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Irreducible representations of $operatornameGL_n(mathbbC)$
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I just started to learn representation theory and I'm interested in the irreducible representations of $operatornameGL_n(mathbbC)$. I looked them up in three different books and apparently, I misunderstood something because for me, they are inconsistent statements.
Stanley (Enumerative Combinatorics II, Theorem A2.4):
The irreducible polynomial representations $varphi^lambda$ of $operatornameGL_n(mathbbC)$ can be indexed by partitions
$lambda$ of length at most $n$.
Bump (Lie Groups, Theorem 38.3):
The irreducible algebraic representations $varphi^lambda$ of $operatornameGL_n(mathbbC)$ are indexed by integer sequences
$lambda_1geqlambda_2geqdotsgeqlambda_n$.
Fulton, Harris (Representation Theory, Proposition 15.47):
Every irreducible complex representation of $operatornameGL_n(mathbbC)$ is isomorphic to $varphi^lambda$
for a unique index $lambda=lambda_1,dots,lambda_n$ with
$lambda_1geqlambda_2geqdotsgeqlambda_n$.
My questions are:
- Is every algebraic irreducible representation indexed by a partition or by an integer sequence with $lambda_1geqlambda_2geqdotsgeqlambda_n$?
- Can every irreducible representation be indexed by an integer partition with $lambda_1geqlambda_2geqdotsgeqlambda_n$ or only the irreducible alegbraic representations?
representation-theory
asked Jul 24 at 7:22
user160919
1218
 |Â
show 4 more comments
up vote
0
down vote
favorite
I just started to learn representation theory and I'm interested in the irreducible representations of $operatornameGL_n(mathbbC)$. I looked them up in three different books and apparently, I misunderstood something because for me, they are inconsistent statements.
Stanley (Enumerative Combinatorics II, Theorem A2.4):
The irreducible polynomial representations $varphi^lambda$ of $operatornameGL_n(mathbbC)$ can be indexed by partitions
$lambda$ of length at most $n$.
Bump (Lie Groups, Theorem 38.3):
The irreducible algebraic representations $varphi^lambda$ of $operatornameGL_n(mathbbC)$ are indexed by integer sequences
$lambda_1geqlambda_2geqdotsgeqlambda_n$.
Fulton, Harris (Representation Theory, Proposition 15.47):
Every irreducible complex representation of $operatornameGL_n(mathbbC)$ is isomorphic to $varphi^lambda$
for a unique index $lambda=lambda_1,dots,lambda_n$ with
$lambda_1geqlambda_2geqdotsgeqlambda_n$.
My questions are:
- Is every algebraic irreducible representation indexed by a partition or by an integer sequence with $lambda_1geqlambda_2geqdotsgeqlambda_n$?
- Can every irreducible representation be indexed by an integer partition with $lambda_1geqlambda_2geqdotsgeqlambda_n$ or only the irreducible alegbraic representations?
representation-theory
asked Jul 24 at 7:22
user160919
1218
What values do Fulton-Harris allow for the $lambda_i$?
– Tobias Kildetoft
Jul 24 at 7:29
I just looked up the result in Stanley, and you misquoted it slightly. It should be "polynomial" instead of "algebraic".
– Tobias Kildetoft
Jul 24 at 7:34
@TobiasKildetoft It says that some of the $lambda_i$ can be negative. So, they're integers.
– user160919
Jul 24 at 7:35
@TobiasKildetoft That's true. But aren't polynomial and algebraic representations not the same thing?
– user160919
Jul 24 at 7:36
3
No, being polynomial is stronger than being algebraic, and here it precisely amounts to whether one allows negative powers of the determinant to appear.
– Tobias Kildetoft
Jul 24 at 7:45
 |Â
show 4 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
I just started to learn representation theory and I'm interested in the irreducible representations of $operatornameGL_n(mathbbC)$. I looked them up in three different books and apparently, I misunderstood something because for me, they are inconsistent statements.
Stanley (Enumerative Combinatorics II, Theorem A2.4):
The irreducible polynomial representations $varphi^lambda$ of $operatornameGL_n(mathbbC)$ can be indexed by partitions
$lambda$ of length at most $n$.
Bump (Lie Groups, Theorem 38.3):
The irreducible algebraic representations $varphi^lambda$ of $operatornameGL_n(mathbbC)$ are indexed by integer sequences
$lambda_1geqlambda_2geqdotsgeqlambda_n$.
Fulton, Harris (Representation Theory, Proposition 15.47):
Every irreducible complex representation of $operatornameGL_n(mathbbC)$ is isomorphic to $varphi^lambda$
for a unique index $lambda=lambda_1,dots,lambda_n$ with
$lambda_1geqlambda_2geqdotsgeqlambda_n$.
My questions are:
- Is every algebraic irreducible representation indexed by a partition or by an integer sequence with $lambda_1geqlambda_2geqdotsgeqlambda_n$?
- Can every irreducible representation be indexed by an integer partition with $lambda_1geqlambda_2geqdotsgeqlambda_n$ or only the irreducible alegbraic representations?
representation-theory
asked Jul 24 at 7:22
user160919
1218
I just started to learn representation theory and I'm interested in the irreducible representations of $operatornameGL_n(mathbbC)$. I looked them up in three different books and apparently, I misunderstood something because for me, they are inconsistent statements.
Stanley (Enumerative Combinatorics II, Theorem A2.4):
The irreducible polynomial representations $varphi^lambda$ of $operatornameGL_n(mathbbC)$ can be indexed by partitions
$lambda$ of length at most $n$.
Bump (Lie Groups, Theorem 38.3):
The irreducible algebraic representations $varphi^lambda$ of $operatornameGL_n(mathbbC)$ are indexed by integer sequences
$lambda_1geqlambda_2geqdotsgeqlambda_n$.
Fulton, Harris (Representation Theory, Proposition 15.47):
Every irreducible complex representation of $operatornameGL_n(mathbbC)$ is isomorphic to $varphi^lambda$
for a unique index $lambda=lambda_1,dots,lambda_n$ with
$lambda_1geqlambda_2geqdotsgeqlambda_n$.
My questions are:
- Is every algebraic irreducible representation indexed by a partition or by an integer sequence with $lambda_1geqlambda_2geqdotsgeqlambda_n$?
- Can every irreducible representation be indexed by an integer partition with $lambda_1geqlambda_2geqdotsgeqlambda_n$ or only the irreducible alegbraic representations?
representation-theory
asked Jul 24 at 7:22
user160919
1218
edited Jul 24 at 7:47
asked Jul 24 at 7:22
user160919
1218
asked Jul 24 at 7:22
user160919
1218
asked Jul 24 at 7:22
user160919
1218
1218
What values do Fulton-Harris allow for the $lambda_i$?
– Tobias Kildetoft
Jul 24 at 7:29
I just looked up the result in Stanley, and you misquoted it slightly. It should be "polynomial" instead of "algebraic".
– Tobias Kildetoft
Jul 24 at 7:34
@TobiasKildetoft It says that some of the $lambda_i$ can be negative. So, they're integers.
– user160919
Jul 24 at 7:35
@TobiasKildetoft That's true. But aren't polynomial and algebraic representations not the same thing?
– user160919
Jul 24 at 7:36
3
No, being polynomial is stronger than being algebraic, and here it precisely amounts to whether one allows negative powers of the determinant to appear.
– Tobias Kildetoft
Jul 24 at 7:45
 |Â
show 4 more comments
What values do Fulton-Harris allow for the $lambda_i$?
– Tobias Kildetoft
Jul 24 at 7:29
I just looked up the result in Stanley, and you misquoted it slightly. It should be "polynomial" instead of "algebraic".
– Tobias Kildetoft
Jul 24 at 7:34
@TobiasKildetoft It says that some of the $lambda_i$ can be negative. So, they're integers.
– user160919
Jul 24 at 7:35
@TobiasKildetoft That's true. But aren't polynomial and algebraic representations not the same thing?
– user160919
Jul 24 at 7:36
3
No, being polynomial is stronger than being algebraic, and here it precisely amounts to whether one allows negative powers of the determinant to appear.
– Tobias Kildetoft
Jul 24 at 7:45
What values do Fulton-Harris allow for the $lambda_i$?
– Tobias Kildetoft
Jul 24 at 7:29
What values do Fulton-Harris allow for the $lambda_i$?
– Tobias Kildetoft
Jul 24 at 7:29
I just looked up the result in Stanley, and you misquoted it slightly. It should be "polynomial" instead of "algebraic".
– Tobias Kildetoft
Jul 24 at 7:34
I just looked up the result in Stanley, and you misquoted it slightly. It should be "polynomial" instead of "algebraic".
– Tobias Kildetoft
Jul 24 at 7:34
@TobiasKildetoft It says that some of the $lambda_i$ can be negative. So, they're integers.
– user160919
Jul 24 at 7:35
@TobiasKildetoft It says that some of the $lambda_i$ can be negative. So, they're integers.
– user160919
Jul 24 at 7:35
@TobiasKildetoft That's true. But aren't polynomial and algebraic representations not the same thing?
– user160919
Jul 24 at 7:36
@TobiasKildetoft That's true. But aren't polynomial and algebraic representations not the same thing?
– user160919
Jul 24 at 7:36
3
3
No, being polynomial is stronger than being algebraic, and here it precisely amounts to whether one allows negative powers of the determinant to appear.
– Tobias Kildetoft
Jul 24 at 7:45
No, being polynomial is stronger than being algebraic, and here it precisely amounts to whether one allows negative powers of the determinant to appear.
– Tobias Kildetoft
Jul 24 at 7:45
 |Â
show 4 more comments
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What values do Fulton-Harris allow for the $lambda_i$?
– Tobias Kildetoft
Jul 24 at 7:29
I just looked up the result in Stanley, and you misquoted it slightly. It should be "polynomial" instead of "algebraic".
– Tobias Kildetoft
Jul 24 at 7:34
@TobiasKildetoft It says that some of the $lambda_i$ can be negative. So, they're integers.
– user160919
Jul 24 at 7:35
@TobiasKildetoft That's true. But aren't polynomial and algebraic representations not the same thing?
– user160919
Jul 24 at 7:36
3
No, being polynomial is stronger than being algebraic, and here it precisely amounts to whether one allows negative powers of the determinant to appear.
– Tobias Kildetoft
Jul 24 at 7:45