Do “$K/k$ twisted” representations exist?
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Given $k$-representations $V,W$ of a group $G$, where $k$ is a field, $K/k$ a field extension, if we have $Votimes_k Kcong Wotimes_k K$ as $K$-representations, do we have that $Vcong W$?
Being more specific, what about in the case of $V,W$ irreps, $G$ a finite group, with $K/k$ finite and galois?
In characteristic $0$, with character theory, the question can be rephrased as: if the characters of $V$ and $W$ agree, then are $V$ and $W$ isomorphic over their field of definition?
Any reference for these questions would be much appreciated.
abstract-algebra representation-theory extension-field characters
asked Dec 23 '18 at 22:52
user277182user277182
431212
$endgroup$
add a comment |
$begingroup$
Given $k$-representations $V,W$ of a group $G$, where $k$ is a field, $K/k$ a field extension, if we have $Votimes_k Kcong Wotimes_k K$ as $K$-representations, do we have that $Vcong W$?
Being more specific, what about in the case of $V,W$ irreps, $G$ a finite group, with $K/k$ finite and galois?
In characteristic $0$, with character theory, the question can be rephrased as: if the characters of $V$ and $W$ agree, then are $V$ and $W$ isomorphic over their field of definition?
Any reference for these questions would be much appreciated.
abstract-algebra representation-theory extension-field characters
asked Dec 23 '18 at 22:52
user277182user277182
431212
$endgroup$
1
$begingroup$
In general, the functor $$- otimes_k K : k[G]mathrm{-Mod} to K[G]mathrm{-Mod}$$ doesn't reflect isomorphisms (even if this is not explained by the existence of irreps that may not be absolutely irreducible). $$ $$ For algebras instead of modules, see math.stackexchange.com/questions/2578592. Notice that for commutative rings, $- otimes_R S : Rtext{-Mod} to Stext{-Mod}$ reflects isomorphisms if $R to S$ is faithfully flat (see here).
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– Watson
Dec 24 '18 at 12:56
2
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You may want to look at Galois cohomology, namely 1.3.6 here : the twists of $(rho, V)$ should be classified by $$H^1( mathrm{Gal}(K/k) ; mathrm{Aut}_K(rho otimes_k K) ).$$
$endgroup$
– Watson
Dec 24 '18 at 12:56
add a comment |
$begingroup$
Given $k$-representations $V,W$ of a group $G$, where $k$ is a field, $K/k$ a field extension, if we have $Votimes_k Kcong Wotimes_k K$ as $K$-representations, do we have that $Vcong W$?
Being more specific, what about in the case of $V,W$ irreps, $G$ a finite group, with $K/k$ finite and galois?
In characteristic $0$, with character theory, the question can be rephrased as: if the characters of $V$ and $W$ agree, then are $V$ and $W$ isomorphic over their field of definition?
Any reference for these questions would be much appreciated.
abstract-algebra representation-theory extension-field characters
asked Dec 23 '18 at 22:52
user277182user277182
431212
$endgroup$
Given $k$-representations $V,W$ of a group $G$, where $k$ is a field, $K/k$ a field extension, if we have $Votimes_k Kcong Wotimes_k K$ as $K$-representations, do we have that $Vcong W$?
Being more specific, what about in the case of $V,W$ irreps, $G$ a finite group, with $K/k$ finite and galois?
In characteristic $0$, with character theory, the question can be rephrased as: if the characters of $V$ and $W$ agree, then are $V$ and $W$ isomorphic over their field of definition?
Any reference for these questions would be much appreciated.
abstract-algebra representation-theory extension-field characters
abstract-algebra representation-theory extension-field characters
asked Dec 23 '18 at 22:52
user277182user277182
431212
asked Dec 23 '18 at 22:52
user277182user277182
431212
asked Dec 23 '18 at 22:52
user277182user277182
431212
asked Dec 23 '18 at 22:52
user277182user277182
431212
asked Dec 23 '18 at 22:52
user277182user277182
431212
431212
1
$begingroup$
In general, the functor $$- otimes_k K : k[G]mathrm{-Mod} to K[G]mathrm{-Mod}$$ doesn't reflect isomorphisms (even if this is not explained by the existence of irreps that may not be absolutely irreducible). $$ $$ For algebras instead of modules, see math.stackexchange.com/questions/2578592. Notice that for commutative rings, $- otimes_R S : Rtext{-Mod} to Stext{-Mod}$ reflects isomorphisms if $R to S$ is faithfully flat (see here).
$endgroup$
– Watson
Dec 24 '18 at 12:56
2
$begingroup$
You may want to look at Galois cohomology, namely 1.3.6 here : the twists of $(rho, V)$ should be classified by $$H^1( mathrm{Gal}(K/k) ; mathrm{Aut}_K(rho otimes_k K) ).$$
$endgroup$
– Watson
Dec 24 '18 at 12:56
add a comment |
1
$begingroup$
In general, the functor $$- otimes_k K : k[G]mathrm{-Mod} to K[G]mathrm{-Mod}$$ doesn't reflect isomorphisms (even if this is not explained by the existence of irreps that may not be absolutely irreducible). $$ $$ For algebras instead of modules, see math.stackexchange.com/questions/2578592. Notice that for commutative rings, $- otimes_R S : Rtext{-Mod} to Stext{-Mod}$ reflects isomorphisms if $R to S$ is faithfully flat (see here).
$endgroup$
– Watson
Dec 24 '18 at 12:56
2
$begingroup$
You may want to look at Galois cohomology, namely 1.3.6 here : the twists of $(rho, V)$ should be classified by $$H^1( mathrm{Gal}(K/k) ; mathrm{Aut}_K(rho otimes_k K) ).$$
$endgroup$
– Watson
Dec 24 '18 at 12:56
1
1
$begingroup$
In general, the functor $$- otimes_k K : k[G]mathrm{-Mod} to K[G]mathrm{-Mod}$$ doesn't reflect isomorphisms (even if this is not explained by the existence of irreps that may not be absolutely irreducible). $$ $$ For algebras instead of modules, see math.stackexchange.com/questions/2578592. Notice that for commutative rings, $- otimes_R S : Rtext{-Mod} to Stext{-Mod}$ reflects isomorphisms if $R to S$ is faithfully flat (see here).
$endgroup$
– Watson
Dec 24 '18 at 12:56
$begingroup$
In general, the functor $$- otimes_k K : k[G]mathrm{-Mod} to K[G]mathrm{-Mod}$$ doesn't reflect isomorphisms (even if this is not explained by the existence of irreps that may not be absolutely irreducible). $$ $$ For algebras instead of modules, see math.stackexchange.com/questions/2578592. Notice that for commutative rings, $- otimes_R S : Rtext{-Mod} to Stext{-Mod}$ reflects isomorphisms if $R to S$ is faithfully flat (see here).
$endgroup$
– Watson
Dec 24 '18 at 12:56
2
2
$begingroup$
You may want to look at Galois cohomology, namely 1.3.6 here : the twists of $(rho, V)$ should be classified by $$H^1( mathrm{Gal}(K/k) ; mathrm{Aut}_K(rho otimes_k K) ).$$
$endgroup$
– Watson
Dec 24 '18 at 12:56
$begingroup$
You may want to look at Galois cohomology, namely 1.3.6 here : the twists of $(rho, V)$ should be classified by $$H^1( mathrm{Gal}(K/k) ; mathrm{Aut}_K(rho otimes_k K) ).$$
$endgroup$
– Watson
Dec 24 '18 at 12:56
add a comment |
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1
$begingroup$
In general, the functor $$- otimes_k K : k[G]mathrm{-Mod} to K[G]mathrm{-Mod}$$ doesn't reflect isomorphisms (even if this is not explained by the existence of irreps that may not be absolutely irreducible). $$ $$ For algebras instead of modules, see math.stackexchange.com/questions/2578592. Notice that for commutative rings, $- otimes_R S : Rtext{-Mod} to Stext{-Mod}$ reflects isomorphisms if $R to S$ is faithfully flat (see here).
$endgroup$
– Watson
Dec 24 '18 at 12:56
2
$begingroup$
You may want to look at Galois cohomology, namely 1.3.6 here : the twists of $(rho, V)$ should be classified by $$H^1( mathrm{Gal}(K/k) ; mathrm{Aut}_K(rho otimes_k K) ).$$
$endgroup$
– Watson
Dec 24 '18 at 12:56