Induced characters of $G$ from a normal subgroup $H$
$begingroup$
Let $H lhd G$ and let $chi$ be a character of $H$. Let $g in G$ and let $H^g = gHg^{-1}$.
Define $chi^g$ to be the class function on $H^g$ given by $chi^{g}(x) = chi(g^{-1}xg)$.
Suppose that $chi$ is an irreducible character of $H$ and we want to see if $Ind_{H}^{G} chi$ is an irreducible character of $G$.
By Frobenius reciprocity, $(Ind_{H}^{G} chi,Ind_{H}^{G} chi)_G = (chi, Res_{H}^{G}Ind_{H}^{G} chi)_H$.
Let $h in H$ and consider $(Res_{H}^{G}Ind_{H}^{G} chi)(h) = Ind_{H}^{G}chi(h) = sum_{g in G/H} chi(g^{-1}hg) = sum_{g in G/H} chi^{g}(h)$.
I have no idea how they go from $Ind_{H}^{G} chi(h)$ to $sum_{g in G/H} chi(g^{-1}hg)$? Where do cosets come in to it? Why are we summing over the cosets?
abstract-algebra representation-theory characters
asked Jan 11 at 22:08
the manthe man
726715
$endgroup$
add a comment |
$begingroup$
Let $H lhd G$ and let $chi$ be a character of $H$. Let $g in G$ and let $H^g = gHg^{-1}$.
Define $chi^g$ to be the class function on $H^g$ given by $chi^{g}(x) = chi(g^{-1}xg)$.
Suppose that $chi$ is an irreducible character of $H$ and we want to see if $Ind_{H}^{G} chi$ is an irreducible character of $G$.
By Frobenius reciprocity, $(Ind_{H}^{G} chi,Ind_{H}^{G} chi)_G = (chi, Res_{H}^{G}Ind_{H}^{G} chi)_H$.
Let $h in H$ and consider $(Res_{H}^{G}Ind_{H}^{G} chi)(h) = Ind_{H}^{G}chi(h) = sum_{g in G/H} chi(g^{-1}hg) = sum_{g in G/H} chi^{g}(h)$.
I have no idea how they go from $Ind_{H}^{G} chi(h)$ to $sum_{g in G/H} chi(g^{-1}hg)$? Where do cosets come in to it? Why are we summing over the cosets?
abstract-algebra representation-theory characters
asked Jan 11 at 22:08
the manthe man
726715
$endgroup$
$begingroup$
The notation means you sum over some transversal set of $H$ in $G$. This is a common definition for any induced class function. groupprops.subwiki.org/wiki/Induced_class_function
$endgroup$
– BWW
Jan 11 at 22:44
add a comment |
$begingroup$
Let $H lhd G$ and let $chi$ be a character of $H$. Let $g in G$ and let $H^g = gHg^{-1}$.
Define $chi^g$ to be the class function on $H^g$ given by $chi^{g}(x) = chi(g^{-1}xg)$.
Suppose that $chi$ is an irreducible character of $H$ and we want to see if $Ind_{H}^{G} chi$ is an irreducible character of $G$.
By Frobenius reciprocity, $(Ind_{H}^{G} chi,Ind_{H}^{G} chi)_G = (chi, Res_{H}^{G}Ind_{H}^{G} chi)_H$.
Let $h in H$ and consider $(Res_{H}^{G}Ind_{H}^{G} chi)(h) = Ind_{H}^{G}chi(h) = sum_{g in G/H} chi(g^{-1}hg) = sum_{g in G/H} chi^{g}(h)$.
I have no idea how they go from $Ind_{H}^{G} chi(h)$ to $sum_{g in G/H} chi(g^{-1}hg)$? Where do cosets come in to it? Why are we summing over the cosets?
abstract-algebra representation-theory characters
asked Jan 11 at 22:08
the manthe man
726715
$endgroup$
Let $H lhd G$ and let $chi$ be a character of $H$. Let $g in G$ and let $H^g = gHg^{-1}$.
Define $chi^g$ to be the class function on $H^g$ given by $chi^{g}(x) = chi(g^{-1}xg)$.
Suppose that $chi$ is an irreducible character of $H$ and we want to see if $Ind_{H}^{G} chi$ is an irreducible character of $G$.
By Frobenius reciprocity, $(Ind_{H}^{G} chi,Ind_{H}^{G} chi)_G = (chi, Res_{H}^{G}Ind_{H}^{G} chi)_H$.
Let $h in H$ and consider $(Res_{H}^{G}Ind_{H}^{G} chi)(h) = Ind_{H}^{G}chi(h) = sum_{g in G/H} chi(g^{-1}hg) = sum_{g in G/H} chi^{g}(h)$.
I have no idea how they go from $Ind_{H}^{G} chi(h)$ to $sum_{g in G/H} chi(g^{-1}hg)$? Where do cosets come in to it? Why are we summing over the cosets?
abstract-algebra representation-theory characters
abstract-algebra representation-theory characters
asked Jan 11 at 22:08
the manthe man
726715
asked Jan 11 at 22:08
the manthe man
726715
asked Jan 11 at 22:08
the manthe man
726715
asked Jan 11 at 22:08
the manthe man
726715
asked Jan 11 at 22:08
the manthe man
726715
726715
$begingroup$
The notation means you sum over some transversal set of $H$ in $G$. This is a common definition for any induced class function. groupprops.subwiki.org/wiki/Induced_class_function
$endgroup$
– BWW
Jan 11 at 22:44
add a comment |
$begingroup$
The notation means you sum over some transversal set of $H$ in $G$. This is a common definition for any induced class function. groupprops.subwiki.org/wiki/Induced_class_function
$endgroup$
– BWW
Jan 11 at 22:44
$begingroup$
The notation means you sum over some transversal set of $H$ in $G$. This is a common definition for any induced class function. groupprops.subwiki.org/wiki/Induced_class_function
$endgroup$
– BWW
Jan 11 at 22:44
$begingroup$
The notation means you sum over some transversal set of $H$ in $G$. This is a common definition for any induced class function. groupprops.subwiki.org/wiki/Induced_class_function
$endgroup$
– BWW
Jan 11 at 22:44
add a comment |
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$begingroup$
The notation means you sum over some transversal set of $H$ in $G$. This is a common definition for any induced class function. groupprops.subwiki.org/wiki/Induced_class_function
$endgroup$
– BWW
Jan 11 at 22:44