Do “$K/k$ twisted” representations exist?












5












$begingroup$


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.










share|cite|improve this question









$endgroup$








  • 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).
    $endgroup$
    – 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
















5












$begingroup$


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.










share|cite|improve this question









$endgroup$








  • 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).
    $endgroup$
    – 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














5












5








5


2



$begingroup$


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.










share|cite|improve this question









$endgroup$




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






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 23 '18 at 22:52









user277182user277182

456212




456212








  • 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




    $begingroup$
    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










2 Answers
2






active

oldest

votes


















1












$begingroup$

In the case of finite group with semisimple group algebra (i.e. $mathrm{char}, k = 0$ or $mathrm{char}, k$ is coprime to $|G|$), the claim should be true, i.e. $Votimes_k K simeq Wotimes_k K$ implies $V simeq W$.



In the case $mathrm{char}, k = 0$, the reason is character theory: Namely, the orthogonality of characters still works (see e.g. here the version for non-alg. closed field). So after decomposing $V$ and $W$ into irreps, comparing the decomposition comes down to computing $langle chi_i, chi rangle$ where $chi$ is the common character of $V, W$ (which is invariant under extension of scalars) and $chi_i$ runs over irreducible chracters, as in the case $k=mathbb{C}$.



Note that this argument does not suffice on its own in positive characteristic coprime to $|G|$, i.e. the other semisimple case, since the character orthogonality formula contains the number $d$ of absolutely irreducible components of an irrep, and it's not obvious that this is necessarily nonzero modulo $mathrm{char}, k$.



However, there is a Brauer-Nesbitt Theorem that can be used: It states that two semisimple representations of $G$ are isomorphic whenever their characteristic polynomials (meaning: functions taking $g$ to the char. polynomial of $rho(g)$ when $rho$ is a representation) agree. Since the char. polynomials are invariant under extension of scalars, the same conclusion as in the previous case follows. (See e.g. these notes of Gabor Wiese, Thm 2.4.6 for a more-general version of the Brauer-Nesbitt theorem.)






share|cite|improve this answer









$endgroup$





















    1












    $begingroup$

    In fact, for any field extension $K/k$ and any $k$-algebra $A$, if $M$ and $N$ are finite dimensional $A$-modules such that $Motimes_kKcong Notimes_kK$ as $Aotimes_kK$-modules, then $Mcong N$ as $A$-modules. This is the Noether-Deuring Theorem (see, for example, (19.25) in Lam's "First Course in Noncommutative Rings"; he assumes $A$ is finite-dimensional, but that's not essential).



    For finite field extensions the proof is very short. If $Motimes_kKcong Notimes_kK$ as $Aotimes_kK$-modules, then, restricting to $A$, $Motimes_kKcong Notimes_kK$ as $A$-modules. But if $[K:k]=n$, then as $A$-modules $Motimes_kKcong M^n$, the direct sum of $n$ copies of $M$, and $Notimes_kKcong N^n$. So $M^ncong N^n$. Now just apply the Krull-Schmidt Theorem to deduce that$Mcong N$.






    share|cite|improve this answer











    $endgroup$













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      2 Answers
      2






      active

      oldest

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      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      1












      $begingroup$

      In the case of finite group with semisimple group algebra (i.e. $mathrm{char}, k = 0$ or $mathrm{char}, k$ is coprime to $|G|$), the claim should be true, i.e. $Votimes_k K simeq Wotimes_k K$ implies $V simeq W$.



      In the case $mathrm{char}, k = 0$, the reason is character theory: Namely, the orthogonality of characters still works (see e.g. here the version for non-alg. closed field). So after decomposing $V$ and $W$ into irreps, comparing the decomposition comes down to computing $langle chi_i, chi rangle$ where $chi$ is the common character of $V, W$ (which is invariant under extension of scalars) and $chi_i$ runs over irreducible chracters, as in the case $k=mathbb{C}$.



      Note that this argument does not suffice on its own in positive characteristic coprime to $|G|$, i.e. the other semisimple case, since the character orthogonality formula contains the number $d$ of absolutely irreducible components of an irrep, and it's not obvious that this is necessarily nonzero modulo $mathrm{char}, k$.



      However, there is a Brauer-Nesbitt Theorem that can be used: It states that two semisimple representations of $G$ are isomorphic whenever their characteristic polynomials (meaning: functions taking $g$ to the char. polynomial of $rho(g)$ when $rho$ is a representation) agree. Since the char. polynomials are invariant under extension of scalars, the same conclusion as in the previous case follows. (See e.g. these notes of Gabor Wiese, Thm 2.4.6 for a more-general version of the Brauer-Nesbitt theorem.)






      share|cite|improve this answer









      $endgroup$


















        1












        $begingroup$

        In the case of finite group with semisimple group algebra (i.e. $mathrm{char}, k = 0$ or $mathrm{char}, k$ is coprime to $|G|$), the claim should be true, i.e. $Votimes_k K simeq Wotimes_k K$ implies $V simeq W$.



        In the case $mathrm{char}, k = 0$, the reason is character theory: Namely, the orthogonality of characters still works (see e.g. here the version for non-alg. closed field). So after decomposing $V$ and $W$ into irreps, comparing the decomposition comes down to computing $langle chi_i, chi rangle$ where $chi$ is the common character of $V, W$ (which is invariant under extension of scalars) and $chi_i$ runs over irreducible chracters, as in the case $k=mathbb{C}$.



        Note that this argument does not suffice on its own in positive characteristic coprime to $|G|$, i.e. the other semisimple case, since the character orthogonality formula contains the number $d$ of absolutely irreducible components of an irrep, and it's not obvious that this is necessarily nonzero modulo $mathrm{char}, k$.



        However, there is a Brauer-Nesbitt Theorem that can be used: It states that two semisimple representations of $G$ are isomorphic whenever their characteristic polynomials (meaning: functions taking $g$ to the char. polynomial of $rho(g)$ when $rho$ is a representation) agree. Since the char. polynomials are invariant under extension of scalars, the same conclusion as in the previous case follows. (See e.g. these notes of Gabor Wiese, Thm 2.4.6 for a more-general version of the Brauer-Nesbitt theorem.)






        share|cite|improve this answer









        $endgroup$
















          1












          1








          1





          $begingroup$

          In the case of finite group with semisimple group algebra (i.e. $mathrm{char}, k = 0$ or $mathrm{char}, k$ is coprime to $|G|$), the claim should be true, i.e. $Votimes_k K simeq Wotimes_k K$ implies $V simeq W$.



          In the case $mathrm{char}, k = 0$, the reason is character theory: Namely, the orthogonality of characters still works (see e.g. here the version for non-alg. closed field). So after decomposing $V$ and $W$ into irreps, comparing the decomposition comes down to computing $langle chi_i, chi rangle$ where $chi$ is the common character of $V, W$ (which is invariant under extension of scalars) and $chi_i$ runs over irreducible chracters, as in the case $k=mathbb{C}$.



          Note that this argument does not suffice on its own in positive characteristic coprime to $|G|$, i.e. the other semisimple case, since the character orthogonality formula contains the number $d$ of absolutely irreducible components of an irrep, and it's not obvious that this is necessarily nonzero modulo $mathrm{char}, k$.



          However, there is a Brauer-Nesbitt Theorem that can be used: It states that two semisimple representations of $G$ are isomorphic whenever their characteristic polynomials (meaning: functions taking $g$ to the char. polynomial of $rho(g)$ when $rho$ is a representation) agree. Since the char. polynomials are invariant under extension of scalars, the same conclusion as in the previous case follows. (See e.g. these notes of Gabor Wiese, Thm 2.4.6 for a more-general version of the Brauer-Nesbitt theorem.)






          share|cite|improve this answer









          $endgroup$



          In the case of finite group with semisimple group algebra (i.e. $mathrm{char}, k = 0$ or $mathrm{char}, k$ is coprime to $|G|$), the claim should be true, i.e. $Votimes_k K simeq Wotimes_k K$ implies $V simeq W$.



          In the case $mathrm{char}, k = 0$, the reason is character theory: Namely, the orthogonality of characters still works (see e.g. here the version for non-alg. closed field). So after decomposing $V$ and $W$ into irreps, comparing the decomposition comes down to computing $langle chi_i, chi rangle$ where $chi$ is the common character of $V, W$ (which is invariant under extension of scalars) and $chi_i$ runs over irreducible chracters, as in the case $k=mathbb{C}$.



          Note that this argument does not suffice on its own in positive characteristic coprime to $|G|$, i.e. the other semisimple case, since the character orthogonality formula contains the number $d$ of absolutely irreducible components of an irrep, and it's not obvious that this is necessarily nonzero modulo $mathrm{char}, k$.



          However, there is a Brauer-Nesbitt Theorem that can be used: It states that two semisimple representations of $G$ are isomorphic whenever their characteristic polynomials (meaning: functions taking $g$ to the char. polynomial of $rho(g)$ when $rho$ is a representation) agree. Since the char. polynomials are invariant under extension of scalars, the same conclusion as in the previous case follows. (See e.g. these notes of Gabor Wiese, Thm 2.4.6 for a more-general version of the Brauer-Nesbitt theorem.)







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Feb 8 at 5:51









          Pavel ČoupekPavel Čoupek

          4,45611126




          4,45611126























              1












              $begingroup$

              In fact, for any field extension $K/k$ and any $k$-algebra $A$, if $M$ and $N$ are finite dimensional $A$-modules such that $Motimes_kKcong Notimes_kK$ as $Aotimes_kK$-modules, then $Mcong N$ as $A$-modules. This is the Noether-Deuring Theorem (see, for example, (19.25) in Lam's "First Course in Noncommutative Rings"; he assumes $A$ is finite-dimensional, but that's not essential).



              For finite field extensions the proof is very short. If $Motimes_kKcong Notimes_kK$ as $Aotimes_kK$-modules, then, restricting to $A$, $Motimes_kKcong Notimes_kK$ as $A$-modules. But if $[K:k]=n$, then as $A$-modules $Motimes_kKcong M^n$, the direct sum of $n$ copies of $M$, and $Notimes_kKcong N^n$. So $M^ncong N^n$. Now just apply the Krull-Schmidt Theorem to deduce that$Mcong N$.






              share|cite|improve this answer











              $endgroup$


















                1












                $begingroup$

                In fact, for any field extension $K/k$ and any $k$-algebra $A$, if $M$ and $N$ are finite dimensional $A$-modules such that $Motimes_kKcong Notimes_kK$ as $Aotimes_kK$-modules, then $Mcong N$ as $A$-modules. This is the Noether-Deuring Theorem (see, for example, (19.25) in Lam's "First Course in Noncommutative Rings"; he assumes $A$ is finite-dimensional, but that's not essential).



                For finite field extensions the proof is very short. If $Motimes_kKcong Notimes_kK$ as $Aotimes_kK$-modules, then, restricting to $A$, $Motimes_kKcong Notimes_kK$ as $A$-modules. But if $[K:k]=n$, then as $A$-modules $Motimes_kKcong M^n$, the direct sum of $n$ copies of $M$, and $Notimes_kKcong N^n$. So $M^ncong N^n$. Now just apply the Krull-Schmidt Theorem to deduce that$Mcong N$.






                share|cite|improve this answer











                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  In fact, for any field extension $K/k$ and any $k$-algebra $A$, if $M$ and $N$ are finite dimensional $A$-modules such that $Motimes_kKcong Notimes_kK$ as $Aotimes_kK$-modules, then $Mcong N$ as $A$-modules. This is the Noether-Deuring Theorem (see, for example, (19.25) in Lam's "First Course in Noncommutative Rings"; he assumes $A$ is finite-dimensional, but that's not essential).



                  For finite field extensions the proof is very short. If $Motimes_kKcong Notimes_kK$ as $Aotimes_kK$-modules, then, restricting to $A$, $Motimes_kKcong Notimes_kK$ as $A$-modules. But if $[K:k]=n$, then as $A$-modules $Motimes_kKcong M^n$, the direct sum of $n$ copies of $M$, and $Notimes_kKcong N^n$. So $M^ncong N^n$. Now just apply the Krull-Schmidt Theorem to deduce that$Mcong N$.






                  share|cite|improve this answer











                  $endgroup$



                  In fact, for any field extension $K/k$ and any $k$-algebra $A$, if $M$ and $N$ are finite dimensional $A$-modules such that $Motimes_kKcong Notimes_kK$ as $Aotimes_kK$-modules, then $Mcong N$ as $A$-modules. This is the Noether-Deuring Theorem (see, for example, (19.25) in Lam's "First Course in Noncommutative Rings"; he assumes $A$ is finite-dimensional, but that's not essential).



                  For finite field extensions the proof is very short. If $Motimes_kKcong Notimes_kK$ as $Aotimes_kK$-modules, then, restricting to $A$, $Motimes_kKcong Notimes_kK$ as $A$-modules. But if $[K:k]=n$, then as $A$-modules $Motimes_kKcong M^n$, the direct sum of $n$ copies of $M$, and $Notimes_kKcong N^n$. So $M^ncong N^n$. Now just apply the Krull-Schmidt Theorem to deduce that$Mcong N$.







                  share|cite|improve this answer














                  share|cite|improve this answer



                  share|cite|improve this answer








                  edited Feb 8 at 13:48

























                  answered Feb 8 at 12:00









                  Jeremy RickardJeremy Rickard

                  16.4k11743




                  16.4k11743






























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