Why is the exterior power $bigwedge^kV$ an irreducible representation of $GL(V)$?
$newcommand{GL}{operatorname{GL}}$
Let $V$ be a real $n$-dimensional vector space. For $1<k<n$ we have a natural representation of $GL(V)$ via the $k$ exterior power:
$rho:GL(V) to GL(bigwedge^kV)$, given by $rho(A)=bigwedge^k A$. I am trying to show $rho$ is an irreducible representation. Let $0neq W le bigwedge^kV$ be a subrepresentation. If we can show $W$ contains a non-zero decomposable element, we are done.
Indeed, suppose $W subsetneq bigwedge^kV$. Then, there exist a decomposable element $sigma=v_1 wedge dots wedge v_k neq 0$, such that $sigma notin W$. We assumed $W$ contains a non-zero decomposable element $sigma'=u_1 wedge dots wedge u_k neq 0$. Define a map $A in GL(V)$ by extending $u_i to v_i$. Then
$$rho(A) (sigma')=bigwedge^k A(u_1 wedge dots wedge u_k)=sigma notin W,$$
while $sigma' in W$, con
So, the question reduces to the following: Why does every non-zero subrepresentation contain a non-zero decomposable element?
I asked an even more naive question here-whether or not every subspace of dimension greater than $1$ contains a non-zero decomposable element?
group-theory representation-theory lie-groups exterior-algebra tensor-decomposition
add a comment |
$newcommand{GL}{operatorname{GL}}$
Let $V$ be a real $n$-dimensional vector space. For $1<k<n$ we have a natural representation of $GL(V)$ via the $k$ exterior power:
$rho:GL(V) to GL(bigwedge^kV)$, given by $rho(A)=bigwedge^k A$. I am trying to show $rho$ is an irreducible representation. Let $0neq W le bigwedge^kV$ be a subrepresentation. If we can show $W$ contains a non-zero decomposable element, we are done.
Indeed, suppose $W subsetneq bigwedge^kV$. Then, there exist a decomposable element $sigma=v_1 wedge dots wedge v_k neq 0$, such that $sigma notin W$. We assumed $W$ contains a non-zero decomposable element $sigma'=u_1 wedge dots wedge u_k neq 0$. Define a map $A in GL(V)$ by extending $u_i to v_i$. Then
$$rho(A) (sigma')=bigwedge^k A(u_1 wedge dots wedge u_k)=sigma notin W,$$
while $sigma' in W$, con
So, the question reduces to the following: Why does every non-zero subrepresentation contain a non-zero decomposable element?
I asked an even more naive question here-whether or not every subspace of dimension greater than $1$ contains a non-zero decomposable element?
group-theory representation-theory lie-groups exterior-algebra tensor-decomposition
3
A nice conceptual way to work with this is to first decompose the representation as a module for the group of diagonal matrices (into so-called weight spaces). Then note what happens to the weight of a vector in one of these subspaces when one acts by suitable upper triangular unipotent matrices.
– Tobias Kildetoft
2 days ago
Thanks. Unfortunately, I really know barely nothing about the machinery of representation theory. Can you please elaborate on this or give me a reference? (I don't know what a weight of a vector is, and naive googling only found something in the context of representations of Lie algebras, not Lie groups).
– Asaf Shachar
2 days ago
The definition is essentially the same. The representation decomposes as a sum of $1$-dimensional subspaces, and a vector in such a subspace will be acted on via a scalar. This scalar depends on the element acting, giving a linear character of the subgroup of diagonal matrices, and this character is what is called the weight of the vector. It may be a bit much to get into if none of this is familiar, but I would still advice you to try writing it up explicitly for $k=1$ when $dim(V) = 2$ to get a feel for what happens.
– Tobias Kildetoft
2 days ago
add a comment |
$newcommand{GL}{operatorname{GL}}$
Let $V$ be a real $n$-dimensional vector space. For $1<k<n$ we have a natural representation of $GL(V)$ via the $k$ exterior power:
$rho:GL(V) to GL(bigwedge^kV)$, given by $rho(A)=bigwedge^k A$. I am trying to show $rho$ is an irreducible representation. Let $0neq W le bigwedge^kV$ be a subrepresentation. If we can show $W$ contains a non-zero decomposable element, we are done.
Indeed, suppose $W subsetneq bigwedge^kV$. Then, there exist a decomposable element $sigma=v_1 wedge dots wedge v_k neq 0$, such that $sigma notin W$. We assumed $W$ contains a non-zero decomposable element $sigma'=u_1 wedge dots wedge u_k neq 0$. Define a map $A in GL(V)$ by extending $u_i to v_i$. Then
$$rho(A) (sigma')=bigwedge^k A(u_1 wedge dots wedge u_k)=sigma notin W,$$
while $sigma' in W$, con
So, the question reduces to the following: Why does every non-zero subrepresentation contain a non-zero decomposable element?
I asked an even more naive question here-whether or not every subspace of dimension greater than $1$ contains a non-zero decomposable element?
group-theory representation-theory lie-groups exterior-algebra tensor-decomposition
$newcommand{GL}{operatorname{GL}}$
Let $V$ be a real $n$-dimensional vector space. For $1<k<n$ we have a natural representation of $GL(V)$ via the $k$ exterior power:
$rho:GL(V) to GL(bigwedge^kV)$, given by $rho(A)=bigwedge^k A$. I am trying to show $rho$ is an irreducible representation. Let $0neq W le bigwedge^kV$ be a subrepresentation. If we can show $W$ contains a non-zero decomposable element, we are done.
Indeed, suppose $W subsetneq bigwedge^kV$. Then, there exist a decomposable element $sigma=v_1 wedge dots wedge v_k neq 0$, such that $sigma notin W$. We assumed $W$ contains a non-zero decomposable element $sigma'=u_1 wedge dots wedge u_k neq 0$. Define a map $A in GL(V)$ by extending $u_i to v_i$. Then
$$rho(A) (sigma')=bigwedge^k A(u_1 wedge dots wedge u_k)=sigma notin W,$$
while $sigma' in W$, con
So, the question reduces to the following: Why does every non-zero subrepresentation contain a non-zero decomposable element?
I asked an even more naive question here-whether or not every subspace of dimension greater than $1$ contains a non-zero decomposable element?
group-theory representation-theory lie-groups exterior-algebra tensor-decomposition
group-theory representation-theory lie-groups exterior-algebra tensor-decomposition
edited yesterday
Zvi
4,960430
4,960430
asked 2 days ago
Asaf Shachar
5,1293941
5,1293941
3
A nice conceptual way to work with this is to first decompose the representation as a module for the group of diagonal matrices (into so-called weight spaces). Then note what happens to the weight of a vector in one of these subspaces when one acts by suitable upper triangular unipotent matrices.
– Tobias Kildetoft
2 days ago
Thanks. Unfortunately, I really know barely nothing about the machinery of representation theory. Can you please elaborate on this or give me a reference? (I don't know what a weight of a vector is, and naive googling only found something in the context of representations of Lie algebras, not Lie groups).
– Asaf Shachar
2 days ago
The definition is essentially the same. The representation decomposes as a sum of $1$-dimensional subspaces, and a vector in such a subspace will be acted on via a scalar. This scalar depends on the element acting, giving a linear character of the subgroup of diagonal matrices, and this character is what is called the weight of the vector. It may be a bit much to get into if none of this is familiar, but I would still advice you to try writing it up explicitly for $k=1$ when $dim(V) = 2$ to get a feel for what happens.
– Tobias Kildetoft
2 days ago
add a comment |
3
A nice conceptual way to work with this is to first decompose the representation as a module for the group of diagonal matrices (into so-called weight spaces). Then note what happens to the weight of a vector in one of these subspaces when one acts by suitable upper triangular unipotent matrices.
– Tobias Kildetoft
2 days ago
Thanks. Unfortunately, I really know barely nothing about the machinery of representation theory. Can you please elaborate on this or give me a reference? (I don't know what a weight of a vector is, and naive googling only found something in the context of representations of Lie algebras, not Lie groups).
– Asaf Shachar
2 days ago
The definition is essentially the same. The representation decomposes as a sum of $1$-dimensional subspaces, and a vector in such a subspace will be acted on via a scalar. This scalar depends on the element acting, giving a linear character of the subgroup of diagonal matrices, and this character is what is called the weight of the vector. It may be a bit much to get into if none of this is familiar, but I would still advice you to try writing it up explicitly for $k=1$ when $dim(V) = 2$ to get a feel for what happens.
– Tobias Kildetoft
2 days ago
3
3
A nice conceptual way to work with this is to first decompose the representation as a module for the group of diagonal matrices (into so-called weight spaces). Then note what happens to the weight of a vector in one of these subspaces when one acts by suitable upper triangular unipotent matrices.
– Tobias Kildetoft
2 days ago
A nice conceptual way to work with this is to first decompose the representation as a module for the group of diagonal matrices (into so-called weight spaces). Then note what happens to the weight of a vector in one of these subspaces when one acts by suitable upper triangular unipotent matrices.
– Tobias Kildetoft
2 days ago
Thanks. Unfortunately, I really know barely nothing about the machinery of representation theory. Can you please elaborate on this or give me a reference? (I don't know what a weight of a vector is, and naive googling only found something in the context of representations of Lie algebras, not Lie groups).
– Asaf Shachar
2 days ago
Thanks. Unfortunately, I really know barely nothing about the machinery of representation theory. Can you please elaborate on this or give me a reference? (I don't know what a weight of a vector is, and naive googling only found something in the context of representations of Lie algebras, not Lie groups).
– Asaf Shachar
2 days ago
The definition is essentially the same. The representation decomposes as a sum of $1$-dimensional subspaces, and a vector in such a subspace will be acted on via a scalar. This scalar depends on the element acting, giving a linear character of the subgroup of diagonal matrices, and this character is what is called the weight of the vector. It may be a bit much to get into if none of this is familiar, but I would still advice you to try writing it up explicitly for $k=1$ when $dim(V) = 2$ to get a feel for what happens.
– Tobias Kildetoft
2 days ago
The definition is essentially the same. The representation decomposes as a sum of $1$-dimensional subspaces, and a vector in such a subspace will be acted on via a scalar. This scalar depends on the element acting, giving a linear character of the subgroup of diagonal matrices, and this character is what is called the weight of the vector. It may be a bit much to get into if none of this is familiar, but I would still advice you to try writing it up explicitly for $k=1$ when $dim(V) = 2$ to get a feel for what happens.
– Tobias Kildetoft
2 days ago
add a comment |
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Pick a basis $e_1, dots e_n$ of $V$ so that we can identify $GL(V)$ with $GL_n(F)$ (we'll start out working with an arbitrary base field $F$ and then restrict $F$ later). Write $T$ for the subgroup of $GL_n(F)$ consisting of diagonal matrices. An element of $T$ consists of some diagonal elements $(t_1, dots t_n)$ and acts on $Lambda^k(V)$ by sending $e_i$ to $t_i e_i$, then extending multiplicatively.
What this means is that each pure tensor $e_{i_1} wedge e_{i_2} wedge dots wedge e_{i_k} in Lambda^k(V)$ is a simultaneous eigenvector for every element of $T$; said another way, it spans a $1$-dimensional (hence simple) subrepresentation of $Lambda^k(V)$, considered as a representation of $T$. (These are the "weight spaces" of this representation.) Since $Lambda^k(V)$ is the direct sum of these $1$-dimensional subspaces, it follows that $Lambda^k(V)$ is semisimple as a representation of $T$.
The significance of semisimplicity is that any $GL(V)$-subrepresentation of $Lambda^k(V)$ is also a $T$-subrepresentation, and subrepresentations of semisimple representations are semisimple; they must also have the same simple components, in the same or smaller multiplicities. Moreover, if $F$ is any field except $mathbb{F}_2$ (over $mathbb{F}_2$, unfortunately, $T$ is the trivial group), the different $1$-dimensional representations above are all nonisomorphic. The conclusion from here is that any $GL(V)$-subrepresentation of $Lambda^k(V)$ must be a direct sum of weight spaces.
But now we're done (again, for any field $F$ except $mathbb{F}_2$), for example because $GL(V)$ acts transitively on these weight spaces.
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Pick a basis $e_1, dots e_n$ of $V$ so that we can identify $GL(V)$ with $GL_n(F)$ (we'll start out working with an arbitrary base field $F$ and then restrict $F$ later). Write $T$ for the subgroup of $GL_n(F)$ consisting of diagonal matrices. An element of $T$ consists of some diagonal elements $(t_1, dots t_n)$ and acts on $Lambda^k(V)$ by sending $e_i$ to $t_i e_i$, then extending multiplicatively.
What this means is that each pure tensor $e_{i_1} wedge e_{i_2} wedge dots wedge e_{i_k} in Lambda^k(V)$ is a simultaneous eigenvector for every element of $T$; said another way, it spans a $1$-dimensional (hence simple) subrepresentation of $Lambda^k(V)$, considered as a representation of $T$. (These are the "weight spaces" of this representation.) Since $Lambda^k(V)$ is the direct sum of these $1$-dimensional subspaces, it follows that $Lambda^k(V)$ is semisimple as a representation of $T$.
The significance of semisimplicity is that any $GL(V)$-subrepresentation of $Lambda^k(V)$ is also a $T$-subrepresentation, and subrepresentations of semisimple representations are semisimple; they must also have the same simple components, in the same or smaller multiplicities. Moreover, if $F$ is any field except $mathbb{F}_2$ (over $mathbb{F}_2$, unfortunately, $T$ is the trivial group), the different $1$-dimensional representations above are all nonisomorphic. The conclusion from here is that any $GL(V)$-subrepresentation of $Lambda^k(V)$ must be a direct sum of weight spaces.
But now we're done (again, for any field $F$ except $mathbb{F}_2$), for example because $GL(V)$ acts transitively on these weight spaces.
add a comment |
Pick a basis $e_1, dots e_n$ of $V$ so that we can identify $GL(V)$ with $GL_n(F)$ (we'll start out working with an arbitrary base field $F$ and then restrict $F$ later). Write $T$ for the subgroup of $GL_n(F)$ consisting of diagonal matrices. An element of $T$ consists of some diagonal elements $(t_1, dots t_n)$ and acts on $Lambda^k(V)$ by sending $e_i$ to $t_i e_i$, then extending multiplicatively.
What this means is that each pure tensor $e_{i_1} wedge e_{i_2} wedge dots wedge e_{i_k} in Lambda^k(V)$ is a simultaneous eigenvector for every element of $T$; said another way, it spans a $1$-dimensional (hence simple) subrepresentation of $Lambda^k(V)$, considered as a representation of $T$. (These are the "weight spaces" of this representation.) Since $Lambda^k(V)$ is the direct sum of these $1$-dimensional subspaces, it follows that $Lambda^k(V)$ is semisimple as a representation of $T$.
The significance of semisimplicity is that any $GL(V)$-subrepresentation of $Lambda^k(V)$ is also a $T$-subrepresentation, and subrepresentations of semisimple representations are semisimple; they must also have the same simple components, in the same or smaller multiplicities. Moreover, if $F$ is any field except $mathbb{F}_2$ (over $mathbb{F}_2$, unfortunately, $T$ is the trivial group), the different $1$-dimensional representations above are all nonisomorphic. The conclusion from here is that any $GL(V)$-subrepresentation of $Lambda^k(V)$ must be a direct sum of weight spaces.
But now we're done (again, for any field $F$ except $mathbb{F}_2$), for example because $GL(V)$ acts transitively on these weight spaces.
add a comment |
Pick a basis $e_1, dots e_n$ of $V$ so that we can identify $GL(V)$ with $GL_n(F)$ (we'll start out working with an arbitrary base field $F$ and then restrict $F$ later). Write $T$ for the subgroup of $GL_n(F)$ consisting of diagonal matrices. An element of $T$ consists of some diagonal elements $(t_1, dots t_n)$ and acts on $Lambda^k(V)$ by sending $e_i$ to $t_i e_i$, then extending multiplicatively.
What this means is that each pure tensor $e_{i_1} wedge e_{i_2} wedge dots wedge e_{i_k} in Lambda^k(V)$ is a simultaneous eigenvector for every element of $T$; said another way, it spans a $1$-dimensional (hence simple) subrepresentation of $Lambda^k(V)$, considered as a representation of $T$. (These are the "weight spaces" of this representation.) Since $Lambda^k(V)$ is the direct sum of these $1$-dimensional subspaces, it follows that $Lambda^k(V)$ is semisimple as a representation of $T$.
The significance of semisimplicity is that any $GL(V)$-subrepresentation of $Lambda^k(V)$ is also a $T$-subrepresentation, and subrepresentations of semisimple representations are semisimple; they must also have the same simple components, in the same or smaller multiplicities. Moreover, if $F$ is any field except $mathbb{F}_2$ (over $mathbb{F}_2$, unfortunately, $T$ is the trivial group), the different $1$-dimensional representations above are all nonisomorphic. The conclusion from here is that any $GL(V)$-subrepresentation of $Lambda^k(V)$ must be a direct sum of weight spaces.
But now we're done (again, for any field $F$ except $mathbb{F}_2$), for example because $GL(V)$ acts transitively on these weight spaces.
Pick a basis $e_1, dots e_n$ of $V$ so that we can identify $GL(V)$ with $GL_n(F)$ (we'll start out working with an arbitrary base field $F$ and then restrict $F$ later). Write $T$ for the subgroup of $GL_n(F)$ consisting of diagonal matrices. An element of $T$ consists of some diagonal elements $(t_1, dots t_n)$ and acts on $Lambda^k(V)$ by sending $e_i$ to $t_i e_i$, then extending multiplicatively.
What this means is that each pure tensor $e_{i_1} wedge e_{i_2} wedge dots wedge e_{i_k} in Lambda^k(V)$ is a simultaneous eigenvector for every element of $T$; said another way, it spans a $1$-dimensional (hence simple) subrepresentation of $Lambda^k(V)$, considered as a representation of $T$. (These are the "weight spaces" of this representation.) Since $Lambda^k(V)$ is the direct sum of these $1$-dimensional subspaces, it follows that $Lambda^k(V)$ is semisimple as a representation of $T$.
The significance of semisimplicity is that any $GL(V)$-subrepresentation of $Lambda^k(V)$ is also a $T$-subrepresentation, and subrepresentations of semisimple representations are semisimple; they must also have the same simple components, in the same or smaller multiplicities. Moreover, if $F$ is any field except $mathbb{F}_2$ (over $mathbb{F}_2$, unfortunately, $T$ is the trivial group), the different $1$-dimensional representations above are all nonisomorphic. The conclusion from here is that any $GL(V)$-subrepresentation of $Lambda^k(V)$ must be a direct sum of weight spaces.
But now we're done (again, for any field $F$ except $mathbb{F}_2$), for example because $GL(V)$ acts transitively on these weight spaces.
answered 10 hours ago
Qiaochu Yuan
277k32581919
277k32581919
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3
A nice conceptual way to work with this is to first decompose the representation as a module for the group of diagonal matrices (into so-called weight spaces). Then note what happens to the weight of a vector in one of these subspaces when one acts by suitable upper triangular unipotent matrices.
– Tobias Kildetoft
2 days ago
Thanks. Unfortunately, I really know barely nothing about the machinery of representation theory. Can you please elaborate on this or give me a reference? (I don't know what a weight of a vector is, and naive googling only found something in the context of representations of Lie algebras, not Lie groups).
– Asaf Shachar
2 days ago
The definition is essentially the same. The representation decomposes as a sum of $1$-dimensional subspaces, and a vector in such a subspace will be acted on via a scalar. This scalar depends on the element acting, giving a linear character of the subgroup of diagonal matrices, and this character is what is called the weight of the vector. It may be a bit much to get into if none of this is familiar, but I would still advice you to try writing it up explicitly for $k=1$ when $dim(V) = 2$ to get a feel for what happens.
– Tobias Kildetoft
2 days ago