Mixing
Is the Hodge dual the unique map which commutes with exterior powers of isometries?
$begingroup$
Let $V$ be a real oriented $d$-dimensional inner product space, $d ge 3$. For $1 le k le d-1$, the Hodge dual map $star: bigwedge^k V to bigwedge^{d-k} V$ commutes with orientation-preserving isometries:
For every $Q in text{SO}(V)$, we have
$$star circ bigwedge^k Q= bigwedge^{d-k} Q circ star tag{1}.$$
Is $star$ the unique linear map $bigwedge^k V to bigwedge^{d-k} V$ satisfying $(1)$ up to scaling?
In the language of representation theory, I ask if the space of equivariant maps w.r.t the natural representations of $ text{SO}(V)$ on $bigwedge^k V,bigwedge^{d-k} V$ is one dimensional.
In $d=2$, $star:V to V$ is of course not the unique map up to scaling which commutes with all isometries, since $text{SO}(2)$ is commutative, we have additional elements... (This is why I restricted $d ge 3$).
differential-geometry representation-theory inner-product-space exterior-algebra isometry
asked Jan 30 at 9:14
Asaf ShacharAsaf Shachar
5,78731145
$endgroup$
add a comment |
$begingroup$
Let $V$ be a real oriented $d$-dimensional inner product space, $d ge 3$. For $1 le k le d-1$, the Hodge dual map $star: bigwedge^k V to bigwedge^{d-k} V$ commutes with orientation-preserving isometries:
For every $Q in text{SO}(V)$, we have
$$star circ bigwedge^k Q= bigwedge^{d-k} Q circ star tag{1}.$$
Is $star$ the unique linear map $bigwedge^k V to bigwedge^{d-k} V$ satisfying $(1)$ up to scaling?
In the language of representation theory, I ask if the space of equivariant maps w.r.t the natural representations of $ text{SO}(V)$ on $bigwedge^k V,bigwedge^{d-k} V$ is one dimensional.
In $d=2$, $star:V to V$ is of course not the unique map up to scaling which commutes with all isometries, since $text{SO}(2)$ is commutative, we have additional elements... (This is why I restricted $d ge 3$).
differential-geometry representation-theory inner-product-space exterior-algebra isometry
asked Jan 30 at 9:14
Asaf ShacharAsaf Shachar
5,78731145
$endgroup$
add a comment |
$begingroup$
Let $V$ be a real oriented $d$-dimensional inner product space, $d ge 3$. For $1 le k le d-1$, the Hodge dual map $star: bigwedge^k V to bigwedge^{d-k} V$ commutes with orientation-preserving isometries:
For every $Q in text{SO}(V)$, we have
$$star circ bigwedge^k Q= bigwedge^{d-k} Q circ star tag{1}.$$
Is $star$ the unique linear map $bigwedge^k V to bigwedge^{d-k} V$ satisfying $(1)$ up to scaling?
In the language of representation theory, I ask if the space of equivariant maps w.r.t the natural representations of $ text{SO}(V)$ on $bigwedge^k V,bigwedge^{d-k} V$ is one dimensional.
In $d=2$, $star:V to V$ is of course not the unique map up to scaling which commutes with all isometries, since $text{SO}(2)$ is commutative, we have additional elements... (This is why I restricted $d ge 3$).
differential-geometry representation-theory inner-product-space exterior-algebra isometry
asked Jan 30 at 9:14
Asaf ShacharAsaf Shachar
5,78731145
$endgroup$
Let $V$ be a real oriented $d$-dimensional inner product space, $d ge 3$. For $1 le k le d-1$, the Hodge dual map $star: bigwedge^k V to bigwedge^{d-k} V$ commutes with orientation-preserving isometries:
For every $Q in text{SO}(V)$, we have
$$star circ bigwedge^k Q= bigwedge^{d-k} Q circ star tag{1}.$$
Is $star$ the unique linear map $bigwedge^k V to bigwedge^{d-k} V$ satisfying $(1)$ up to scaling?
In the language of representation theory, I ask if the space of equivariant maps w.r.t the natural representations of $ text{SO}(V)$ on $bigwedge^k V,bigwedge^{d-k} V$ is one dimensional.
In $d=2$, $star:V to V$ is of course not the unique map up to scaling which commutes with all isometries, since $text{SO}(2)$ is commutative, we have additional elements... (This is why I restricted $d ge 3$).
differential-geometry representation-theory inner-product-space exterior-algebra isometry
differential-geometry representation-theory inner-product-space exterior-algebra isometry
asked Jan 30 at 9:14
Asaf ShacharAsaf Shachar
5,78731145
asked Jan 30 at 9:14
Asaf ShacharAsaf Shachar
5,78731145
asked Jan 30 at 9:14
Asaf ShacharAsaf Shachar
5,78731145
asked Jan 30 at 9:14
Asaf ShacharAsaf Shachar
5,78731145
asked Jan 30 at 9:14
Asaf ShacharAsaf Shachar
5,78731145
5,78731145
add a comment |
add a comment |
1 Answer
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This is true unless $d$ is even and $k=d-k=frac{d}2$. This follows directly from the representation theory description you give in the question using Schur's lemma. Unless $d$ is even and $k=frac{d}2$ the representation $Lambda^kV$ is irreducible and so the isomorphism to $Lambda^{d-k}V$ is unique up to a scalar multiple.If $d$ is even and $k=frac{d}2$, then $Lambda^kV$ is the direct sum of two non-isomorphic irreducible representations (the two eigenspaces of $*$) and you can choose independently choose sclalar factors on the two components (so there is a two parameter family of homomorphisms).
answered Jan 30 at 10:09
Andreas CapAndreas Cap
11.4k923
$endgroup$
$begingroup$
Thanks. Do you have a reference for these facts? (The irreducibility of $bigwedge^k V$ for $k neq d/2$ and the direct sum decomposition in the case $k=d/2$). Also, in the case where $d$ is not a multiple of $4$, the eigenvalues of $star$ are $pm i$, so the maps $omega to omega^+,omega to omega^-$ are not real. Doesn't this create problems, if we restrict the discussion to the real case? (which is what I asked about). Thank you for your help and patience...I am weak on representation theory.
$endgroup$
– Asaf Shachar
Jan 30 at 15:04
$begingroup$
You are right about the real vs. complex issue, I wasn't careful enough there. In a real setting uniqueness indeed holds unless $d$ is a multiple of $4$. You can certainly find the results in the complex case in Fulton-Harris. They are fundamental for the representation theory of SO and $mathfrak{so}$ since these are the so-called fundamental representations. So they should show up whethever the representation theory of orthogonal groups and algebras is discussed.
$endgroup$
– Andreas Cap
Jan 31 at 8:23
add a comment |
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1 Answer
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1 Answer
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$begingroup$
This is true unless $d$ is even and $k=d-k=frac{d}2$. This follows directly from the representation theory description you give in the question using Schur's lemma. Unless $d$ is even and $k=frac{d}2$ the representation $Lambda^kV$ is irreducible and so the isomorphism to $Lambda^{d-k}V$ is unique up to a scalar multiple.If $d$ is even and $k=frac{d}2$, then $Lambda^kV$ is the direct sum of two non-isomorphic irreducible representations (the two eigenspaces of $*$) and you can choose independently choose sclalar factors on the two components (so there is a two parameter family of homomorphisms).
answered Jan 30 at 10:09
Andreas CapAndreas Cap
11.4k923
$endgroup$
$begingroup$
Thanks. Do you have a reference for these facts? (The irreducibility of $bigwedge^k V$ for $k neq d/2$ and the direct sum decomposition in the case $k=d/2$). Also, in the case where $d$ is not a multiple of $4$, the eigenvalues of $star$ are $pm i$, so the maps $omega to omega^+,omega to omega^-$ are not real. Doesn't this create problems, if we restrict the discussion to the real case? (which is what I asked about). Thank you for your help and patience...I am weak on representation theory.
$endgroup$
– Asaf Shachar
Jan 30 at 15:04
$begingroup$
You are right about the real vs. complex issue, I wasn't careful enough there. In a real setting uniqueness indeed holds unless $d$ is a multiple of $4$. You can certainly find the results in the complex case in Fulton-Harris. They are fundamental for the representation theory of SO and $mathfrak{so}$ since these are the so-called fundamental representations. So they should show up whethever the representation theory of orthogonal groups and algebras is discussed.
$endgroup$
– Andreas Cap
Jan 31 at 8:23
add a comment |
$begingroup$
This is true unless $d$ is even and $k=d-k=frac{d}2$. This follows directly from the representation theory description you give in the question using Schur's lemma. Unless $d$ is even and $k=frac{d}2$ the representation $Lambda^kV$ is irreducible and so the isomorphism to $Lambda^{d-k}V$ is unique up to a scalar multiple.If $d$ is even and $k=frac{d}2$, then $Lambda^kV$ is the direct sum of two non-isomorphic irreducible representations (the two eigenspaces of $*$) and you can choose independently choose sclalar factors on the two components (so there is a two parameter family of homomorphisms).
answered Jan 30 at 10:09
Andreas CapAndreas Cap
11.4k923
$endgroup$
$begingroup$
Thanks. Do you have a reference for these facts? (The irreducibility of $bigwedge^k V$ for $k neq d/2$ and the direct sum decomposition in the case $k=d/2$). Also, in the case where $d$ is not a multiple of $4$, the eigenvalues of $star$ are $pm i$, so the maps $omega to omega^+,omega to omega^-$ are not real. Doesn't this create problems, if we restrict the discussion to the real case? (which is what I asked about). Thank you for your help and patience...I am weak on representation theory.
$endgroup$
– Asaf Shachar
Jan 30 at 15:04
$begingroup$
You are right about the real vs. complex issue, I wasn't careful enough there. In a real setting uniqueness indeed holds unless $d$ is a multiple of $4$. You can certainly find the results in the complex case in Fulton-Harris. They are fundamental for the representation theory of SO and $mathfrak{so}$ since these are the so-called fundamental representations. So they should show up whethever the representation theory of orthogonal groups and algebras is discussed.
$endgroup$
– Andreas Cap
Jan 31 at 8:23
add a comment |
$begingroup$
This is true unless $d$ is even and $k=d-k=frac{d}2$. This follows directly from the representation theory description you give in the question using Schur's lemma. Unless $d$ is even and $k=frac{d}2$ the representation $Lambda^kV$ is irreducible and so the isomorphism to $Lambda^{d-k}V$ is unique up to a scalar multiple.If $d$ is even and $k=frac{d}2$, then $Lambda^kV$ is the direct sum of two non-isomorphic irreducible representations (the two eigenspaces of $*$) and you can choose independently choose sclalar factors on the two components (so there is a two parameter family of homomorphisms).
answered Jan 30 at 10:09
Andreas CapAndreas Cap
11.4k923
$endgroup$
This is true unless $d$ is even and $k=d-k=frac{d}2$. This follows directly from the representation theory description you give in the question using Schur's lemma. Unless $d$ is even and $k=frac{d}2$ the representation $Lambda^kV$ is irreducible and so the isomorphism to $Lambda^{d-k}V$ is unique up to a scalar multiple.If $d$ is even and $k=frac{d}2$, then $Lambda^kV$ is the direct sum of two non-isomorphic irreducible representations (the two eigenspaces of $*$) and you can choose independently choose sclalar factors on the two components (so there is a two parameter family of homomorphisms).
answered Jan 30 at 10:09
Andreas CapAndreas Cap
11.4k923
answered Jan 30 at 10:09
Andreas CapAndreas Cap
11.4k923
answered Jan 30 at 10:09
Andreas CapAndreas Cap
11.4k923
answered Jan 30 at 10:09
Andreas CapAndreas Cap
11.4k923
11.4k923
$begingroup$
Thanks. Do you have a reference for these facts? (The irreducibility of $bigwedge^k V$ for $k neq d/2$ and the direct sum decomposition in the case $k=d/2$). Also, in the case where $d$ is not a multiple of $4$, the eigenvalues of $star$ are $pm i$, so the maps $omega to omega^+,omega to omega^-$ are not real. Doesn't this create problems, if we restrict the discussion to the real case? (which is what I asked about). Thank you for your help and patience...I am weak on representation theory.
$endgroup$
– Asaf Shachar
Jan 30 at 15:04
$begingroup$
You are right about the real vs. complex issue, I wasn't careful enough there. In a real setting uniqueness indeed holds unless $d$ is a multiple of $4$. You can certainly find the results in the complex case in Fulton-Harris. They are fundamental for the representation theory of SO and $mathfrak{so}$ since these are the so-called fundamental representations. So they should show up whethever the representation theory of orthogonal groups and algebras is discussed.
$endgroup$
– Andreas Cap
Jan 31 at 8:23
add a comment |
$begingroup$
Thanks. Do you have a reference for these facts? (The irreducibility of $bigwedge^k V$ for $k neq d/2$ and the direct sum decomposition in the case $k=d/2$). Also, in the case where $d$ is not a multiple of $4$, the eigenvalues of $star$ are $pm i$, so the maps $omega to omega^+,omega to omega^-$ are not real. Doesn't this create problems, if we restrict the discussion to the real case? (which is what I asked about). Thank you for your help and patience...I am weak on representation theory.
$endgroup$
– Asaf Shachar
Jan 30 at 15:04
$begingroup$
You are right about the real vs. complex issue, I wasn't careful enough there. In a real setting uniqueness indeed holds unless $d$ is a multiple of $4$. You can certainly find the results in the complex case in Fulton-Harris. They are fundamental for the representation theory of SO and $mathfrak{so}$ since these are the so-called fundamental representations. So they should show up whethever the representation theory of orthogonal groups and algebras is discussed.
$endgroup$
– Andreas Cap
Jan 31 at 8:23
$begingroup$
Thanks. Do you have a reference for these facts? (The irreducibility of $bigwedge^k V$ for $k neq d/2$ and the direct sum decomposition in the case $k=d/2$). Also, in the case where $d$ is not a multiple of $4$, the eigenvalues of $star$ are $pm i$, so the maps $omega to omega^+,omega to omega^-$ are not real. Doesn't this create problems, if we restrict the discussion to the real case? (which is what I asked about). Thank you for your help and patience...I am weak on representation theory.
$endgroup$
– Asaf Shachar
Jan 30 at 15:04
$begingroup$
Thanks. Do you have a reference for these facts? (The irreducibility of $bigwedge^k V$ for $k neq d/2$ and the direct sum decomposition in the case $k=d/2$). Also, in the case where $d$ is not a multiple of $4$, the eigenvalues of $star$ are $pm i$, so the maps $omega to omega^+,omega to omega^-$ are not real. Doesn't this create problems, if we restrict the discussion to the real case? (which is what I asked about). Thank you for your help and patience...I am weak on representation theory.
$endgroup$
– Asaf Shachar
Jan 30 at 15:04
$begingroup$
You are right about the real vs. complex issue, I wasn't careful enough there. In a real setting uniqueness indeed holds unless $d$ is a multiple of $4$. You can certainly find the results in the complex case in Fulton-Harris. They are fundamental for the representation theory of SO and $mathfrak{so}$ since these are the so-called fundamental representations. So they should show up whethever the representation theory of orthogonal groups and algebras is discussed.
$endgroup$
– Andreas Cap
Jan 31 at 8:23
$begingroup$
You are right about the real vs. complex issue, I wasn't careful enough there. In a real setting uniqueness indeed holds unless $d$ is a multiple of $4$. You can certainly find the results in the complex case in Fulton-Harris. They are fundamental for the representation theory of SO and $mathfrak{so}$ since these are the so-called fundamental representations. So they should show up whethever the representation theory of orthogonal groups and algebras is discussed.
$endgroup$
– Andreas Cap
Jan 31 at 8:23
add a comment |
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