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edit approved Sep 26, 2013 at 7:44
Sergiy Kozerenko
edit approved Sep 26, 2013 at 7:44
Sergiy Kozerenko
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Let X$X$ be a compact Riemann surface and G = Aut(X)$G = Aut(X)$ be its group of automorphisms (biholomorphisms between X$X$ and X$X$). We know GIt is known that $G$ acts on the space Harm(X)$Harm(X)$ of all harmonic forms and also on the space Omega(X)$Omega(X)$ of all holomorphic forms.

We know that Harm(X)$Harm(X)$ is a direct sum of Omega(X)$Omega(X)$ and its conjugate.

Now, if we know the representation of G$G$ on Harm(X)$Harm(X)$ (by this I mean we have a matrix for each element of G$G$), how can we find matrices for the representation of G$G$ on Omega(X)?$Omega(X)\ ?$

ADDED: I am assuming we have no information about Omega(X)$Omega(X)$ or Harm(X)$Harm(X)$. We just know the genus of X$X$, G$G$ (with a multiplication table) and the representation of G$G$ on Harm(X)$Harm(X)$ (the matrices associated to each element of the group).

Let X be a compact Riemann surface and G = Aut(X) be its group of automorphisms (biholomorphisms between X and X). We know G acts on the space Harm(X) of harmonic forms and also on the space Omega(X) of holomorphic forms.

We know that Harm(X) is a direct sum of Omega(X) and its conjugate.

Now, if we know the representation of G on Harm(X) (by this I mean we have a matrix for each element of G), how can we find matrices for the representation of G on Omega(X)?

ADDED: I am assuming we have no information about Omega(X) or Harm(X). We just know the genus of X, G (with a multiplication table) and the representation of G on Harm(X) (the matrices associated to each element of the group).

Let $X$ be a compact Riemann surface and $G = Aut(X)$ be its group of automorphisms (biholomorphisms between $X$ and $X$). It is known that $G$ acts on the space $Harm(X)$ of all harmonic forms and also on the space $Omega(X)$ of all holomorphic forms.

We know that $Harm(X)$ is a direct sum of $Omega(X)$ and its conjugate.

Now, if we know the representation of $G$ on $Harm(X)$ (by this I mean we have a matrix for each element of $G$), how can we find matrices for the representation of $G$ on $Omega(X)\ ?$

ADDED: I am assuming we have no information about $Omega(X)$ or $Harm(X)$. We just know the genus of $X$, $G$ (with a multiplication table) and the representation of $G$ on $Harm(X)$ (the matrices associated to each element of the group).

added 274 characters in body; deleted 93 characters in body
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expmat
expmat
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Let X be a compact Riemann surface and G = Aut(X) be its group of automorphisms (biholomorphisms between X and X). We know G acts on the space Harm(X) of harmonic forms and also on the space Omega(X) of holomorphic forms.

We know that the space of harmonic formsHarm(X) is a direct sum of the space of holomorphic formsOmega(X) and its conjugate.

Now, if we know the representation of G on the harmonic formsHarm(X) (by this I mean we have a matrix for each element of G), how can we find matrices for the representation of G on the holomorphic formsOmega(X)?

ADDED: I am assuming we have no information about Omega(X) or Harm(X). We just know the genus of X, G (with a multiplication table) and the representation of G on Harm(X) (the matrices associated to each element of the group).

Let X be a compact Riemann surface and G = Aut(X) be its group of automorphisms (biholomorphisms between X and X). We know G acts on the space of harmonic forms and also on the space of holomorphic forms.

We know that the space of harmonic forms is a direct sum of the space of holomorphic forms and its conjugate.

Now, if we know the representation of G on the harmonic forms (by this I mean we have a matrix for each element of G), how can we find matrices for the representation of G on the holomorphic forms?

Let X be a compact Riemann surface and G = Aut(X) be its group of automorphisms (biholomorphisms between X and X). We know G acts on the space Harm(X) of harmonic forms and also on the space Omega(X) of holomorphic forms.

We know that Harm(X) is a direct sum of Omega(X) and its conjugate.

Now, if we know the representation of G on Harm(X) (by this I mean we have a matrix for each element of G), how can we find matrices for the representation of G on Omega(X)?

ADDED: I am assuming we have no information about Omega(X) or Harm(X). We just know the genus of X, G (with a multiplication table) and the representation of G on Harm(X) (the matrices associated to each element of the group).

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expmat
expmat
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Representation of the group of automorphisms on the holomorphic forms

Let X be a compact Riemann surface and G = Aut(X) be its group of automorphisms (biholomorphisms between X and X). We know G acts on the space of harmonic forms and also on the space of holomorphic forms.

We know that the space of harmonic forms is a direct sum of the space of holomorphic forms and its conjugate.

Now, if we know the representation of G on the harmonic forms (by this I mean we have a matrix for each element of G), how can we find matrices for the representation of G on the holomorphic forms?