Tensor product of two irreducible representations is not irreducible [closed]
$begingroup$
I have seen that if $rho: G longrightarrow text{GL}_{mathbb{C}}(V)$ and $alpha: G longrightarrow text{GL}_{mathbb{C}}(W)$ are two irreducible representations of a finite grup $G$, then its tensor product representation $rho otimes alpha: G longrightarrow text{GL}_{mathbb{C}}(V otimes W)$ is usually not irreducible.
Can anyone tell me an example?
abstract-algebra group-theory finite-groups representation-theory examples-counterexamples
edited Dec 11 '18 at 16:07
Batominovski
1
asked Dec 10 '18 at 14:46
idriskameniidriskameni
585318
$endgroup$
closed as off-topic by Derek Holt, user26857, Saad, Eevee Trainer, amWhy Dec 23 '18 at 9:58
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Derek Holt, user26857, Saad, Eevee Trainer, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.
add a comment |
$begingroup$
I have seen that if $rho: G longrightarrow text{GL}_{mathbb{C}}(V)$ and $alpha: G longrightarrow text{GL}_{mathbb{C}}(W)$ are two irreducible representations of a finite grup $G$, then its tensor product representation $rho otimes alpha: G longrightarrow text{GL}_{mathbb{C}}(V otimes W)$ is usually not irreducible.
Can anyone tell me an example?
abstract-algebra group-theory finite-groups representation-theory examples-counterexamples
edited Dec 11 '18 at 16:07
Batominovski
1
asked Dec 10 '18 at 14:46
idriskameniidriskameni
585318
$endgroup$
closed as off-topic by Derek Holt, user26857, Saad, Eevee Trainer, amWhy Dec 23 '18 at 9:58
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Derek Holt, user26857, Saad, Eevee Trainer, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
Since it is usually not irreducible, you might perhaps have tried a few examples yourself before asking.
$endgroup$
– Derek Holt
Dec 10 '18 at 15:06
add a comment |
$begingroup$
I have seen that if $rho: G longrightarrow text{GL}_{mathbb{C}}(V)$ and $alpha: G longrightarrow text{GL}_{mathbb{C}}(W)$ are two irreducible representations of a finite grup $G$, then its tensor product representation $rho otimes alpha: G longrightarrow text{GL}_{mathbb{C}}(V otimes W)$ is usually not irreducible.
Can anyone tell me an example?
abstract-algebra group-theory finite-groups representation-theory examples-counterexamples
edited Dec 11 '18 at 16:07
Batominovski
1
asked Dec 10 '18 at 14:46
idriskameniidriskameni
585318
$endgroup$
I have seen that if $rho: G longrightarrow text{GL}_{mathbb{C}}(V)$ and $alpha: G longrightarrow text{GL}_{mathbb{C}}(W)$ are two irreducible representations of a finite grup $G$, then its tensor product representation $rho otimes alpha: G longrightarrow text{GL}_{mathbb{C}}(V otimes W)$ is usually not irreducible.
Can anyone tell me an example?
abstract-algebra group-theory finite-groups representation-theory examples-counterexamples
abstract-algebra group-theory finite-groups representation-theory examples-counterexamples
edited Dec 11 '18 at 16:07
Batominovski
1
asked Dec 10 '18 at 14:46
idriskameniidriskameni
585318
edited Dec 11 '18 at 16:07
Batominovski
1
asked Dec 10 '18 at 14:46
idriskameniidriskameni
585318
edited Dec 11 '18 at 16:07
Batominovski
1
edited Dec 11 '18 at 16:07
Batominovski
1
edited Dec 11 '18 at 16:07
Batominovski
1
1
asked Dec 10 '18 at 14:46
idriskameniidriskameni
585318
asked Dec 10 '18 at 14:46
idriskameniidriskameni
585318
asked Dec 10 '18 at 14:46
idriskameniidriskameni
585318
585318
closed as off-topic by Derek Holt, user26857, Saad, Eevee Trainer, amWhy Dec 23 '18 at 9:58
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Derek Holt, user26857, Saad, Eevee Trainer, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.
closed as off-topic by Derek Holt, user26857, Saad, Eevee Trainer, amWhy Dec 23 '18 at 9:58
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please provide additional context, which ideally explains why the question is relevant to you and our community. Some forms of context include: background and motivation, relevant definitions, source, possible strategies, your current progress, why the question is interesting or important, etc." – Derek Holt, user26857, Saad, Eevee Trainer, amWhy
If this question can be reworded to fit the rules in the help center, please edit the question.
2
$begingroup$
Since it is usually not irreducible, you might perhaps have tried a few examples yourself before asking.
$endgroup$
– Derek Holt
Dec 10 '18 at 15:06
add a comment |
2
$begingroup$
Since it is usually not irreducible, you might perhaps have tried a few examples yourself before asking.
$endgroup$
– Derek Holt
Dec 10 '18 at 15:06
2
2
$begingroup$
Since it is usually not irreducible, you might perhaps have tried a few examples yourself before asking.
$endgroup$
– Derek Holt
Dec 10 '18 at 15:06
$begingroup$
Since it is usually not irreducible, you might perhaps have tried a few examples yourself before asking.
$endgroup$
– Derek Holt
Dec 10 '18 at 15:06
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Take the standard representation $rho$ of $S_3$ on $V={(z_1,z_2,z_3)inmathbb{C}^3,|,z_1+z_2+z_3=0}$:$$rho(sigma)(z_1,z_2,z_3)=(z_{sigma^{-1}(1)},z_{sigma^{-1}(2)},z_{sigma^{-1}(3)}).$$Then $rho$ is irreducible, but $rhootimesrho$ is not ($S_3$ has no irreducible representation whose dimension is greater than $2$).
answered Dec 10 '18 at 14:53
José Carlos SantosJosé Carlos Santos
159k22126231
$endgroup$
$begingroup$
And do you know any example with $rho neq alpha$?
$endgroup$
– idriskameni
Dec 10 '18 at 14:54
2
$begingroup$
Sure. The group $S_4$ has two and (up to isomorphism) only two irreducible representations of dimension $3$, $rho$ and $rho^star$, and no irreducible representation of dimension $9$. Therefore, $rhootimesrho^star$ is reducible.
$endgroup$
– José Carlos Santos
Dec 10 '18 at 14:59
4
$begingroup$
Come @idriskameni, if you take an irreducible of highest degree and tensor it with some other irreducible of dimension greater than $1$ that's not going to be irreducible, is it?
$endgroup$
– ancientmathematician
Dec 10 '18 at 15:00
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Take the standard representation $rho$ of $S_3$ on $V={(z_1,z_2,z_3)inmathbb{C}^3,|,z_1+z_2+z_3=0}$:$$rho(sigma)(z_1,z_2,z_3)=(z_{sigma^{-1}(1)},z_{sigma^{-1}(2)},z_{sigma^{-1}(3)}).$$Then $rho$ is irreducible, but $rhootimesrho$ is not ($S_3$ has no irreducible representation whose dimension is greater than $2$).
answered Dec 10 '18 at 14:53
José Carlos SantosJosé Carlos Santos
159k22126231
$endgroup$
$begingroup$
And do you know any example with $rho neq alpha$?
$endgroup$
– idriskameni
Dec 10 '18 at 14:54
2
$begingroup$
Sure. The group $S_4$ has two and (up to isomorphism) only two irreducible representations of dimension $3$, $rho$ and $rho^star$, and no irreducible representation of dimension $9$. Therefore, $rhootimesrho^star$ is reducible.
$endgroup$
– José Carlos Santos
Dec 10 '18 at 14:59
4
$begingroup$
Come @idriskameni, if you take an irreducible of highest degree and tensor it with some other irreducible of dimension greater than $1$ that's not going to be irreducible, is it?
$endgroup$
– ancientmathematician
Dec 10 '18 at 15:00
add a comment |
$begingroup$
Take the standard representation $rho$ of $S_3$ on $V={(z_1,z_2,z_3)inmathbb{C}^3,|,z_1+z_2+z_3=0}$:$$rho(sigma)(z_1,z_2,z_3)=(z_{sigma^{-1}(1)},z_{sigma^{-1}(2)},z_{sigma^{-1}(3)}).$$Then $rho$ is irreducible, but $rhootimesrho$ is not ($S_3$ has no irreducible representation whose dimension is greater than $2$).
answered Dec 10 '18 at 14:53
José Carlos SantosJosé Carlos Santos
159k22126231
$endgroup$
$begingroup$
And do you know any example with $rho neq alpha$?
$endgroup$
– idriskameni
Dec 10 '18 at 14:54
2
$begingroup$
Sure. The group $S_4$ has two and (up to isomorphism) only two irreducible representations of dimension $3$, $rho$ and $rho^star$, and no irreducible representation of dimension $9$. Therefore, $rhootimesrho^star$ is reducible.
$endgroup$
– José Carlos Santos
Dec 10 '18 at 14:59
4
$begingroup$
Come @idriskameni, if you take an irreducible of highest degree and tensor it with some other irreducible of dimension greater than $1$ that's not going to be irreducible, is it?
$endgroup$
– ancientmathematician
Dec 10 '18 at 15:00
add a comment |
$begingroup$
Take the standard representation $rho$ of $S_3$ on $V={(z_1,z_2,z_3)inmathbb{C}^3,|,z_1+z_2+z_3=0}$:$$rho(sigma)(z_1,z_2,z_3)=(z_{sigma^{-1}(1)},z_{sigma^{-1}(2)},z_{sigma^{-1}(3)}).$$Then $rho$ is irreducible, but $rhootimesrho$ is not ($S_3$ has no irreducible representation whose dimension is greater than $2$).
answered Dec 10 '18 at 14:53
José Carlos SantosJosé Carlos Santos
159k22126231
$endgroup$
Take the standard representation $rho$ of $S_3$ on $V={(z_1,z_2,z_3)inmathbb{C}^3,|,z_1+z_2+z_3=0}$:$$rho(sigma)(z_1,z_2,z_3)=(z_{sigma^{-1}(1)},z_{sigma^{-1}(2)},z_{sigma^{-1}(3)}).$$Then $rho$ is irreducible, but $rhootimesrho$ is not ($S_3$ has no irreducible representation whose dimension is greater than $2$).
answered Dec 10 '18 at 14:53
José Carlos SantosJosé Carlos Santos
159k22126231
edited Dec 10 '18 at 14:55
answered Dec 10 '18 at 14:53
José Carlos SantosJosé Carlos Santos
159k22126231
answered Dec 10 '18 at 14:53
José Carlos SantosJosé Carlos Santos
159k22126231
answered Dec 10 '18 at 14:53
José Carlos SantosJosé Carlos Santos
159k22126231
159k22126231
$begingroup$
And do you know any example with $rho neq alpha$?
$endgroup$
– idriskameni
Dec 10 '18 at 14:54
2
$begingroup$
Sure. The group $S_4$ has two and (up to isomorphism) only two irreducible representations of dimension $3$, $rho$ and $rho^star$, and no irreducible representation of dimension $9$. Therefore, $rhootimesrho^star$ is reducible.
$endgroup$
– José Carlos Santos
Dec 10 '18 at 14:59
4
$begingroup$
Come @idriskameni, if you take an irreducible of highest degree and tensor it with some other irreducible of dimension greater than $1$ that's not going to be irreducible, is it?
$endgroup$
– ancientmathematician
Dec 10 '18 at 15:00
add a comment |
$begingroup$
And do you know any example with $rho neq alpha$?
$endgroup$
– idriskameni
Dec 10 '18 at 14:54
2
$begingroup$
Sure. The group $S_4$ has two and (up to isomorphism) only two irreducible representations of dimension $3$, $rho$ and $rho^star$, and no irreducible representation of dimension $9$. Therefore, $rhootimesrho^star$ is reducible.
$endgroup$
– José Carlos Santos
Dec 10 '18 at 14:59
4
$begingroup$
Come @idriskameni, if you take an irreducible of highest degree and tensor it with some other irreducible of dimension greater than $1$ that's not going to be irreducible, is it?
$endgroup$
– ancientmathematician
Dec 10 '18 at 15:00
$begingroup$
And do you know any example with $rho neq alpha$?
$endgroup$
– idriskameni
Dec 10 '18 at 14:54
$begingroup$
And do you know any example with $rho neq alpha$?
$endgroup$
– idriskameni
Dec 10 '18 at 14:54
2
2
$begingroup$
Sure. The group $S_4$ has two and (up to isomorphism) only two irreducible representations of dimension $3$, $rho$ and $rho^star$, and no irreducible representation of dimension $9$. Therefore, $rhootimesrho^star$ is reducible.
$endgroup$
– José Carlos Santos
Dec 10 '18 at 14:59
$begingroup$
Sure. The group $S_4$ has two and (up to isomorphism) only two irreducible representations of dimension $3$, $rho$ and $rho^star$, and no irreducible representation of dimension $9$. Therefore, $rhootimesrho^star$ is reducible.
$endgroup$
– José Carlos Santos
Dec 10 '18 at 14:59
4
4
$begingroup$
Come @idriskameni, if you take an irreducible of highest degree and tensor it with some other irreducible of dimension greater than $1$ that's not going to be irreducible, is it?
$endgroup$
– ancientmathematician
Dec 10 '18 at 15:00
$begingroup$
Come @idriskameni, if you take an irreducible of highest degree and tensor it with some other irreducible of dimension greater than $1$ that's not going to be irreducible, is it?
$endgroup$
– ancientmathematician
Dec 10 '18 at 15:00
add a comment |
2
$begingroup$
Since it is usually not irreducible, you might perhaps have tried a few examples yourself before asking.
$endgroup$
– Derek Holt
Dec 10 '18 at 15:06