Are there mixed-symmetry, primitive invariant tensors for simple Lie algebras?
I am interested in $mathfrak{g}$-invariant tensors for a simple Lie algebra $mathfrak{g}$. That is, in tensors $kappa_{i_1dots i_n}$ such that
$$
sumlimits_{s=1}^m f^rho_{nu i_s} kappa_{i_1dotshat{i_s}rho i_{s+1}dots i_m} = 0~,
$$
where $f^rho_{nu i_s}$ are the structure constants of $mathfrak{g}$. These correspond to the invariant polynomials often denoted as $mathcal{P}(mathfrak{g})^{mathfrak{g}}$.
I can only find sources (Humphreys, Tauvel-Yu etc.) that treat the completely symmetric or skew-symmetric cases. It is, e.g., known that there are $r$ symmetric, primitive invariant polynomials corresponding to the Casimirs of $mathcal{G}$, where $mathcal{G}$ is the Lie group integrating $mathfrak{g}$ and $r$ is the Lie algebra's rank. There are also $r$ skew-symmetric primitive invariant polynomials that determine the non-trivial cocycles of the Lie algebra cohomology. See, e.g., this paper for nice and explicit expressions for these.
My question is this: Are there any mixed-symmetry invariant polynomials that cannot be written as a product of completely symmetric and anti-symmetric ones? If not, do you have any an idea on how to show this? If so, do you have an example, say for $mathfrak{su}(n)$? Also, would there be restrictions on the order of those polynomials (the skew-symmetric ones, e.g., have order 2m-1)?
This feels like something that is just classically known, but I cannot find a reference, find an example nor show the converse, so I thought I'd ask here. Thank you very much for any help!
representation-theory lie-groups lie-algebras invariant-theory
asked Nov 18 at 16:19
user119921
516
add a comment |
I am interested in $mathfrak{g}$-invariant tensors for a simple Lie algebra $mathfrak{g}$. That is, in tensors $kappa_{i_1dots i_n}$ such that
$$
sumlimits_{s=1}^m f^rho_{nu i_s} kappa_{i_1dotshat{i_s}rho i_{s+1}dots i_m} = 0~,
$$
where $f^rho_{nu i_s}$ are the structure constants of $mathfrak{g}$. These correspond to the invariant polynomials often denoted as $mathcal{P}(mathfrak{g})^{mathfrak{g}}$.
I can only find sources (Humphreys, Tauvel-Yu etc.) that treat the completely symmetric or skew-symmetric cases. It is, e.g., known that there are $r$ symmetric, primitive invariant polynomials corresponding to the Casimirs of $mathcal{G}$, where $mathcal{G}$ is the Lie group integrating $mathfrak{g}$ and $r$ is the Lie algebra's rank. There are also $r$ skew-symmetric primitive invariant polynomials that determine the non-trivial cocycles of the Lie algebra cohomology. See, e.g., this paper for nice and explicit expressions for these.
My question is this: Are there any mixed-symmetry invariant polynomials that cannot be written as a product of completely symmetric and anti-symmetric ones? If not, do you have any an idea on how to show this? If so, do you have an example, say for $mathfrak{su}(n)$? Also, would there be restrictions on the order of those polynomials (the skew-symmetric ones, e.g., have order 2m-1)?
This feels like something that is just classically known, but I cannot find a reference, find an example nor show the converse, so I thought I'd ask here. Thank you very much for any help!
representation-theory lie-groups lie-algebras invariant-theory
asked Nov 18 at 16:19
user119921
516
I don't know what the indices are supposed to be ranging over here. Do you only want to consider tensor powers of $mathfrak{g}$ itself or also other representations? There's a very nice answer for $mathfrak{g} = mathfrak{su}(2)$ and the standard representation.
– Qiaochu Yuan
Nov 18 at 20:47
Apologies, the indices are supposed to range over the Lie algebra. I.e., I'm interested in tensor powers of $mathfrak{g}$ itself. However, the nice answer you mention still sounds interesting!
– user119921
Nov 18 at 22:24
add a comment |
I am interested in $mathfrak{g}$-invariant tensors for a simple Lie algebra $mathfrak{g}$. That is, in tensors $kappa_{i_1dots i_n}$ such that
$$
sumlimits_{s=1}^m f^rho_{nu i_s} kappa_{i_1dotshat{i_s}rho i_{s+1}dots i_m} = 0~,
$$
where $f^rho_{nu i_s}$ are the structure constants of $mathfrak{g}$. These correspond to the invariant polynomials often denoted as $mathcal{P}(mathfrak{g})^{mathfrak{g}}$.
I can only find sources (Humphreys, Tauvel-Yu etc.) that treat the completely symmetric or skew-symmetric cases. It is, e.g., known that there are $r$ symmetric, primitive invariant polynomials corresponding to the Casimirs of $mathcal{G}$, where $mathcal{G}$ is the Lie group integrating $mathfrak{g}$ and $r$ is the Lie algebra's rank. There are also $r$ skew-symmetric primitive invariant polynomials that determine the non-trivial cocycles of the Lie algebra cohomology. See, e.g., this paper for nice and explicit expressions for these.
My question is this: Are there any mixed-symmetry invariant polynomials that cannot be written as a product of completely symmetric and anti-symmetric ones? If not, do you have any an idea on how to show this? If so, do you have an example, say for $mathfrak{su}(n)$? Also, would there be restrictions on the order of those polynomials (the skew-symmetric ones, e.g., have order 2m-1)?
This feels like something that is just classically known, but I cannot find a reference, find an example nor show the converse, so I thought I'd ask here. Thank you very much for any help!
representation-theory lie-groups lie-algebras invariant-theory
asked Nov 18 at 16:19
user119921
516
I am interested in $mathfrak{g}$-invariant tensors for a simple Lie algebra $mathfrak{g}$. That is, in tensors $kappa_{i_1dots i_n}$ such that
$$
sumlimits_{s=1}^m f^rho_{nu i_s} kappa_{i_1dotshat{i_s}rho i_{s+1}dots i_m} = 0~,
$$
where $f^rho_{nu i_s}$ are the structure constants of $mathfrak{g}$. These correspond to the invariant polynomials often denoted as $mathcal{P}(mathfrak{g})^{mathfrak{g}}$.
I can only find sources (Humphreys, Tauvel-Yu etc.) that treat the completely symmetric or skew-symmetric cases. It is, e.g., known that there are $r$ symmetric, primitive invariant polynomials corresponding to the Casimirs of $mathcal{G}$, where $mathcal{G}$ is the Lie group integrating $mathfrak{g}$ and $r$ is the Lie algebra's rank. There are also $r$ skew-symmetric primitive invariant polynomials that determine the non-trivial cocycles of the Lie algebra cohomology. See, e.g., this paper for nice and explicit expressions for these.
My question is this: Are there any mixed-symmetry invariant polynomials that cannot be written as a product of completely symmetric and anti-symmetric ones? If not, do you have any an idea on how to show this? If so, do you have an example, say for $mathfrak{su}(n)$? Also, would there be restrictions on the order of those polynomials (the skew-symmetric ones, e.g., have order 2m-1)?
This feels like something that is just classically known, but I cannot find a reference, find an example nor show the converse, so I thought I'd ask here. Thank you very much for any help!
representation-theory lie-groups lie-algebras invariant-theory
representation-theory lie-groups lie-algebras invariant-theory
asked Nov 18 at 16:19
user119921
516
asked Nov 18 at 16:19
user119921
516
asked Nov 18 at 16:19
user119921
516
asked Nov 18 at 16:19
user119921
516
asked Nov 18 at 16:19
user119921
516
516
I don't know what the indices are supposed to be ranging over here. Do you only want to consider tensor powers of $mathfrak{g}$ itself or also other representations? There's a very nice answer for $mathfrak{g} = mathfrak{su}(2)$ and the standard representation.
– Qiaochu Yuan
Nov 18 at 20:47
Apologies, the indices are supposed to range over the Lie algebra. I.e., I'm interested in tensor powers of $mathfrak{g}$ itself. However, the nice answer you mention still sounds interesting!
– user119921
Nov 18 at 22:24
add a comment |
I don't know what the indices are supposed to be ranging over here. Do you only want to consider tensor powers of $mathfrak{g}$ itself or also other representations? There's a very nice answer for $mathfrak{g} = mathfrak{su}(2)$ and the standard representation.
– Qiaochu Yuan
Nov 18 at 20:47
Apologies, the indices are supposed to range over the Lie algebra. I.e., I'm interested in tensor powers of $mathfrak{g}$ itself. However, the nice answer you mention still sounds interesting!
– user119921
Nov 18 at 22:24
I don't know what the indices are supposed to be ranging over here. Do you only want to consider tensor powers of $mathfrak{g}$ itself or also other representations? There's a very nice answer for $mathfrak{g} = mathfrak{su}(2)$ and the standard representation.
– Qiaochu Yuan
Nov 18 at 20:47
I don't know what the indices are supposed to be ranging over here. Do you only want to consider tensor powers of $mathfrak{g}$ itself or also other representations? There's a very nice answer for $mathfrak{g} = mathfrak{su}(2)$ and the standard representation.
– Qiaochu Yuan
Nov 18 at 20:47
Apologies, the indices are supposed to range over the Lie algebra. I.e., I'm interested in tensor powers of $mathfrak{g}$ itself. However, the nice answer you mention still sounds interesting!
– user119921
Nov 18 at 22:24
Apologies, the indices are supposed to range over the Lie algebra. I.e., I'm interested in tensor powers of $mathfrak{g}$ itself. However, the nice answer you mention still sounds interesting!
– user119921
Nov 18 at 22:24
add a comment |
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I don't know what the indices are supposed to be ranging over here. Do you only want to consider tensor powers of $mathfrak{g}$ itself or also other representations? There's a very nice answer for $mathfrak{g} = mathfrak{su}(2)$ and the standard representation.
– Qiaochu Yuan
Nov 18 at 20:47
Apologies, the indices are supposed to range over the Lie algebra. I.e., I'm interested in tensor powers of $mathfrak{g}$ itself. However, the nice answer you mention still sounds interesting!
– user119921
Nov 18 at 22:24