By Peter Müller
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Extra info for Darstellungstheorie endlicher Gruppen
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 , .
6 ✼✳✸ ✷✳ ❖rt❤♦❣♦♥❛❧✐tätsr❡❧❛t✐♦♥ Sym3 ❦♦♥♥t❡♥ ✇✐r ❞✐❡ ❈❤❛r❛❦t❡rt❛❢❡❧ ♠ü❤❡❧♦s ❜❡st✐♠♠❡♥✳ ❋ür ❣röÿ❡r❡ ●r✉♣♣❡♥ ❛❧❧❡r❞✐♥❣s ❜❡♥öt✐❣❡♥ ✇✐r ✇❡✐t❡r❡ ❍✐❧❢s♠✐tt❡❧✳ ❉❡r ❢♦❧❣❡♥❞❡ ❙❛t③ ✐st ❋ür ❞✐❡ s❡❤r ❦❧❡✐♥❡ ●r✉♣♣❡ s♦✇♦❤❧ ❢ür ♣r❛❦t✐s❝❤❡ ❛❧s ❛✉❝❤ ❢ür t❤❡♦r❡t✐s❝❤❡ ❩✇❡❝❦❡ ✇✐❝❤t✐❣✳ ■♥s❜❡s♦♥❞❡r❡ ❦ö♥♥❡♥ ✇✐r ♠✐t ❍✐❧❢❡ ❞❡r ✐rr❡❞✉③✐❜❧❡♥ ❈❤❛r❛❦t❡r❡ ❞✐❡ ❑♦♥❥✉❣✐❡rt❤❡✐t ✈♦♥ ❊❧❡♠❡♥t❡♥ ❡r❦❡♥♥❡♥✳ ❙❛t③ ✼✳✶ ✭✷✳ ❖rt❤♦❣♦♥❛❧✐tätsr❡❧❛t✐♦♥✮ χ(g)χ(h) ¯ = χ∈■rr(G) ✳ ❋ür g, h ∈ G ❣✐❧t |CG (h)| ❢❛❧❧s g ✉♥❞ h ❦♦♥❥✉❣✐❡rt s✐♥❞ 0 s♦♥st ✸✼ Pr♦♦❢✳ g, h ∈ G ❣❡❣❡❜❡♥✳ ❙❡✐ C ❞✐❡ ❑♦♥❥✉❣❛t✐♦♥s❦❧❛ss❡ ✈♦♥ h✳ ❲✐r ❞❡✜♥✐❡r❡♥ ❡✐♥❡ ❑❧❛ss❡♥❢✉♥❦t✐♦♥ ψ ∈ C(G, C) ❞✉r❝❤ ψ(x) = |CG (h)| ❢❛❧❧s x ∈ C ✱ ✉♥❞ ψ(x) = 0✱ ✇❡♥♥ x∈ / C ✳ ❊s ❣✐❧t ψ= [ψ, χ]χ.
N✱ ❊❧❡♠❡♥t❡ ❞❡r ❖r❞♥✉♥❣ 2✳ ❲✐r ❜❡❤❛✉♣t❡♥✱ ❞❛ss τi τj ∈ F ❢ür i, j ✳ ❉❛s ✐st ❦❧❛r ❢ür i = j ✳ ❙❡✐ ♥✉♥ i = j ✱ ✉♥❞ τi τj ∈ / F ✳ ❉❛♥♥ ❣✐❧t τi τj ∈ Hk ❢ür ❡✐♥ −1 k ✳ ◆❛tür❧✐❝❤ ❣✐❧t ❞❛♥♥ ❛✉❝❤ (τi τj ) ∈ Hk ✳ ❆♥❞❡r❡rs❡✐ts ❣✐❧t (τi τj )−1 = τj τi = (τi τj )τi ✱ ❙❡✐ ❛❧s♦ [G : H]✱ G ❡✐♥❡ ❡♥❞❧✐❝❤❡ ❋r♦❜❡♥✐✉s❣r✉♣♣❡ ♠✐t ❑♦♠♣❧❡♠❡♥t ✉♥❞ ❛❧s♦ e = (τi τj )−1 ∈ Hk ∩ Hkτi , ✉♥❞ ❞❛❤❡r τi ∈ Hk ✳ ❆❜❡r ❞❛♥♥ ❣✐❧t ❛✉❝❤ τj ∈ Hk ✱ ❛❧s♦ i = k = j✱ ✐♠ ❲✐❞❡rs♣r✉❝❤ ③✉ i = j✳ j = 1, 2, . . , n s✐♥❞ ❞✐❡ n ❊❧❡♠❡♥t❡ τ1 τj ♣❛❛r✇❡✐s❡ ✈❡rs❝❤✐❡❞❡♥✳ ❆❜❡r |F | = n✱ ❞❛❤❡r ❜❡st❡❤t F ❣❡♥❛✉ ❛✉s ❞✐❡s❡♥ ❊❧❡♠❡♥t❡♥✳ ❆❜❡r ❞❛♠✐t s❡❤❡♥ ✇✐r✱ ❞❛ss F ♠✉❧t✐♣❧✐❦❛t✐✈ −1 ❛❜❣❡s❝❤❧♦ss❡♥ ✐st✿ ❙❡✐❡♥ a, b ∈ F ✳ ❉❛♥♥ ❣✐❧t ❛✉❝❤ a ∈ F ✱ ✉♥❞ ♥❛❝❤ ❞❡♠ ❣❡r❛❞❡ −1 ❣❡③❡✐❣t❡♥ ❣✐❜t ❡s j, k ♠✐t a = τ1 τj ✉♥❞ b = τ1 τk ✳ ❆✉s a = (τ1 τj )−1 = τj τ1 ❢♦❧❣t ❋ür ab = τj τ1 τ1 τk = τj τk ∈ F.