Mixing
Is the number of irreducible unitary representations of a finitely-generated group of a given dimension finite?
Clash Royale CLAN TAG#URR8PPP
up vote
-3
down vote
favorite
Let $Gamma$ be a finitely-generated group. Fix an integer $ngeq 1$. Is it possible that $Gamma$ has infinitely many pairwise non-isomorphic irreducible unitary representations of dimension $n$?
I tried to show that it is not possible:
We can fix a generating set $S$, and think of a unitary representation $rho$ as a function $Srightarrow U(n)$. Since $U(n)^S$ is compact, it would suffice to show that every element of $U(n)^S$ which corresponds to an irreducible representation has an open neighborhood $U$, such that all unitary representations that are mapped into $U$ are isomorphic to $rho$. I don't know if that's true.
representation-theory finitely-generated
asked Jul 22 at 15:03
Scott Harris
1
add a comment |Â
up vote
-3
down vote
favorite
Let $Gamma$ be a finitely-generated group. Fix an integer $ngeq 1$. Is it possible that $Gamma$ has infinitely many pairwise non-isomorphic irreducible unitary representations of dimension $n$?
I tried to show that it is not possible:
We can fix a generating set $S$, and think of a unitary representation $rho$ as a function $Srightarrow U(n)$. Since $U(n)^S$ is compact, it would suffice to show that every element of $U(n)^S$ which corresponds to an irreducible representation has an open neighborhood $U$, such that all unitary representations that are mapped into $U$ are isomorphic to $rho$. I don't know if that's true.
representation-theory finitely-generated
asked Jul 22 at 15:03
Scott Harris
1
2
Doesn't $Gamma=Bbb Z$, $n=1$ give a counterexample? I'm pretty sure the unit circel in the complex plane is infinite.
– David C. Ullrich
Jul 22 at 15:05
add a comment |Â
up vote
-3
down vote
favorite
up vote
-3
down vote
favorite
Let $Gamma$ be a finitely-generated group. Fix an integer $ngeq 1$. Is it possible that $Gamma$ has infinitely many pairwise non-isomorphic irreducible unitary representations of dimension $n$?
I tried to show that it is not possible:
We can fix a generating set $S$, and think of a unitary representation $rho$ as a function $Srightarrow U(n)$. Since $U(n)^S$ is compact, it would suffice to show that every element of $U(n)^S$ which corresponds to an irreducible representation has an open neighborhood $U$, such that all unitary representations that are mapped into $U$ are isomorphic to $rho$. I don't know if that's true.
representation-theory finitely-generated
asked Jul 22 at 15:03
Scott Harris
1
Let $Gamma$ be a finitely-generated group. Fix an integer $ngeq 1$. Is it possible that $Gamma$ has infinitely many pairwise non-isomorphic irreducible unitary representations of dimension $n$?
I tried to show that it is not possible:
We can fix a generating set $S$, and think of a unitary representation $rho$ as a function $Srightarrow U(n)$. Since $U(n)^S$ is compact, it would suffice to show that every element of $U(n)^S$ which corresponds to an irreducible representation has an open neighborhood $U$, such that all unitary representations that are mapped into $U$ are isomorphic to $rho$. I don't know if that's true.
representation-theory finitely-generated
asked Jul 22 at 15:03
Scott Harris
1
asked Jul 22 at 15:03
Scott Harris
1
asked Jul 22 at 15:03
Scott Harris
1
asked Jul 22 at 15:03
Scott Harris
1
1
2
Doesn't $Gamma=Bbb Z$, $n=1$ give a counterexample? I'm pretty sure the unit circel in the complex plane is infinite.
– David C. Ullrich
Jul 22 at 15:05
add a comment |Â
2
Doesn't $Gamma=Bbb Z$, $n=1$ give a counterexample? I'm pretty sure the unit circel in the complex plane is infinite.
– David C. Ullrich
Jul 22 at 15:05
2
2
Doesn't $Gamma=Bbb Z$, $n=1$ give a counterexample? I'm pretty sure the unit circel in the complex plane is infinite.
– David C. Ullrich
Jul 22 at 15:05
Doesn't $Gamma=Bbb Z$, $n=1$ give a counterexample? I'm pretty sure the unit circel in the complex plane is infinite.
– David C. Ullrich
Jul 22 at 15:05
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2859485%2fis-the-number-of-irreducible-unitary-representations-of-a-finitely-generated-gro%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
2
Doesn't $Gamma=Bbb Z$, $n=1$ give a counterexample? I'm pretty sure the unit circel in the complex plane is infinite.
– David C. Ullrich
Jul 22 at 15:05