Download PDF by Peter Müller: Darstellungstheorie endlicher Gruppen

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Ch )] ❡✐♥❡ ❣❛♥③③❛❤❧✐❣❡ ▲✐♥❡❛r❦♦♠❜✐♥❛t✐♦♥ ❞❡r ❩❛❤❧❡♥ ωχ (C1 ), . . , ωχ (Ch ) ✐st✳ ◆❛❝❤ ❙❛t③ ✺✳✸ s✐♥❞ ❛❧❧❡ ❊❧❡♠❡♥t❡ ❞❡s ❘✐♥❣s R ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤✱ ❛❧s♦ ✐♥s❜❡s♦♥❞❡r❡ ❛✉❝❤ ❞✐❡ ❩❛❤❧❡♥ ωχ (C1 ), . . , ωχ (Ch )✳ ✸✸ ❊✐♥❡ ❡rst❡ ✉♥❞ ✇✐❝❤t✐❣❡ ❆♥✇❡♥❞✉♥❣ ✐st ❞✐❡ ❢♦❧❣❡♥❞❡ ❆✉ss❛❣❡✱ ❢ür ❞✐❡ ❡s ✇♦❤❧ ❦❡✐♥❡♥ ❞✐r❡❦t❡♥ ❇❡✇❡✐s ❣✐❜t✳ ❙❛t③ ✺✳✶✶✳ ❙❡✐ V ❡✐♥ ✐rr❡❞✉③✐❜❧❡r G✲▼♦❞✉❧ ❞❡r ●r✉♣♣❡ G ✭ü❜❡r ❞❡♠ ●r✉♥❞❦ör♣❡r C✮✳ ❉❛♥♥ ✐st dim(V ) ❡✐♥ ❚❡✐❧❡r ✈♦♥ |G|✳ Pr♦♦❢✳ ❙❡✐ χ ❞❡r ❈❤❛r❛❦t❡r ✈♦♥ V ✳ ❲❡❣❡♥ ❞❡r ■rr❡❞✉③✐❜✐❧✐tät ✈♦♥ V ❣✐❧t [χ, χ] = 1✳ ❙❡✐❡♥ C1 , C2 , .

Ch ❞✐❡ ❑♦♥❥✉❣❛t✐♦♥s❦❧❛ss❡♥ ✈♦♥ G✱ ✉♥❞ gi ∈ Ci ✳ ❆✉s ❇❡♠❡r❦✉♥❣ ✸✳✸✻ ✉♥❞ ▲❡♠♠❛ ✺✳✼ ❢♦❧❣t |G| |G| = [χ, χ] = χ(e) χ(e) h i=1 |Ci |χ(gi ) χ(g ¯ i ). χ(e) |Ci |χ(gi ) ◆❛❝❤ ❞❡♠ ✈♦r✐❣❡♥ ❙❛t③ ✐st ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤✳ ❉❛ ❛✉❝❤ χ(g ¯ i ) ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤ ✐st✱ χ(e) |G| ✐st s♦✇♦❤❧ ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤✱ ❛❧s ❛✉❝❤ r❛t✐♦♥❛❧✱ ❛❧s♦ ❣❛♥③③❛❤❧✐❣ ♥❛❝❤ ▲❡♠♠❛ ✺✳✻✳ χ(e) ❙❛t③ ✺✳✶✷ ✭❇✉r♥s✐❞❡✮✳ ❙❡✐ χ ∈ ■rr(G) ✉♥❞ g ∈ G✱ s♦ ❞❛ss ❞✐❡ ●röÿ❡ ❞❡r ❑♦♥❥✉❣❛t✐♦♥✲ s❦❧❛ss❡ ✈♦♥ g t❡✐❧❡r❢r❡♠❞ ✐st ③✉ χ(e)✳ ❉❛♥♥ ❣✐❧t ❡♥t✇❡❞❡r χ(g) = 0 ♦❞❡r |χ(g)| = χ(e)✳ Pr♦♦❢✳ ❙❡✐ C ❞✐❡ ❑♦♥❥✉❣❛t✐♦♥s❦❧❛ss❡ ✈♦♥ g ✳ ❉❛ |C| ✉♥❞ χ(e) u, v ∈ Z ♠✐t u|C| + vχ(e) = 1✳ ❲❡❣❡♥ ❙❛t③ ✺✳✶✵ ✉♥❞ t❡✐❧❡r❢r❡♠❞ s✐♥❞✱ ❣✐❜t ❡s χ(g) χ(g)|C| χ(g) = (u|C| + vχ(e)) = u + vχ(g) χ(e) χ(e) χ(e) ✐st α= χ(g) ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤✳ ◆❛❝❤ ▲❡♠♠❛ ✸✳✸✸ ❣✐❧t χ(e) ❞✐❡ ③✇❡✐t❡ ▼ö❣❧✐❝❤❦❡✐t ✐♠ ❙❛t③✳ ❙❡✐ ❛❧s♦ |α| ≤ 1✳ ❋❛❧❧s |α| = 1✱ ❞❛♥♥ ❣✐❧t |α| < 1✳ m ❞✐❡ ❖r❞♥✉♥❣ ✈♦♥ g ✱ ✉♥❞ ζ ❡✐♥❡ ♣r✐♠✐t✐✈❡ m✕t❡ ❊✐♥❤❡✐ts✇✉r③❡❧✳ ❉❛♥♥ ❧✐❡❣❡♥ χ(g) ✐♥ K = Q(ζ)✳ ❙❡✐ Γ = Gal(K/Q) ❞✐❡ ●❛❧♦✐s❣r✉♣♣❡ ✈♦♥ K/Q✳ ❉❛ χ(g)γ γ γ ❢ür γ ∈ Γ ❡✐♥❡ ❙✉♠♠❡ ✈♦♥ χ(e) ❊✐♥❤❡✐t✇✉r③❡❧♥ ✐st✱ ❣✐❧t |χ(g) | ≤ χ(e)✱ ❛❧s♦ |α | ≤ 1✳ ❩✉s❛♠♠❡♥ ♠✐t |α| < 1 ❢♦❧❣t | αγ | < 1.

H✱ h ∈ C ✱ ✈♦♥ ϕC ❞✐❡ ❙♣✉r χ(g)✱ ❛❧s♦ ❣✐❧t ❉✐❡ ❇❡❤❛✉♣t✉♥❣ ❢♦❧❣t✳ χ(g)|C| ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤ χ(e) ♥❛❝❤ ❞❡♥ ♦❜✐❣❡♥ ❘❡s✉❧t❛t❡♥ ❯♥s❡r ♥ä❝❤st❡s ❩✐❡❧ ✐st ❡s ③✉ ③❡✐❣❡♥✱ ❞❛ss ❞✐❡ ❦♦♠♣❧❡①❡♥ ❩❛❤❧❡♥ χ(g) ❡✐♥❡ ❙✉♠♠❡ ✈♦♥ ❊✐♥❤❡✐ts✇✉r③❡❧♥ ✐st✱ ✐st χ(g) ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤✱ ✉♥❞ ❞❛♥♥ ✐st ❛✉❝❤ χ(g)|C| ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤✳ ❞✐❡ ❣❛♥③❡ ❩❛❤❧ χ(e) ❦ö♥♥t❡ ❞✐❡ ●❛♥③❤❡✐t ③❡rstör❡♥✳ s✐♥❞✳ ❉❛ ❆❜❡r ❞✐❡ ❉✐✈✐s✐♦♥ ❞✉r❝❤ ▲❡♠♠❛ ✺✳✽✳ ❙❡✐❡♥ C ✉♥❞ C ③✇❡✐ ✭♥✐❝❤t ♥♦t✇❡♥❞✐❣ ✈❡rs❝❤✐❡❞❡♥❡✮ ❑♦♥❥✉❣❛t✐♦♥s❦❧❛ss❡♥ ✈♦♥ G✳ ❙❡✐ ψ(g) ❞✐❡ ❆♥③❛❤❧ ❞❡r ❋❛❦t♦r✐s✐❡r✉♥❣❡♥ g = cc ♠✐t c ∈ C ✱ c ∈ C ✳ ❉❛♥♥ ✐st ψ ❡✐♥❡ ❑❧❛ss❡♥❢✉♥❦t✐♦♥✳ Pr♦♦❢✳ ❙❡✐❡♥ g ✉♥❞ h ❦♦♥❥✉❣✐❡rt✱ ❛❧s♦ h = g x ♠✐t x ∈ G✳ ❲❡❣❡♥ h = g x = (cc )x = cx c x x x ✐st (c, c ) → (c , c ) ❡✐♥❡ ❇✐❥❡❦t✐♦♥ ❞❡r P❛❛r❡ c, c ♠✐t g = cc ✉♥❞ ❞❡♥ ❡♥ts♣r❡❝❤❡♥❞❡♥ P❛❛r❡♥ ❢ür h✳ ▲❡♠♠❛ ✺✳✾✳ ❊s s❡✐❡♥ C1 , C2 , .

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Darstellungstheorie endlicher Gruppen by Peter Müller


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