Induced characters of $G$ from a normal subgroup $H$












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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?










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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
















0












$begingroup$


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?










share|cite|improve this question









$endgroup$












  • $begingroup$
    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














0












0








0





$begingroup$


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?










share|cite|improve this question









$endgroup$




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






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asked Jan 11 at 22:08









the manthe man

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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


















  • $begingroup$
    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










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