Mixing
Let $V$ and $W$ be real representations of a torus $T$ s.t. $dim V^H=dim W^H$, $forall H<T$. Show that...
$begingroup$
$V^H:={vin V:hv=v,,forall hin H}$ 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
asked Jan 23 at 19:14
Andre GomesAndre Gomes
920516
$endgroup$
add a comment |
$begingroup$
$V^H:={vin V:hv=v,,forall hin H}$ 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
asked Jan 23 at 19:14
Andre GomesAndre Gomes
920516
$endgroup$
add a comment |
$begingroup$
$V^H:={vin V:hv=v,,forall hin H}$ 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
asked Jan 23 at 19:14
Andre GomesAndre Gomes
920516
$endgroup$
$V^H:={vin V:hv=v,,forall hin H}$ 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
asked Jan 23 at 19:14
Andre GomesAndre Gomes
920516
asked Jan 23 at 19:14
Andre GomesAndre Gomes
920516
asked Jan 23 at 19:14
Andre GomesAndre Gomes
920516
asked Jan 23 at 19:14
Andre GomesAndre Gomes
920516
asked Jan 23 at 19:14
Andre GomesAndre Gomes
920516
920516
add a comment |
add a comment |
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