sum_{x in G} dot{chi}(x^{-1}gx).
$$
representation-theory characters
asked Nov 20 at 17:52
Sigurd
480211
Suppose $varphi: Z(G) to GL(V)$ is an irreducible representation of the center of a non-abelian group $G$. I want to show that $Ind^G_{Z(G)} varphi$ is not irreducible. Any hints?
So far, I have consider that $Z(G)$ is a normal subgroup. Moreover, the character is given by
$$
chi_{Ind^G_{Z(G)}} varphi(g) = Ind^G_{Z(G)}(g) = frac{1}{|Z(G)|} sum_{x in G} dot{chi}(x^{-1}gx).
$$
representation-theory characters
representation-theory characters
asked Nov 20 at 17:52
Sigurd
480211
asked Nov 20 at 17:52
Sigurd
480211
asked Nov 20 at 17:52
Sigurd
480211
asked Nov 20 at 17:52
Sigurd
480211
asked Nov 20 at 17:52
Sigurd
480211
480211
It might be easiest to calculate the inner product of this character with itself. First, you should note that $x^{-1}gxin Z(G)$ implies $gin Z(G)$, which should simplify the calculation.
– Tobias Kildetoft
Nov 20 at 18:48
Thanks for the hint, that's quite helpful! I'm working on the computations right now.
– Sigurd
Nov 20 at 19:27
add a comment |
It might be easiest to calculate the inner product of this character with itself. First, you should note that $x^{-1}gxin Z(G)$ implies $gin Z(G)$, which should simplify the calculation.
– Tobias Kildetoft
Nov 20 at 18:48
Thanks for the hint, that's quite helpful! I'm working on the computations right now.
– Sigurd
Nov 20 at 19:27
It might be easiest to calculate the inner product of this character with itself. First, you should note that $x^{-1}gxin Z(G)$ implies $gin Z(G)$, which should simplify the calculation.
– Tobias Kildetoft
Nov 20 at 18:48
It might be easiest to calculate the inner product of this character with itself. First, you should note that $x^{-1}gxin Z(G)$ implies $gin Z(G)$, which should simplify the calculation.
– Tobias Kildetoft
Nov 20 at 18:48
Thanks for the hint, that's quite helpful! I'm working on the computations right now.
– Sigurd
Nov 20 at 19:27
Thanks for the hint, that's quite helpful! I'm working on the computations right now.
– Sigurd
Nov 20 at 19:27
add a comment |
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It might be easiest to calculate the inner product of this character with itself. First, you should note that $x^{-1}gxin Z(G)$ implies $gin Z(G)$, which should simplify the calculation.
– Tobias Kildetoft
Nov 20 at 18:48
Thanks for the hint, that's quite helpful! I'm working on the computations right now.
– Sigurd
Nov 20 at 19:27