Let $V$ and $W$ be real representations of a torus $T$ s.t. $dim V^H=dim W^H$, $forall H<T$. Show that...












1












$begingroup$


$V^H:={vin V:hv=v,,forall hin H}$ is the fixed point set.



I'm trying to show this result first for the irreducible real representations, which are the trivial (one dimensional) ones and those given by



$$x=[x_1,cdots,x_n]mapstoleft(begin{array}{cc}cos2pilangle a,xrangle & sin 2pilangle a,xrangle \ -sinpilangle a,xrangle & cos2pilangle a,xrangle end{array}right);$$
with $ainmathbb{Z}^n$ and $langle a,xrangle=sum_ja_jx_j$.



Also, every subgroup of $T$ is compact and abelian, thus its irreducible real representations are either one-dimensional and of real type or two dimensional and of complex type.



I'm also trying to approach it via decompositions in weight spaces, but I'm kinda stuck.






representation-theory abelian-groups







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$endgroup$

















    1












    $begingroup$


    $V^H:={vin V:hv=v,,forall hin H}$ is the fixed point set.



    I'm trying to show this result first for the irreducible real representations, which are the trivial (one dimensional) ones and those given by



    $$x=[x_1,cdots,x_n]mapstoleft(begin{array}{cc}cos2pilangle a,xrangle & sin 2pilangle a,xrangle \ -sinpilangle a,xrangle & cos2pilangle a,xrangle end{array}right);$$
    with $ainmathbb{Z}^n$ and $langle a,xrangle=sum_ja_jx_j$.



    Also, every subgroup of $T$ is compact and abelian, thus its irreducible real representations are either one-dimensional and of real type or two dimensional and of complex type.



    I'm also trying to approach it via decompositions in weight spaces, but I'm kinda stuck.






    representation-theory abelian-groups







    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      $V^H:={vin V:hv=v,,forall hin H}$ is the fixed point set.



      I'm trying to show this result first for the irreducible real representations, which are the trivial (one dimensional) ones and those given by



      $$x=[x_1,cdots,x_n]mapstoleft(begin{array}{cc}cos2pilangle a,xrangle & sin 2pilangle a,xrangle \ -sinpilangle a,xrangle & cos2pilangle a,xrangle end{array}right);$$
      with $ainmathbb{Z}^n$ and $langle a,xrangle=sum_ja_jx_j$.



      Also, every subgroup of $T$ is compact and abelian, thus its irreducible real representations are either one-dimensional and of real type or two dimensional and of complex type.



      I'm also trying to approach it via decompositions in weight spaces, but I'm kinda stuck.






      representation-theory abelian-groups







      share|cite|improve this question









      $endgroup$




      $V^H:={vin V:hv=v,,forall hin H}$ is the fixed point set.



      I'm trying to show this result first for the irreducible real representations, which are the trivial (one dimensional) ones and those given by



      $$x=[x_1,cdots,x_n]mapstoleft(begin{array}{cc}cos2pilangle a,xrangle & sin 2pilangle a,xrangle \ -sinpilangle a,xrangle & cos2pilangle a,xrangle end{array}right);$$
      with $ainmathbb{Z}^n$ and $langle a,xrangle=sum_ja_jx_j$.



      Also, every subgroup of $T$ is compact and abelian, thus its irreducible real representations are either one-dimensional and of real type or two dimensional and of complex type.



      I'm also trying to approach it via decompositions in weight spaces, but I'm kinda stuck.






      representation-theory abelian-groups




      representation-theory abelian-groups






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Jan 23 at 19:14









      Andre GomesAndre Gomes

      920516




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