Darstellungstheorie endlicher Gruppen by Peter Müller

By Peter Müller

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T∈T ●✐❧t ❞✐❡ ❣❧❡✐❝❤❡ ❋♦r♠❡❧ ❛✉❝❤ ❞❛♥♥✱ ✇❡♥♥ ✈♦♥ H ✐♥ G T ❡✐♥ ❱❡rtr❡t❡rs②st❡♠ ❞❡r ❘❡❝❤ts♥❡❜❡♥❦❧❛ss❡♥ ✐st❄ ❊✐♥❡ ❢✉♥❞❛♠❡♥t❛❧❡ ❊✐❣❡♥s❝❤❛❢t ❞❡r ■♥❞✉❦t✐♦♥ ✈♦♥ ❑❧❛ss❡♥❢✉♥❦t✐♦♥❡♥ ✐st ❙❛t③ ✽✳✺ ✳ ✭❋r♦❜❡♥✐✉s✕❘❡③✐♣r♦③✐tät✮ θ ∈ C(G, C)✳ ❉❛♥♥ ❣✐❧t ❙❡✐ H ❡✐♥❡ ❯♥t❡r❣r✉♣♣❡ ✈♦♥ G✱ ψ ∈ C(H, C) ✉♥❞ [ψ G , θ]G = [ψ, θH ]H . ✹✷ Pr♦♦❢✳ [ψ G , θ]G = 1 ¯ ψ G (g)θ(g) |G| g∈G = 1 1 ¯ ψ 0 (g a )θ(g) |G| |H| g∈G a∈G = 1 1 ¯ ψ 0 (g a )θ(g) |G| |H| a∈G g∈G = 1 1 ¯ a−1 ) ψ 0 (g)θ(g |G| |H| a∈G g∈G = 1 1 ¯ ψ(g)θ(g) |G| |H| a∈G g∈H = 1 ¯ ψ(g)θ(g) |H| g∈H = [ψ, θH ]H ❑♦r♦❧❧❛r ✽✳✻✳ ❙❡✐ ψ ❡✐♥ ❈❤❛r❛❦t❡r ❞❡r ❯♥t❡r❣r✉♣♣❡ H ✈♦♥ G✳ ❉❛♥♥ ✐st ψ G ❡✐♥ ❈❤❛r❛❦✲ t❡r ✈♦♥ G✱ ♠✐t ψ G (e) = [G : H]ψ(e)✳ Pr♦♦❢✳ ψG aχ ∈ C ψG = χ∈■rr(G) aχ χ✳ ❉✐❡ ❖r✲ t❤♦♥♦r♠❛❧✐tät ❞❡r ✐rr❡❞✉③✐❜❧❡♥ ❈❤❛r❛❦t❡r❡✱ ③✉s❛♠♠❡♥ ♠✐t ❞❡r ❋r♦❜❡♥✐✉s✕❘❡③✐♣r♦③✐tät✱ ❉❛ ❡✐♥❡ ❑❧❛ss❡♥❢✉♥❦t✐♦♥ ✐st✱ ❣✐❜t ❡s ♠✐t ❡r❣✐❜t aχ = [ψ G , χ]G = [ψ, χH ]H ∈ N0 .

Rn ✱ ❛❧s♦ rri = nj=1 aij rj ✳ ❙❡✐ A ❞✐❡ ▼❛tr✐① ♠✐t ❊✐♥tr❛❣ aij ✐♥ P♦s✐t✐♦♥ (i, j)✱ ✉♥❞ v ❞❡r ❙♣❛❧t❡♥✈❡❦t♦r ♠✐t i✕t❡♠ ❊✐♥tr❛❣ ri ✳ ❉❛♥♥ ❣✐❧t rv = Av ✱ ❛❧s♦ det(rEn − A) = 0✳ ■st x ❡✐♥❡ ❱❛r✐❛❜❧❡✱ ❞❛♥♥ s✐❡❤t ♠❛♥ ✭③✳❇✳ ♠✐t ❞❡r ▲❡✐❜♥✐③✕❋♦r♠❡❧✮✱ ❞❛ss ❞❛s ❝❤❛r❛❦t❡r✐st✐s❝❤❡ P♦❧②♥♦♠ f (x) = det(xEn − A) ♥♦r♠✐❡rt ✐st ✉♥❞ ❣❛♥③③❛❤❧✐❣❡ ❑♦❡✣③✐❡♥t❡♥ ❤❛t✳ ❲❡❣❡♥ f (r) = 0 ❢♦❧❣t ❞✐❡ ❇❡❤❛✉♣t✉♥❣✳ ✸✶ ❙❛t③ ✺✳✹✳ ❊s s❡✐❡♥ α, β ∈ C ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤✳ ❉❛♥♥ ✐st Z[α, β] ❡♥❞❧✐❝❤ ü❜❡r Z✳ Pr♦♦❢✳ ❉❛ α, β ❣❛♥③ ❛❧❣❡❜r❛✐s❝❤ s✐♥❞✱ ❡①✐st✐❡r❡♥ m, n ≥ 1 ✉♥❞ P♦❧②♥♦♠❡ a, b ∈ Z[X] ♠✐t grad(a) < m✱ grad(b) < n ✉♥❞ αm = a(α)✱ β n = b(β)✳ ❍✐❡r❛✉s s✐❡❤t ♠❛♥ ✐♥❞✉❦t✐✈✱ ❞❛ss k 0 1 m−1 ❥❡❞❡ P♦t❡♥③ α ✭k ∈ N0 ✮ ❡✐♥❡ ❣❛♥③③❛❤❧✐❣❡ ▲✐♥❡❛r❦♦♠❜✐♥❛t✐♦♥ ✈♦♥ α , α , .

Rr(G) µG (e) = [G : P ]µ(e) = [G : P ] ❣✐❧t p µG (e)✱ ✉♥❞ ❞❛❤❡r ❣✐❧t ❢ür ♠✐♥❞❡st❡♥s ❡✐♥ χ ∈ ■rr(G)✱ ❞❛ss aχ = 0✱ ✉♥❞ p χ(e)✳ ❲❡❣❡♥ 0 = aχ = [µG , χ]G = [µ, χP ]P ✐st µ ❡✐♥ ❇❡st❛♥❞t❡✐❧ ✈♦♥ χP ✳ ❉❛ µ ❛❧s♦ ❡✐♥ ❣❡♠❡✐♥s❛♠❡r ❇❡st❛♥❞t❡✐❧ ✈♦♥ χP ✉♥❞ λP ✐st✱ ❣✐❧t 0 = [λP , χP ]P = [λ, χU ]U ✳ ❲✐r s❡❤❡♥✱ ❞❛ss λ ❡✐♥ ❇❡st❛♥❞t❡✐❧ ✈♦♥ χU ✐st✳ ❆❜❡r U ≤ Z(G)✱ ❛❧s♦ χU = χ(e)λ ♥❛❝❤ ❆✉❢❣❛❜❡ ✽✳✼✳ ■st ♥✉♥ φ : U → GL(V ) ❡✐♥❡ ❉❛rst❡❧❧✉♥❣ ♠✐t ❈❤❛r❛❦t❡r χU ✱ ❞❛♥♥ ❤❛t φ(u) ❢ür u ∈ U ❞❡♥ χ(e)✕❢❛❝❤❡♥ ❊✐❣❡♥✇❡rt λ(u)✱ ❛❧s♦ det(φ(u)) = λ(u)χ(e) ✳ ❆♥❞❡r❡rs❡✐ts ❣✐❧t φ(U ) ≤ SL(V ) ✇❡❣❡♥ U ≤ G ✱ ❛❧s♦ λ(u)χ(e) = det(φ(u)) = 1✳ ❋ür u = e ✐st λ(u) ❡✐♥❡ ♣r✐♠✐t✐✈❡ p✕t❡ ❊✐♥❤❡✐ts✇✉r③❡❧✱ ❛❧s♦ χ(e) ❞✉r❝❤ p t❡✐❧❜❛r✱ ✐♠ ●❡❣❡♥s❛t③ ③✉r ❲❛❤❧ ✈♦♥ χ✳ ❲❡❣❡♥ ❉❛s ♥ä❝❤st❡ ▲❡♠♠❛ ✉♥t❡rs✉❝❤t ❞❡♥ ❩✉s❛♠♠❡♥❤❛♥❣ ③✇✐s❝❤❡♥ ❞❡♥ ❑❡r♥❡♥ ✐♥❞✉③✐❡rt❡r ❈❤❛r❛❦t❡r❡✳ ▲❡♠♠❛ ✽✳✾✳ ❙❡✐ ψ ❡✐♥ ❈❤❛r❛❦t❡r ❞❡r ❯♥t❡r❣r✉♣♣❡ H ✈♦♥ G✳ ❉❛♥♥ ❣✐❧t ❑❡r♥(ψ G ) = ❑❡r♥(ψ)g .

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