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rs s rt r - - PowerPoint PPT Presentation

Jan 19, 2024 477 likes •1.12k views

rs s rt r trss r s t rt t

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SLIDE 1

■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

◆ór❛ ❙③❛❦á❝s ✭✇✐t❤ ❘♦❜❡rt ❏❛❥❝❛②✱ ❚❛t✐❛♥❛ ❏❛❥❝❛②♦✈❛ ❛♥❞ ▼ár✐❛ ❙③❡♥❞r❡✐✮

❯♥✐✈❡rs✐t② ♦❢ ❙③❡❣❡❞✱ ❇♦❧②❛✐ ■♥st✐t✉t❡

■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❙❡♠✐❣r♦✉♣s ✷✵✶✽✱ ▲✐s❜♦♥

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-2
SLIDE 2

P❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❣r❛♣❤s

❉❡✜♥✐t✐♦♥

  • r❛♣❤✿

◮ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡✿ Γ = (V , E)✱ ✇❤❡r❡ E ⊆

V

✷

  • ❞✐r❡❝t❡❞✿

✱ ✇❤❡r❡ ♠❛♣s ❛r❡ ❣✐✈❡♥ ❞✐r❡❝t❡❞✱ ❡❞❣❡✲❝♦❧♦r❡❞✿

✶

  • r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ❜✐❥❡❝t✐♦♥

✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ P❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ♣❛rt✐❛❧ ♦♥❡✲t♦✲♦♥❡ ♠❛♣ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 3

P❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❣r❛♣❤s

❉❡✜♥✐t✐♦♥

  • r❛♣❤✿

◮ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡✿ Γ = (V , E)✱ ✇❤❡r❡ E ⊆

V

✷

  • ◮ ❞✐r❡❝t❡❞✿ Γ = (V , E)✱ ✇❤❡r❡ ♠❛♣s α, ω: E → V ❛r❡ ❣✐✈❡♥

❞✐r❡❝t❡❞✱ ❡❞❣❡✲❝♦❧♦r❡❞✿

✶

  • r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ❜✐❥❡❝t✐♦♥

✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ P❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ♣❛rt✐❛❧ ♦♥❡✲t♦✲♦♥❡ ♠❛♣ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 4

P❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❣r❛♣❤s

❉❡✜♥✐t✐♦♥

  • r❛♣❤✿

◮ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡✿ Γ = (V , E)✱ ✇❤❡r❡ E ⊆

V

✷

  • ◮ ❞✐r❡❝t❡❞✿ Γ = (V , E)✱ ✇❤❡r❡ ♠❛♣s α, ω: E → V ❛r❡ ❣✐✈❡♥

◮ ❞✐r❡❝t❡❞✱ ❡❞❣❡✲❝♦❧♦r❡❞✿ Γ = (V ; E✶, · · · , En)

  • r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ❜✐❥❡❝t✐♦♥

✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ P❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ♣❛rt✐❛❧ ♦♥❡✲t♦✲♦♥❡ ♠❛♣ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 5

P❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❣r❛♣❤s

❉❡✜♥✐t✐♦♥

  • r❛♣❤✿

◮ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡✿ Γ = (V , E)✱ ✇❤❡r❡ E ⊆

V

✷

  • ◮ ❞✐r❡❝t❡❞✿ Γ = (V , E)✱ ✇❤❡r❡ ♠❛♣s α, ω: E → V ❛r❡ ❣✐✈❡♥

◮ ❞✐r❡❝t❡❞✱ ❡❞❣❡✲❝♦❧♦r❡❞✿ Γ = (V ; E✶, · · · , En)

  • r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ❜✐❥❡❝t✐♦♥ ϕ: V → V ✇❤✐❝❤ ♣r❡s❡r✈❡s

❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ P❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ♣❛rt✐❛❧ ♦♥❡✲t♦✲♦♥❡ ♠❛♣ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-6
SLIDE 6

P❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ ❣r❛♣❤s

❉❡✜♥✐t✐♦♥

  • r❛♣❤✿

◮ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡✿ Γ = (V , E)✱ ✇❤❡r❡ E ⊆

V

✷

  • ◮ ❞✐r❡❝t❡❞✿ Γ = (V , E)✱ ✇❤❡r❡ ♠❛♣s α, ω: E → V ❛r❡ ❣✐✈❡♥

◮ ❞✐r❡❝t❡❞✱ ❡❞❣❡✲❝♦❧♦r❡❞✿ Γ = (V ; E✶, · · · , En)

  • r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ❜✐❥❡❝t✐♦♥ ϕ: V → V ✇❤✐❝❤ ♣r❡s❡r✈❡s

❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳ P❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠✿ ❛ ♣❛rt✐❛❧ ♦♥❡✲t♦✲♦♥❡ ♠❛♣ ψ: V → V ✇❤✐❝❤ ♣r❡s❡r✈❡s ❡❞❣❡s ✭✇✐t❤ ❞✐r❡❝t✐♦♥✴❝♦❧♦r✴♠✉❧t✐♣❧✐❝✐t②✮ ❛♥❞ ♥♦♥✲❡❞❣❡s✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 7

❲❤② ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s❄

◮ t♦ ❤❛✈❡ ❛♥ ❛❧❣❡❜r❛✐❝ t♦♦❧ t♦ ✉♥❞❡rst❛♥❞ ❣r❛♣❤s ✇✐t❤ tr✐✈✐❛❧

❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s t❤❡ ❣r❛♣❤ r❡❝♦♥str✉❝t✐♦♥ ❝♦♥❥❡❝t✉r❡✿ ▲❡t ❜❡ ❛♥ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡ ❣r❛♣❤ ♦♥ ♣♦✐♥ts✳ ❉❡❝❦ ✿ ♠✉❧t✐s❡t ♦❢ s♣❛♥♥❡❞ s✉❜❣r❛♣❤s ♦❢ ♦♥ ✶ ♣♦✐♥ts ❈♦♥❥❡❝t✉r❡✿ ❉❡❝❦ ❞❡t❡r♠✐♥❡s ✳ ❚❤❡ r❡❛s♦♥ t❤✐s ✐s ❤❛r❞✿ s♦♠❡ ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ s✉❜❣r❛♣❤s ♦♥ ✶ ♣♦✐♥ts ❞♦♥✬t ❡①t❡♥❞ t♦ ❛✉t♦♠♦r♣❤✐s♠s✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 8

❲❤② ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s❄

◮ t♦ ❤❛✈❡ ❛♥ ❛❧❣❡❜r❛✐❝ t♦♦❧ t♦ ✉♥❞❡rst❛♥❞ ❣r❛♣❤s ✇✐t❤ tr✐✈✐❛❧

❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s

◮ t❤❡ ❣r❛♣❤ r❡❝♦♥str✉❝t✐♦♥ ❝♦♥❥❡❝t✉r❡✿

▲❡t Γ ❜❡ ❛♥ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡ ❣r❛♣❤ ♦♥ n ♣♦✐♥ts✳ ❉❡❝❦(Γ)✿ ♠✉❧t✐s❡t ♦❢ s♣❛♥♥❡❞ s✉❜❣r❛♣❤s ♦❢ Γ ♦♥ n − ✶ ♣♦✐♥ts ❈♦♥❥❡❝t✉r❡✿ ❉❡❝❦(Γ) ❞❡t❡r♠✐♥❡s Γ✳ ❚❤❡ r❡❛s♦♥ t❤✐s ✐s ❤❛r❞✿ s♦♠❡ ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ s✉❜❣r❛♣❤s ♦♥ ✶ ♣♦✐♥ts ❞♦♥✬t ❡①t❡♥❞ t♦ ❛✉t♦♠♦r♣❤✐s♠s✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-9
SLIDE 9

❲❤② ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s❄

◮ t♦ ❤❛✈❡ ❛♥ ❛❧❣❡❜r❛✐❝ t♦♦❧ t♦ ✉♥❞❡rst❛♥❞ ❣r❛♣❤s ✇✐t❤ tr✐✈✐❛❧

❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s

◮ t❤❡ ❣r❛♣❤ r❡❝♦♥str✉❝t✐♦♥ ❝♦♥❥❡❝t✉r❡✿

▲❡t Γ ❜❡ ❛♥ ✉♥❞✐r❡❝t❡❞✱ s✐♠♣❧❡ ❣r❛♣❤ ♦♥ n ♣♦✐♥ts✳ ❉❡❝❦(Γ)✿ ♠✉❧t✐s❡t ♦❢ s♣❛♥♥❡❞ s✉❜❣r❛♣❤s ♦❢ Γ ♦♥ n − ✶ ♣♦✐♥ts ❈♦♥❥❡❝t✉r❡✿ ❉❡❝❦(Γ) ❞❡t❡r♠✐♥❡s Γ✳ ❚❤❡ r❡❛s♦♥ t❤✐s ✐s ❤❛r❞✿ s♦♠❡ ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ❜❡t✇❡❡♥ s✉❜❣r❛♣❤s ♦♥ n − ✶ ♣♦✐♥ts ❞♦♥✬t ❡①t❡♥❞ t♦ ❛✉t♦♠♦r♣❤✐s♠s✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 10

❚❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣

Aut(Γ)✿ t❤❡ ❣r♦✉♣ ♦❢ ❛❧❧ ❛✉t♦♠♦r♣✐s♠s ♦❢ Γ✳

❚❤❡♦r❡♠ ✭❋r✉❝❤t✱ ✶✾✸✾✮

❋♦r ❛❧❧ ✜♥✐t❡ ❣r♦✉♣s G✱ t❤❡r❡ ❡①✐sts ❛ ❣r❛♣❤ Γ s✉❝❤ t❤❛t G ∼ = Aut(Γ) ✭♠♦r❡♦✈❡r✱ Γ ❝❛♥ ❜❡ ❝❤♦s❡♥ t♦ ❜❡ ✸✲r❡❣✉❧❛r✮✳ ❆ ♠♦r❡ ❞✐✣❝✉❧t q✉❡st✐♦♥✿ ■❢ ✱ t❤❡♥ ✳ ●✐✈❡♥ ❛ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ ✱ ❞♦❡s t❤❡r❡ ❡①✐st ❛ ❣r❛♣❤ ♦♥ ✈❡rt✐❝❡s ❢♦r ✇❤✐❝❤ ❄ ❚❤❡r❡ ✐s ❛ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r ❡❞❣❡✲❝♦❧♦r❡❞ ❣r❛♣❤s✱ ✐♥ ❣❡♥❡r❛❧ ✭t♦ ♠② ❦♥♦✇❧❡❞❣❡✮ s✉❝❤ ✐s ♥♦t ❦♥♦✇♥✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 11

❚❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣

Aut(Γ)✿ t❤❡ ❣r♦✉♣ ♦❢ ❛❧❧ ❛✉t♦♠♦r♣✐s♠s ♦❢ Γ✳

❚❤❡♦r❡♠ ✭❋r✉❝❤t✱ ✶✾✸✾✮

❋♦r ❛❧❧ ✜♥✐t❡ ❣r♦✉♣s G✱ t❤❡r❡ ❡①✐sts ❛ ❣r❛♣❤ Γ s✉❝❤ t❤❛t G ∼ = Aut(Γ) ✭♠♦r❡♦✈❡r✱ Γ ❝❛♥ ❜❡ ❝❤♦s❡♥ t♦ ❜❡ ✸✲r❡❣✉❧❛r✮✳ ❆ ♠♦r❡ ❞✐✣❝✉❧t q✉❡st✐♦♥✿ ■❢ G = Aut(Γ)✱ t❤❡♥ G ≤ SV ✳ ●✐✈❡♥ ❛ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ G ≤ Sn✱ ❞♦❡s t❤❡r❡ ❡①✐st ❛ ❣r❛♣❤ Γ ♦♥ n ✈❡rt✐❝❡s ❢♦r ✇❤✐❝❤ G = Aut(Γ)❄ ❚❤❡r❡ ✐s ❛ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r ❡❞❣❡✲❝♦❧♦r❡❞ ❣r❛♣❤s✱ ✐♥ ❣❡♥❡r❛❧ ✭t♦ ♠② ❦♥♦✇❧❡❞❣❡✮ s✉❝❤ ✐s ♥♦t ❦♥♦✇♥✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 12

❚❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣

Aut(Γ)✿ t❤❡ ❣r♦✉♣ ♦❢ ❛❧❧ ❛✉t♦♠♦r♣✐s♠s ♦❢ Γ✳

❚❤❡♦r❡♠ ✭❋r✉❝❤t✱ ✶✾✸✾✮

❋♦r ❛❧❧ ✜♥✐t❡ ❣r♦✉♣s G✱ t❤❡r❡ ❡①✐sts ❛ ❣r❛♣❤ Γ s✉❝❤ t❤❛t G ∼ = Aut(Γ) ✭♠♦r❡♦✈❡r✱ Γ ❝❛♥ ❜❡ ❝❤♦s❡♥ t♦ ❜❡ ✸✲r❡❣✉❧❛r✮✳ ❆ ♠♦r❡ ❞✐✣❝✉❧t q✉❡st✐♦♥✿ ■❢ G = Aut(Γ)✱ t❤❡♥ G ≤ SV ✳ ●✐✈❡♥ ❛ ♣❡r♠✉t❛t✐♦♥ ❣r♦✉♣ G ≤ Sn✱ ❞♦❡s t❤❡r❡ ❡①✐st ❛ ❣r❛♣❤ Γ ♦♥ n ✈❡rt✐❝❡s ❢♦r ✇❤✐❝❤ G = Aut(Γ)❄ ❚❤❡r❡ ✐s ❛ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r ❡❞❣❡✲❝♦❧♦r❡❞ ❣r❛♣❤s✱ ✐♥ ❣❡♥❡r❛❧ ✭t♦ ♠② ❦♥♦✇❧❡❞❣❡✮ s✉❝❤ ✐s ♥♦t ❦♥♦✇♥✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 13

❚❤❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦❢ ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s

PAut(Γ)✿ t❤❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦❢ ❛❧❧ ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ Γ✳ ◆♦t❡✿ PAut(Γ) ≤ IV ✱ t❤❡ s②♠♠❡tr✐❝ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦♥ V ✳ ◗✉❡st✐♦♥s✿ ❋♦r ✇❤✐❝❤ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ❞♦❡s t❤❡r❡ ❡①✐st ❛ ❣r❛♣❤ s✉❝❤ t❤❛t ❄ ❋♦r ✇❤✐❝❤ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞s ♦❢ ❞♦❡s t❤❡r❡ ❡①✐st ❛ ❣r❛♣❤ ♦♥ s✉❝❤ t❤❛t ❄

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 14

❚❤❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦❢ ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s

PAut(Γ)✿ t❤❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦❢ ❛❧❧ ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ Γ✳ ◆♦t❡✿ PAut(Γ) ≤ IV ✱ t❤❡ s②♠♠❡tr✐❝ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦♥ V ✳ ◗✉❡st✐♦♥s✿

◮ ❋♦r ✇❤✐❝❤ ✐♥✈❡rs❡ ♠♦♥♦✐❞s S ❞♦❡s t❤❡r❡ ❡①✐st ❛ ❣r❛♣❤ Γ s✉❝❤

t❤❛t PAut(Γ) ∼ = S❄ ❋♦r ✇❤✐❝❤ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞s ♦❢ ❞♦❡s t❤❡r❡ ❡①✐st ❛ ❣r❛♣❤ ♦♥ s✉❝❤ t❤❛t ❄

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 15

❚❤❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦❢ ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s

PAut(Γ)✿ t❤❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦❢ ❛❧❧ ♣❛rt✐❛❧ ❛✉t♦♠♦r♣❤✐s♠s ♦❢ Γ✳ ◆♦t❡✿ PAut(Γ) ≤ IV ✱ t❤❡ s②♠♠❡tr✐❝ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦♥ V ✳ ◗✉❡st✐♦♥s✿

◮ ❋♦r ✇❤✐❝❤ ✐♥✈❡rs❡ ♠♦♥♦✐❞s S ❞♦❡s t❤❡r❡ ❡①✐st ❛ ❣r❛♣❤ Γ s✉❝❤

t❤❛t PAut(Γ) ∼ = S❄

◮ ❋♦r ✇❤✐❝❤ ✐♥✈❡rs❡ s✉❜♠♦♥♦✐❞s S ♦❢ IV ❞♦❡s t❤❡r❡ ❡①✐st ❛ ❣r❛♣❤

Γ ♦♥ V s✉❝❤ t❤❛t PAut(Γ) = S❄

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 16

❖✈❡r✈✐❡✇✿ t❤❡ str✉❝t✉r❡ ♦❢ IX

■❞❡♠♣♦t❡♥ts✿ ♣❛rt✐❛❧ ✐❞❡♥t✐❝❛❧ ♠❛♣s ❘❛♥❦ ♦❢ ❛ ♠❛♣✿ ❝❛r❞✐♥❛❧✐t② ♦❢ ✐ts ❞♦♠❛✐♥✴✐♠❛❣❡

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 17

❖✈❡r✈✐❡✇✿ t❤❡ str✉❝t✉r❡ ♦❢ IX

■❞❡♠♣♦t❡♥ts✿ ♣❛rt✐❛❧ ✐❞❡♥t✐❝❛❧ ♠❛♣s

◮ a L b ⇐

⇒ dom(a) = dom(b)

◮ a R b ⇐

⇒ ran(a) = ran(b)

◮ a H b ⇐

⇒ dom(a) = dom(b) ∧ ran(a) = ran(b) ❘❛♥❦ ♦❢ ❛ ♠❛♣✿ ❝❛r❞✐♥❛❧✐t② ♦❢ ✐ts ❞♦♠❛✐♥✴✐♠❛❣❡

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 18

❖✈❡r✈✐❡✇✿ t❤❡ str✉❝t✉r❡ ♦❢ IX

■❞❡♠♣♦t❡♥ts✿ ♣❛rt✐❛❧ ✐❞❡♥t✐❝❛❧ ♠❛♣s

◮ a L b ⇐

⇒ dom(a) = dom(b)

◮ a R b ⇐

⇒ ran(a) = ran(b)

◮ a H b ⇐

⇒ dom(a) = dom(b) ∧ ran(a) = ran(b)

◮ a D b ⇐

⇒ |dom(a)| = |dom(b)| ❘❛♥❦ ♦❢ ❛ ♠❛♣✿ ❝❛r❞✐♥❛❧✐t② ♦❢ ✐ts ❞♦♠❛✐♥✴✐♠❛❣❡

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 19

❖✈❡r✈✐❡✇✿ t❤❡ str✉❝t✉r❡ ♦❢ IX

■❞❡♠♣♦t❡♥ts✿ ♣❛rt✐❛❧ ✐❞❡♥t✐❝❛❧ ♠❛♣s

◮ a L b ⇐

⇒ dom(a) = dom(b)

◮ a R b ⇐

⇒ ran(a) = ran(b)

◮ a H b ⇐

⇒ dom(a) = dom(b) ∧ ran(a) = ran(b)

◮ a D b ⇐

⇒ |dom(a)| = |dom(b)| ❘❛♥❦ ♦❢ ❛ ♠❛♣✿ ❝❛r❞✐♥❛❧✐t② ♦❢ ✐ts ❞♦♠❛✐♥✴✐♠❛❣❡

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 20

❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ PAut(Γ)

❙✉♣♣♦s❡ Γ ✐s ❛♥ ✉♥❞✐r❡❝t❡❞✱ ✜♥✐t❡ ❣r❛♣❤✳ ❚❤❡♥ PAut(Γ) ≤ IV ✱ ❛♥❞ ❡✈❡r② ✐❞❡♠♣♦t❡♥t ♦❢ ✐s ✐♥ ✱ ❡✈❡r② ♠❛♣ ♦❢ r❛♥❦ ✶ ✐♥ ✐s ✐♥ ✱ ✲✱ ✲ ❛♥❞ ✲❝❧❛ss❡s ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ ✱ ✲❝❧❛ss❡s ♦❢ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ♦❢ ✱ t❤❡ ❣r♦✉♣ ✲❝❧❛ss❡s ♦❢ ❛ ✲❝❧❛ss ❛r❡ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❣r♦✉♣s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✉❜❣r❛♣❤s✳ ◆♦t❡ t❤❛t t❤❡ r❛♥❦ ✷ ♠❛♣s ♦❢ ❞❡t❡r♠✐♥❡ ✉♥✐q✉❡❧② ✉♣ t♦ t❛❦✐♥❣ t❤❡ ❝♦♠♣❧❡♠❡♥t✳

❚❤❡♦r❡♠

■❢ ❢♦r ❣r❛♣❤s

✶ ✷ ✇❡ ❤❛✈❡ ✶ ✷ ✱ t❤❡♥ ✶ ✷ ♦r ✶ ✷✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 21

❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ PAut(Γ)

❙✉♣♣♦s❡ Γ ✐s ❛♥ ✉♥❞✐r❡❝t❡❞✱ ✜♥✐t❡ ❣r❛♣❤✳ ❚❤❡♥ PAut(Γ) ≤ IV ✱ ❛♥❞

◮ ❡✈❡r② ✐❞❡♠♣♦t❡♥t ♦❢ IV ✐s ✐♥ PAut(Γ)✱

❡✈❡r② ♠❛♣ ♦❢ r❛♥❦ ✶ ✐♥ ✐s ✐♥ ✱ ✲✱ ✲ ❛♥❞ ✲❝❧❛ss❡s ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ ✱ ✲❝❧❛ss❡s ♦❢ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ♦❢ ✱ t❤❡ ❣r♦✉♣ ✲❝❧❛ss❡s ♦❢ ❛ ✲❝❧❛ss ❛r❡ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❣r♦✉♣s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✉❜❣r❛♣❤s✳ ◆♦t❡ t❤❛t t❤❡ r❛♥❦ ✷ ♠❛♣s ♦❢ ❞❡t❡r♠✐♥❡ ✉♥✐q✉❡❧② ✉♣ t♦ t❛❦✐♥❣ t❤❡ ❝♦♠♣❧❡♠❡♥t✳

❚❤❡♦r❡♠

■❢ ❢♦r ❣r❛♣❤s

✶ ✷ ✇❡ ❤❛✈❡ ✶ ✷ ✱ t❤❡♥ ✶ ✷ ♦r ✶ ✷✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-22
SLIDE 22

❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ PAut(Γ)

❙✉♣♣♦s❡ Γ ✐s ❛♥ ✉♥❞✐r❡❝t❡❞✱ ✜♥✐t❡ ❣r❛♣❤✳ ❚❤❡♥ PAut(Γ) ≤ IV ✱ ❛♥❞

◮ ❡✈❡r② ✐❞❡♠♣♦t❡♥t ♦❢ IV ✐s ✐♥ PAut(Γ)✱ ◮ ❡✈❡r② ♠❛♣ ♦❢ r❛♥❦ ✶ ✐♥ IV ✐s ✐♥ PAut(Γ)✱

✲✱ ✲ ❛♥❞ ✲❝❧❛ss❡s ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ ✱ ✲❝❧❛ss❡s ♦❢ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ♦❢ ✱ t❤❡ ❣r♦✉♣ ✲❝❧❛ss❡s ♦❢ ❛ ✲❝❧❛ss ❛r❡ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❣r♦✉♣s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✉❜❣r❛♣❤s✳ ◆♦t❡ t❤❛t t❤❡ r❛♥❦ ✷ ♠❛♣s ♦❢ ❞❡t❡r♠✐♥❡ ✉♥✐q✉❡❧② ✉♣ t♦ t❛❦✐♥❣ t❤❡ ❝♦♠♣❧❡♠❡♥t✳

❚❤❡♦r❡♠

■❢ ❢♦r ❣r❛♣❤s

✶ ✷ ✇❡ ❤❛✈❡ ✶ ✷ ✱ t❤❡♥ ✶ ✷ ♦r ✶ ✷✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-23
SLIDE 23

❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ PAut(Γ)

❙✉♣♣♦s❡ Γ ✐s ❛♥ ✉♥❞✐r❡❝t❡❞✱ ✜♥✐t❡ ❣r❛♣❤✳ ❚❤❡♥ PAut(Γ) ≤ IV ✱ ❛♥❞

◮ ❡✈❡r② ✐❞❡♠♣♦t❡♥t ♦❢ IV ✐s ✐♥ PAut(Γ)✱ ◮ ❡✈❡r② ♠❛♣ ♦❢ r❛♥❦ ✶ ✐♥ IV ✐s ✐♥ PAut(Γ)✱ ◮ R✲✱ L✲ ❛♥❞ H −✲❝❧❛ss❡s ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ IV ✱

✲❝❧❛ss❡s ♦❢ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ♦❢ ✱ t❤❡ ❣r♦✉♣ ✲❝❧❛ss❡s ♦❢ ❛ ✲❝❧❛ss ❛r❡ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❣r♦✉♣s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✉❜❣r❛♣❤s✳ ◆♦t❡ t❤❛t t❤❡ r❛♥❦ ✷ ♠❛♣s ♦❢ ❞❡t❡r♠✐♥❡ ✉♥✐q✉❡❧② ✉♣ t♦ t❛❦✐♥❣ t❤❡ ❝♦♠♣❧❡♠❡♥t✳

❚❤❡♦r❡♠

■❢ ❢♦r ❣r❛♣❤s

✶ ✷ ✇❡ ❤❛✈❡ ✶ ✷ ✱ t❤❡♥ ✶ ✷ ♦r ✶ ✷✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-24
SLIDE 24

❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ PAut(Γ)

❙✉♣♣♦s❡ Γ ✐s ❛♥ ✉♥❞✐r❡❝t❡❞✱ ✜♥✐t❡ ❣r❛♣❤✳ ❚❤❡♥ PAut(Γ) ≤ IV ✱ ❛♥❞

◮ ❡✈❡r② ✐❞❡♠♣♦t❡♥t ♦❢ IV ✐s ✐♥ PAut(Γ)✱ ◮ ❡✈❡r② ♠❛♣ ♦❢ r❛♥❦ ✶ ✐♥ IV ✐s ✐♥ PAut(Γ)✱ ◮ R✲✱ L✲ ❛♥❞ H −✲❝❧❛ss❡s ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ IV ✱ ◮ D✲❝❧❛ss❡s ♦❢ PAut(Γ) ←

→ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ♦❢ PAut(Γ)✱ t❤❡ ❣r♦✉♣ ✲❝❧❛ss❡s ♦❢ ❛ ✲❝❧❛ss ❛r❡ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❣r♦✉♣s ♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✉❜❣r❛♣❤s✳ ◆♦t❡ t❤❛t t❤❡ r❛♥❦ ✷ ♠❛♣s ♦❢ ❞❡t❡r♠✐♥❡ ✉♥✐q✉❡❧② ✉♣ t♦ t❛❦✐♥❣ t❤❡ ❝♦♠♣❧❡♠❡♥t✳

❚❤❡♦r❡♠

■❢ ❢♦r ❣r❛♣❤s

✶ ✷ ✇❡ ❤❛✈❡ ✶ ✷ ✱ t❤❡♥ ✶ ✷ ♦r ✶ ✷✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-25
SLIDE 25

❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ PAut(Γ)

❙✉♣♣♦s❡ Γ ✐s ❛♥ ✉♥❞✐r❡❝t❡❞✱ ✜♥✐t❡ ❣r❛♣❤✳ ❚❤❡♥ PAut(Γ) ≤ IV ✱ ❛♥❞

◮ ❡✈❡r② ✐❞❡♠♣♦t❡♥t ♦❢ IV ✐s ✐♥ PAut(Γ)✱ ◮ ❡✈❡r② ♠❛♣ ♦❢ r❛♥❦ ✶ ✐♥ IV ✐s ✐♥ PAut(Γ)✱ ◮ R✲✱ L✲ ❛♥❞ H −✲❝❧❛ss❡s ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ IV ✱ ◮ D✲❝❧❛ss❡s ♦❢ PAut(Γ) ←

→ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ♦❢ PAut(Γ)✱

◮ t❤❡ ❣r♦✉♣ H✲❝❧❛ss❡s ♦❢ ❛ D✲❝❧❛ss ❛r❡ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❣r♦✉♣s

♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✉❜❣r❛♣❤s✳ ◆♦t❡ t❤❛t t❤❡ r❛♥❦ ✷ ♠❛♣s ♦❢ ❞❡t❡r♠✐♥❡ ✉♥✐q✉❡❧② ✉♣ t♦ t❛❦✐♥❣ t❤❡ ❝♦♠♣❧❡♠❡♥t✳

❚❤❡♦r❡♠

■❢ ❢♦r ❣r❛♣❤s

✶ ✷ ✇❡ ❤❛✈❡ ✶ ✷ ✱ t❤❡♥ ✶ ✷ ♦r ✶ ✷✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-26
SLIDE 26

❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ PAut(Γ)

❙✉♣♣♦s❡ Γ ✐s ❛♥ ✉♥❞✐r❡❝t❡❞✱ ✜♥✐t❡ ❣r❛♣❤✳ ❚❤❡♥ PAut(Γ) ≤ IV ✱ ❛♥❞

◮ ❡✈❡r② ✐❞❡♠♣♦t❡♥t ♦❢ IV ✐s ✐♥ PAut(Γ)✱ ◮ ❡✈❡r② ♠❛♣ ♦❢ r❛♥❦ ✶ ✐♥ IV ✐s ✐♥ PAut(Γ)✱ ◮ R✲✱ L✲ ❛♥❞ H −✲❝❧❛ss❡s ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ IV ✱ ◮ D✲❝❧❛ss❡s ♦❢ PAut(Γ) ←

→ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ♦❢ PAut(Γ)✱

◮ t❤❡ ❣r♦✉♣ H✲❝❧❛ss❡s ♦❢ ❛ D✲❝❧❛ss ❛r❡ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❣r♦✉♣s

♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✉❜❣r❛♣❤s✳ ◆♦t❡ t❤❛t t❤❡ r❛♥❦ ≤ ✷ ♠❛♣s ♦❢ PAut(Γ) ❞❡t❡r♠✐♥❡ Γ ✉♥✐q✉❡❧② ✉♣ t♦ t❛❦✐♥❣ t❤❡ ❝♦♠♣❧❡♠❡♥t✳

❚❤❡♦r❡♠

■❢ ❢♦r ❣r❛♣❤s

✶ ✷ ✇❡ ❤❛✈❡ ✶ ✷ ✱ t❤❡♥ ✶ ✷ ♦r ✶ ✷✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-27
SLIDE 27

❙♦♠❡ ♣r♦♣❡rt✐❡s ♦❢ PAut(Γ)

❙✉♣♣♦s❡ Γ ✐s ❛♥ ✉♥❞✐r❡❝t❡❞✱ ✜♥✐t❡ ❣r❛♣❤✳ ❚❤❡♥ PAut(Γ) ≤ IV ✱ ❛♥❞

◮ ❡✈❡r② ✐❞❡♠♣♦t❡♥t ♦❢ IV ✐s ✐♥ PAut(Γ)✱ ◮ ❡✈❡r② ♠❛♣ ♦❢ r❛♥❦ ✶ ✐♥ IV ✐s ✐♥ PAut(Γ)✱ ◮ R✲✱ L✲ ❛♥❞ H −✲❝❧❛ss❡s ❛r❡ t❤❡ s❛♠❡ ❛s ✐♥ IV ✱ ◮ D✲❝❧❛ss❡s ♦❢ PAut(Γ) ←

→ ✐s♦♠♦r♣❤✐s♠ ❝❧❛ss❡s ♦❢ PAut(Γ)✱

◮ t❤❡ ❣r♦✉♣ H✲❝❧❛ss❡s ♦❢ ❛ D✲❝❧❛ss ❛r❡ t❤❡ ✐s♦♠♦r♣❤✐s♠ ❣r♦✉♣s

♦❢ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ s✉❜❣r❛♣❤s✳ ◆♦t❡ t❤❛t t❤❡ r❛♥❦ ≤ ✷ ♠❛♣s ♦❢ PAut(Γ) ❞❡t❡r♠✐♥❡ Γ ✉♥✐q✉❡❧② ✉♣ t♦ t❛❦✐♥❣ t❤❡ ❝♦♠♣❧❡♠❡♥t✳

❚❤❡♦r❡♠

■❢ ❢♦r ❣r❛♣❤s Γ✶, Γ✷ ✇❡ ❤❛✈❡ PAut(Γ✶) = PAut(Γ✷)✱ t❤❡♥ Γ✶ = Γ✷ ♦r Γ✶ = Γ✷✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-28
SLIDE 28

(1) [12) [21) [31) [32) [34) [41) [42) [43) [23) [24) [13) [14) (2) (3) (4) (1)(2) (1)(3) (1)(4) (2)(4) (3)(4) (2)(3) (12) (13) (14) (24) (34) (23) [123) [314) [241) [341) [342) [413) [142) [143) [243) [431) [134) [321) (2)[13) (1)[34) (4)[21) (4)[31) (4)[32) (1)[43) (4)[12) (3)[41) (3)[14) (4)[13) (4)[23) (2)[31) [12)[34) [21)[43) [14)[32) [41)[23) (1)(2)(3) (2)(13) (1)(2)(4) (2)(3)(4) (4)(12) (4)(23) (2)(4)[13) (2)(4)[31) (4)[123) (4)[321) (1)(3)(4) (134) (431) (1)(34) (3)(14) (4)(13) (2)(4)(13) (1)(2)(3)(4)

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-29
SLIDE 29

❈♦♠♣❛t✐❜❧❡ ♠❛♣s

❍♦✇ ❝❛♥ ♦♥❡ ✭❛❧❣❡❜r❛✐❝❛❧❧②✮ ♦❜t❛✐♥ PAut(Γ) ❢r♦♠ t❤❡ ❧♦✇✲r❛♥❦ ♠❛♣s❄

❉❡✜♥✐t✐♦♥

❊❧❡♠❡♥ts ♦❢ ❛r❡ ❝❛❧❧❡❞ ❝♦♠♣❛t✐❜❧❡ ✭♥♦t✳✿ ✮ ✐❢ ✳

Pr♦♣♦s✐t✐♦♥

✶ ❛♥❞ ✶

❛r❡ ✐❞❡♠♣♦t❡♥ts

Pr♦♣♦s✐t✐♦♥

❙✉♣♣♦s❡ ✐s ♦❢ r❛♥❦ ✸✳ ❚❤❡♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ ✐ts r❛♥❦ ✷ r❡str✐❝t✐♦♥s ❛r❡ ✐♥ ✳

Pr♦♣♦s✐t✐♦♥

❋♦r ❛❧❧ ♣❛✐r✇✐s❡ ❝♦♠♣❛t✐❜❧❡ r❛♥❦ ✶ ♠❛♣s✱ ✐✛ ❢♦r ❛❧❧

✶ ✷

✇❡ ❤❛✈❡

✶ ✷

✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-30
SLIDE 30

❈♦♠♣❛t✐❜❧❡ ♠❛♣s

❍♦✇ ❝❛♥ ♦♥❡ ✭❛❧❣❡❜r❛✐❝❛❧❧②✮ ♦❜t❛✐♥ PAut(Γ) ❢r♦♠ t❤❡ ❧♦✇✲r❛♥❦ ♠❛♣s❄

❉❡✜♥✐t✐♦♥

❊❧❡♠❡♥ts ♦❢ a, b ∈ IX ❛r❡ ❝❛❧❧❡❞ ❝♦♠♣❛t✐❜❧❡ ✭♥♦t✳✿ a ∼ b✮ ✐❢ a ∪ b ∈ IX✳

Pr♦♣♦s✐t✐♦♥

✶ ❛♥❞ ✶

❛r❡ ✐❞❡♠♣♦t❡♥ts

Pr♦♣♦s✐t✐♦♥

❙✉♣♣♦s❡ ✐s ♦❢ r❛♥❦ ✸✳ ❚❤❡♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ ✐ts r❛♥❦ ✷ r❡str✐❝t✐♦♥s ❛r❡ ✐♥ ✳

Pr♦♣♦s✐t✐♦♥

❋♦r ❛❧❧ ♣❛✐r✇✐s❡ ❝♦♠♣❛t✐❜❧❡ r❛♥❦ ✶ ♠❛♣s✱ ✐✛ ❢♦r ❛❧❧

✶ ✷

✇❡ ❤❛✈❡

✶ ✷

✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 31

❈♦♠♣❛t✐❜❧❡ ♠❛♣s

❍♦✇ ❝❛♥ ♦♥❡ ✭❛❧❣❡❜r❛✐❝❛❧❧②✮ ♦❜t❛✐♥ PAut(Γ) ❢r♦♠ t❤❡ ❧♦✇✲r❛♥❦ ♠❛♣s❄

❉❡✜♥✐t✐♦♥

❊❧❡♠❡♥ts ♦❢ a, b ∈ IX ❛r❡ ❝❛❧❧❡❞ ❝♦♠♣❛t✐❜❧❡ ✭♥♦t✳✿ a ∼ b✮ ✐❢ a ∪ b ∈ IX✳

Pr♦♣♦s✐t✐♦♥

a ∼ b ⇐ ⇒ ab−✶ ❛♥❞ a−✶b ❛r❡ ✐❞❡♠♣♦t❡♥ts

Pr♦♣♦s✐t✐♦♥

❙✉♣♣♦s❡ ✐s ♦❢ r❛♥❦ ✸✳ ❚❤❡♥ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ ✐ts r❛♥❦ ✷ r❡str✐❝t✐♦♥s ❛r❡ ✐♥ ✳

Pr♦♣♦s✐t✐♦♥

❋♦r ❛❧❧ ♣❛✐r✇✐s❡ ❝♦♠♣❛t✐❜❧❡ r❛♥❦ ✶ ♠❛♣s✱ ✐✛ ❢♦r ❛❧❧

✶ ✷

✇❡ ❤❛✈❡

✶ ✷

✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 32

❈♦♠♣❛t✐❜❧❡ ♠❛♣s

❍♦✇ ❝❛♥ ♦♥❡ ✭❛❧❣❡❜r❛✐❝❛❧❧②✮ ♦❜t❛✐♥ PAut(Γ) ❢r♦♠ t❤❡ ❧♦✇✲r❛♥❦ ♠❛♣s❄

❉❡✜♥✐t✐♦♥

❊❧❡♠❡♥ts ♦❢ a, b ∈ IX ❛r❡ ❝❛❧❧❡❞ ❝♦♠♣❛t✐❜❧❡ ✭♥♦t✳✿ a ∼ b✮ ✐❢ a ∪ b ∈ IX✳

Pr♦♣♦s✐t✐♦♥

a ∼ b ⇐ ⇒ ab−✶ ❛♥❞ a−✶b ❛r❡ ✐❞❡♠♣♦t❡♥ts

Pr♦♣♦s✐t✐♦♥

❙✉♣♣♦s❡ a ∈ IV ✐s ♦❢ r❛♥❦ ≥ ✸✳ ❚❤❡♥ a ∈ PAut(Γ) ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ ✐ts r❛♥❦ ✷ r❡str✐❝t✐♦♥s ❛r❡ ✐♥ PAut(Γ)✳

Pr♦♣♦s✐t✐♦♥

❋♦r ❛❧❧ ♣❛✐r✇✐s❡ ❝♦♠♣❛t✐❜❧❡ r❛♥❦ ✶ ♠❛♣s✱ ✐✛ ❢♦r ❛❧❧

✶ ✷

✇❡ ❤❛✈❡

✶ ✷

✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 33

❈♦♠♣❛t✐❜❧❡ ♠❛♣s

❍♦✇ ❝❛♥ ♦♥❡ ✭❛❧❣❡❜r❛✐❝❛❧❧②✮ ♦❜t❛✐♥ PAut(Γ) ❢r♦♠ t❤❡ ❧♦✇✲r❛♥❦ ♠❛♣s❄

❉❡✜♥✐t✐♦♥

❊❧❡♠❡♥ts ♦❢ a, b ∈ IX ❛r❡ ❝❛❧❧❡❞ ❝♦♠♣❛t✐❜❧❡ ✭♥♦t✳✿ a ∼ b✮ ✐❢ a ∪ b ∈ IX✳

Pr♦♣♦s✐t✐♦♥

a ∼ b ⇐ ⇒ ab−✶ ❛♥❞ a−✶b ❛r❡ ✐❞❡♠♣♦t❡♥ts

Pr♦♣♦s✐t✐♦♥

❙✉♣♣♦s❡ a ∈ IV ✐s ♦❢ r❛♥❦ ≥ ✸✳ ❚❤❡♥ a ∈ PAut(Γ) ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❛❧❧ ✐ts r❛♥❦ ✷ r❡str✐❝t✐♦♥s ❛r❡ ✐♥ PAut(Γ)✳

Pr♦♣♦s✐t✐♦♥

❋♦r ❛❧❧ A ⊆ IV ♣❛✐r✇✐s❡ ❝♦♠♣❛t✐❜❧❡ r❛♥❦ ✶ ♠❛♣s✱ A ∈ PAut(Γ) ✐✛ ❢♦r ❛❧❧ a✶, a✷ ∈ A ✇❡ ❤❛✈❡ a✶ ∪ a✷ ∈ PAut(Γ)✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 34

❚❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ PAut(Γ) ✐♥ IV

❚❤❡♦r❡♠

  • ✐✈❡♥ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ S ⊆ IV ✭V ✐s ✜♥✐t❡✮✱ t❤❡r❡ ❡①✐sts ❛ s✐♠♣❧❡✱

✉♥❞✐r❡❝t❡❞ ❣r❛♣❤ Γ ♦♥ V s✉❝❤ t❤❛t PAut(Γ) = S ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✿ ✶✳ E(IV ) ⊆ S✱ ✷✳ t❤❡ r❛♥❦ ✷ ❡❧❡♠❡♥ts ♦❢ S ❢♦r♠ ❛t ♠♦st t✇♦ D✲❝❧❛ss❡s✱ ✸✳ t❤❡ r❛♥❦ ✷ H✲❝❧❛ss❡s ♦❢ S ❛r❡ ♥♦♥tr✐✈✐❛❧✱ ✹✳ ❢♦r ❛♥② ❝♦♠♣❛t✐❜❧❡ s✉❜s❡t A ⊆ S ♦❢ r❛♥❦ ✶ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s✱ ✐❢ S ❝♦♥t❛✐♥s t❤❡ ❥♦✐♥ ♦❢ ❛♥② t✇♦ ❡❧❡♠❡♥ts ♦❢ A✱ t❤❡♥ S ❝♦♥t❛✐♥s t❤❡ ❥♦✐♥ ♦❢ t❤❡ s❡t A✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 35

❚❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ PAut(Γ) ✐♥ IV

❚❤❡♦r❡♠

  • ✐✈❡♥ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ S ⊆ IV ✭V ✐s ✜♥✐t❡✮✱ t❤❡r❡ ❡①✐sts ❛♥

❡❞❣❡✲❝♦❧♦r❡❞ ❞✐❣r❛♣❤ Γ ♦♥ V s✉❝❤ t❤❛t PAut(Γ) = S ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✿ ✶✳ E(IV ) ⊆ S✱ ✷✳ ❢♦r ❛♥② ❝♦♠♣❛t✐❜❧❡ s✉❜s❡t A ⊆ S ♦❢ r❛♥❦ ✶ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s✱ ✐❢ S ❝♦♥t❛✐♥s t❤❡ ❥♦✐♥ ♦❢ ❛♥② t✇♦ ❡❧❡♠❡♥ts ♦❢ A✱ t❤❡♥ S ❝♦♥t❛✐♥s t❤❡ ❥♦✐♥ ♦❢ t❤❡ s❡t A✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-36
SLIDE 36

❇♦♦❧❡❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❉❡✜♥✐t✐♦♥

❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ S ✇✐t❤ ③❡r♦ ✐s ❝❛❧❧❡❞ ❇♦♦❧❡❛♥ ✐❢ t❤❡ s❡♠✐❧❛tt✐❝❡ E(S) ✐s t❤❡ ♠❡❡t s❡♠✐❧❛tt✐❝❡ ♦❢ ❛ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛✳ ■♥ ♣❛rt✐❝✉❧❛r ✐❢ S ✐s ✜♥✐t❡ ❇♦♦❧❡❛♥✱ t❤❡♥ E(S) ∼ = ✷X ❢♦r s♦♠❡ X✳ ◆♦t❡✿ ✐s ❇♦♦❧❡❛♥ ❢♦r ❛♥② ❣r❛♣❤✱ ❛♥❞ t❤❡ ❛t♦♠s ♦❢ ✳ ❤❛s ❛ ❢❛✐t❤❢✉❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦♥ t❤❡ ❛t♦♠s ♦❢ ✱ ❛♥❞ ✐t ✐s ❡①❛❝t❧② ✇❤❛t ♦♥❡ ❣❡ts r❡str✐❝t✐♥❣ t❤❡ ✭✐♥ t❤✐s ❝❛s❡✱ ❢❛✐t❤❢✉❧✮ ▼✉♥♥ r❡♣r❡s❡♥t❛t✐♦♥✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 37

❇♦♦❧❡❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❉❡✜♥✐t✐♦♥

❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ S ✇✐t❤ ③❡r♦ ✐s ❝❛❧❧❡❞ ❇♦♦❧❡❛♥ ✐❢ t❤❡ s❡♠✐❧❛tt✐❝❡ E(S) ✐s t❤❡ ♠❡❡t s❡♠✐❧❛tt✐❝❡ ♦❢ ❛ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛✳ ■♥ ♣❛rt✐❝✉❧❛r ✐❢ S ✐s ✜♥✐t❡ ❇♦♦❧❡❛♥✱ t❤❡♥ E(S) ∼ = ✷X ❢♦r s♦♠❡ X✳ ◆♦t❡✿ PAut(Γ) ✐s ❇♦♦❧❡❛♥ ❢♦r ❛♥② ❣r❛♣❤✱ ❛♥❞ V ← → t❤❡ ❛t♦♠s ♦❢ E(PAut(Γ))✳ ❤❛s ❛ ❢❛✐t❤❢✉❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦♥ t❤❡ ❛t♦♠s ♦❢ ✱ ❛♥❞ ✐t ✐s ❡①❛❝t❧② ✇❤❛t ♦♥❡ ❣❡ts r❡str✐❝t✐♥❣ t❤❡ ✭✐♥ t❤✐s ❝❛s❡✱ ❢❛✐t❤❢✉❧✮ ▼✉♥♥ r❡♣r❡s❡♥t❛t✐♦♥✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 38

❇♦♦❧❡❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❉❡✜♥✐t✐♦♥

❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ S ✇✐t❤ ③❡r♦ ✐s ❝❛❧❧❡❞ ❇♦♦❧❡❛♥ ✐❢ t❤❡ s❡♠✐❧❛tt✐❝❡ E(S) ✐s t❤❡ ♠❡❡t s❡♠✐❧❛tt✐❝❡ ♦❢ ❛ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛✳ ■♥ ♣❛rt✐❝✉❧❛r ✐❢ S ✐s ✜♥✐t❡ ❇♦♦❧❡❛♥✱ t❤❡♥ E(S) ∼ = ✷X ❢♦r s♦♠❡ X✳ ◆♦t❡✿ PAut(Γ) ✐s ❇♦♦❧❡❛♥ ❢♦r ❛♥② ❣r❛♣❤✱ ❛♥❞ V ← → t❤❡ ❛t♦♠s ♦❢ E(PAut(Γ))✳ = ⇒ PAut(Γ) ❤❛s ❛ ❢❛✐t❤❢✉❧ r❡♣r❡s❡♥t❛t✐♦♥ ♦♥ t❤❡ ❛t♦♠s ♦❢ E(PAut(Γ))✱ ❛♥❞ ✐t ✐s ❡①❛❝t❧② ✇❤❛t ♦♥❡ ❣❡ts r❡str✐❝t✐♥❣ t❤❡ ✭✐♥ t❤✐s ❝❛s❡✱ ❢❛✐t❤❢✉❧✮ ▼✉♥♥ r❡♣r❡s❡♥t❛t✐♦♥✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 39

❘❡str✐❝t❡❞ ▼✉♥♥ r❡♣r❡s❡♥t❛t✐♦♥

▲❡t S ❜❡ ❛ ✜♥✐t❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞✱ ❛♥❞ ❧❡t X ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❛t♦♠s ♦❢ E(S)✳ ❚❤❡ r❡str✐❝t❡❞ ▼✉♥♥ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ S ✐s t❤❡ ▼✉♥♥ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ S r❡str✐❝t❡❞ t♦ t❤❡ ❛t♦♠s A ♦❢ E(S)✿ αS : s → ˆ ms✱ ˆ ms : ss−✶ ∩ A → s−✶s ∩ A✱ e → s−✶es✳

Pr♦♣♦s✐t✐♦♥

❚❤❡ ❛❜♦✈❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐s ❢❛✐t❤❢✉❧ ✐s ❇♦♦❧❡❛♥ ❛♥❞ ❢✉♥❞❛♠❡♥t❛❧✳ ■❢ ✱ t❤❡♥ ✭✉♥❞❡r t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ❛♥❞ ✮✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-40
SLIDE 40

❘❡str✐❝t❡❞ ▼✉♥♥ r❡♣r❡s❡♥t❛t✐♦♥

▲❡t S ❜❡ ❛ ✜♥✐t❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞✱ ❛♥❞ ❧❡t X ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❛t♦♠s ♦❢ E(S)✳ ❚❤❡ r❡str✐❝t❡❞ ▼✉♥♥ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ S ✐s t❤❡ ▼✉♥♥ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ S r❡str✐❝t❡❞ t♦ t❤❡ ❛t♦♠s A ♦❢ E(S)✿ αS : s → ˆ ms✱ ˆ ms : ss−✶ ∩ A → s−✶s ∩ A✱ e → s−✶es✳

Pr♦♣♦s✐t✐♦♥

❚❤❡ ❛❜♦✈❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐s ❢❛✐t❤❢✉❧ ⇐ ⇒ S ✐s ❇♦♦❧❡❛♥ ❛♥❞ ❢✉♥❞❛♠❡♥t❛❧✳ ■❢ ✱ t❤❡♥ ✭✉♥❞❡r t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ ❛♥❞ ✮✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 41

❘❡str✐❝t❡❞ ▼✉♥♥ r❡♣r❡s❡♥t❛t✐♦♥

▲❡t S ❜❡ ❛ ✜♥✐t❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞✱ ❛♥❞ ❧❡t X ❞❡♥♦t❡ t❤❡ s❡t ♦❢ ❛t♦♠s ♦❢ E(S)✳ ❚❤❡ r❡str✐❝t❡❞ ▼✉♥♥ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ S ✐s t❤❡ ▼✉♥♥ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ S r❡str✐❝t❡❞ t♦ t❤❡ ❛t♦♠s A ♦❢ E(S)✿ αS : s → ˆ ms✱ ˆ ms : ss−✶ ∩ A → s−✶s ∩ A✱ e → s−✶es✳

Pr♦♣♦s✐t✐♦♥

❚❤❡ ❛❜♦✈❡ r❡♣r❡s❡♥t❛t✐♦♥ ✐s ❢❛✐t❤❢✉❧ ⇐ ⇒ S ✐s ❇♦♦❧❡❛♥ ❛♥❞ ❢✉♥❞❛♠❡♥t❛❧✳ ■❢ S = PAut(Γ)✱ t❤❡♥ αS(S) = S ✭✉♥❞❡r t❤❡ ✐❞❡♥t✐✜❝❛t✐♦♥ ♦❢ idv ❛♥❞ v✮✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 42

S ❛s ❛ ♣♦s❡t

■❢ S ✐s ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞✱ t❤❡♥ (S, ≤) ✐s ❛ ♣♦s❡t ✇rt t❤❡ ♥❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r✳ ■♥ ✇❡ ❝❛♥ t❛❧❦ ❛❜♦✉t ❛ ♣❛rt✐❛❧❧② ❞❡✜♥❡❞ ❥♦✐♥ ✭ ✮✳

❉❡✜♥✐t✐♦♥

❲❡ ❝❛❧❧ ❝♦♠♣❛t✐❜❧❡ ✐❢

✶ ✶

❛r❡ ✐❞❡♠♣♦t❡♥ts✳ ❋❛❝t✿ ■❢ ❤❛s ❛ ❥♦✐♥✱ t❤❡♥ ❡❧❡♠❡♥ts ♦❢ ❛r❡ ♣❛✐r✇✐s❡ ❝♦♠♣❛t✐❜❧❡ ✭t❤❡ ❝♦♥✈❡rs❡ ✐s ♥♦t tr✉❡✱ ❤♦✇❡✈❡r✮✳

❉❡✜♥✐t✐♦♥

❙✉♣♣♦s❡ ✐s ❛ ✜♥✐t❡ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ ✇✐t❤ ✵✳ ❚❤❡ ❤❡✐❣❤t ♦❢ ❛♥ ❡❧❡♠❡♥t ✐♥ ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝❤❛✐♥ ✵ ✳ ◆♦t❡✿ ✐♥ ✱ ❤❡✐❣❤t ❂ r❛♥❦✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-43
SLIDE 43

S ❛s ❛ ♣♦s❡t

■❢ S ✐s ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞✱ t❤❡♥ (S, ≤) ✐s ❛ ♣♦s❡t ✇rt t❤❡ ♥❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r✳ ■♥ (S, ≤) ✇❡ ❝❛♥ t❛❧❦ ❛❜♦✉t ❛ ♣❛rt✐❛❧❧② ❞❡✜♥❡❞ ❥♦✐♥ ✭∨✮✳

❉❡✜♥✐t✐♦♥

❲❡ ❝❛❧❧ ❝♦♠♣❛t✐❜❧❡ ✐❢

✶ ✶

❛r❡ ✐❞❡♠♣♦t❡♥ts✳ ❋❛❝t✿ ■❢ ❤❛s ❛ ❥♦✐♥✱ t❤❡♥ ❡❧❡♠❡♥ts ♦❢ ❛r❡ ♣❛✐r✇✐s❡ ❝♦♠♣❛t✐❜❧❡ ✭t❤❡ ❝♦♥✈❡rs❡ ✐s ♥♦t tr✉❡✱ ❤♦✇❡✈❡r✮✳

❉❡✜♥✐t✐♦♥

❙✉♣♣♦s❡ ✐s ❛ ✜♥✐t❡ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ ✇✐t❤ ✵✳ ❚❤❡ ❤❡✐❣❤t ♦❢ ❛♥ ❡❧❡♠❡♥t ✐♥ ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝❤❛✐♥ ✵ ✳ ◆♦t❡✿ ✐♥ ✱ ❤❡✐❣❤t ❂ r❛♥❦✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-44
SLIDE 44

S ❛s ❛ ♣♦s❡t

■❢ S ✐s ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞✱ t❤❡♥ (S, ≤) ✐s ❛ ♣♦s❡t ✇rt t❤❡ ♥❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r✳ ■♥ (S, ≤) ✇❡ ❝❛♥ t❛❧❦ ❛❜♦✉t ❛ ♣❛rt✐❛❧❧② ❞❡✜♥❡❞ ❥♦✐♥ ✭∨✮✳

❉❡✜♥✐t✐♦♥

❲❡ ❝❛❧❧ a, b ∈ S ❝♦♠♣❛t✐❜❧❡ ✐❢ ab−✶, a−✶b ❛r❡ ✐❞❡♠♣♦t❡♥ts✳ ❋❛❝t✿ ■❢ A ⊆ S ❤❛s ❛ ❥♦✐♥✱ t❤❡♥ ❡❧❡♠❡♥ts ♦❢ A ❛r❡ ♣❛✐r✇✐s❡ ❝♦♠♣❛t✐❜❧❡ ✭t❤❡ ❝♦♥✈❡rs❡ ✐s ♥♦t tr✉❡✱ ❤♦✇❡✈❡r✮✳

❉❡✜♥✐t✐♦♥

❙✉♣♣♦s❡ ✐s ❛ ✜♥✐t❡ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ ✇✐t❤ ✵✳ ❚❤❡ ❤❡✐❣❤t ♦❢ ❛♥ ❡❧❡♠❡♥t ✐♥ ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝❤❛✐♥ ✵ ✳ ◆♦t❡✿ ✐♥ ✱ ❤❡✐❣❤t ❂ r❛♥❦✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 45

S ❛s ❛ ♣♦s❡t

■❢ S ✐s ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞✱ t❤❡♥ (S, ≤) ✐s ❛ ♣♦s❡t ✇rt t❤❡ ♥❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r✳ ■♥ (S, ≤) ✇❡ ❝❛♥ t❛❧❦ ❛❜♦✉t ❛ ♣❛rt✐❛❧❧② ❞❡✜♥❡❞ ❥♦✐♥ ✭∨✮✳

❉❡✜♥✐t✐♦♥

❲❡ ❝❛❧❧ a, b ∈ S ❝♦♠♣❛t✐❜❧❡ ✐❢ ab−✶, a−✶b ❛r❡ ✐❞❡♠♣♦t❡♥ts✳ ❋❛❝t✿ ■❢ A ⊆ S ❤❛s ❛ ❥♦✐♥✱ t❤❡♥ ❡❧❡♠❡♥ts ♦❢ A ❛r❡ ♣❛✐r✇✐s❡ ❝♦♠♣❛t✐❜❧❡ ✭t❤❡ ❝♦♥✈❡rs❡ ✐s ♥♦t tr✉❡✱ ❤♦✇❡✈❡r✮✳

❉❡✜♥✐t✐♦♥

❙✉♣♣♦s❡ S ✐s ❛ ✜♥✐t❡ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ ✇✐t❤ ✵✳ ❚❤❡ ❤❡✐❣❤t ♦❢ ❛♥ ❡❧❡♠❡♥t s ✐♥ (S, ≤) ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝❤❛✐♥ [✵, s]✳ ◆♦t❡✿ ✐♥ ✱ ❤❡✐❣❤t ❂ r❛♥❦✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-46
SLIDE 46

S ❛s ❛ ♣♦s❡t

■❢ S ✐s ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞✱ t❤❡♥ (S, ≤) ✐s ❛ ♣♦s❡t ✇rt t❤❡ ♥❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r✳ ■♥ (S, ≤) ✇❡ ❝❛♥ t❛❧❦ ❛❜♦✉t ❛ ♣❛rt✐❛❧❧② ❞❡✜♥❡❞ ❥♦✐♥ ✭∨✮✳

❉❡✜♥✐t✐♦♥

❲❡ ❝❛❧❧ a, b ∈ S ❝♦♠♣❛t✐❜❧❡ ✐❢ ab−✶, a−✶b ❛r❡ ✐❞❡♠♣♦t❡♥ts✳ ❋❛❝t✿ ■❢ A ⊆ S ❤❛s ❛ ❥♦✐♥✱ t❤❡♥ ❡❧❡♠❡♥ts ♦❢ A ❛r❡ ♣❛✐r✇✐s❡ ❝♦♠♣❛t✐❜❧❡ ✭t❤❡ ❝♦♥✈❡rs❡ ✐s ♥♦t tr✉❡✱ ❤♦✇❡✈❡r✮✳

❉❡✜♥✐t✐♦♥

❙✉♣♣♦s❡ S ✐s ❛ ✜♥✐t❡ ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ ✇✐t❤ ✵✳ ❚❤❡ ❤❡✐❣❤t ♦❢ ❛♥ ❡❧❡♠❡♥t s ✐♥ (S, ≤) ✐s t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝❤❛✐♥ [✵, s]✳ ◆♦t❡✿ ✐♥ PAut(Γ)✱ ❤❡✐❣❤t ❂ r❛♥❦✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-47
SLIDE 47

❚❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ PAut(Γ) ✐♥ t❤❡ ❛❜str❛❝t ❝❛s❡

❚❤❡♦r❡♠

  • ✐✈❡♥ ❛ ✜♥✐t❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ S t❤❡r❡ ❡①✐sts ❛ s✐♠♣❧❡✱ ✉♥❞✐r❡❝t❡❞

❣r❛♣❤ Γ s✉❝❤ t❤❛t PAut(Γ) ∼ = S ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✿ ✶✳ S ✐s ❇♦♦❧❡❛♥ ✭❤❡♥❝❡ ✐t ❤❛s ❛ ✵✮✱ ✷✳ S ✐s ❢✉♥❞❛♠❡♥t❛❧✱ ✸✳ t❤❡ ❡❧❡♠❡♥ts ♦❢ ❤❡✐❣❤t ✶ ❢♦r♠ ❛ s✐♥❣❧❡ D✲❝❧❛ss✱ ✹✳ ❡❧❡♠❡♥ts ♦❢ ❤❡✐❣❤t ✷ ❢♦r♠ ❛t ♠♦st t✇♦ D✲❝❧❛ss❡s✱ ✇✐t❤ t✇♦✲❡❧❡♠❡♥t H✲❝❧❛ss❡s✱ ✺✳ ✐❢ X ⊆ S ✐s ❛ s❡t ♦❢ ♣❛✐r✇✐s❡ ❝♦♠♣❛t✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ ❤❡✐❣❤t ✶✱ ❛♥❞ ❢♦r ❛❧❧ a✶, a✷ ∈ A✱ a✶ ∨ a✷ ❡①✐sts✱ t❤❡♥ A ❡①✐sts✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-48
SLIDE 48

❚❤❡ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ PAut(Γ) ✐♥ t❤❡ ❛❜str❛❝t ❝❛s❡

❚❤❡♦r❡♠

  • ✐✈❡♥ ❛ ✜♥✐t❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞ S t❤❡r❡ ❡①✐sts ❛♥ ❡❞❣❡✲❝♦❧♦r❡❞ ❞✐❣r❛♣❤

Γ s✉❝❤ t❤❛t PAut(Γ) ∼ = S ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ❤♦❧❞✿ ✶✳ S ✐s ❇♦♦❧❡❛♥ ✭❤❡♥❝❡ ✐t ❤❛s ❛ ✵✮✱ ✷✳ S ✐s ❢✉♥❞❛♠❡♥t❛❧✱ ✸✳ ✐❢ X ⊆ S ✐s ❛ s❡t ♦❢ ♣❛✐r✇✐s❡ ❝♦♠♣❛t✐❜❧❡ ❡❧❡♠❡♥ts ♦❢ ❤❡✐❣❤t ✶✱ ❛♥❞ ❢♦r ❛❧❧ a✶, a✷ ∈ A✱ a✶ ∨ a✷ ❡①✐sts✱ t❤❡♥ A ❡①✐sts✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-49
SLIDE 49

❚❤❡ ❈❛②❧❡② ❣r❛♣❤

❚❤❡♦r❡♠

■❢ G ✐s ❛ ❣r♦✉♣✱ ❛♥❞ Γ(G) ✐s ✐ts ❈❛②❧❡② ❣r❛♣❤✱ t❤❡♥ t❤❡ ✭❡❞❣❡✲❝♦❧♦r❡❞✱ ❞✐r❡❝t❡❞✮ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ Aut(Γ(G)) ∼ = G✳ Pr♦♦❢✿ ✐s ❡①❛❝t❧② ✇❤❛t ♦♥❡ ♦❜t❛✐♥s ❛s t❤❡ ✭❧❡❢t✮ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ ✳ ❈❧❡❛r❧②✱ ✐❢ ✐s ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❛♥❞ ✐s ✐ts ❈❛②❧❡② ❣r❛♣❤✱ ✐♥ ❣❡♥❡r❛❧ ✳ ❍♦✇❡✈❡r✱ ♦♥❡ ❝❛♥ r❡❛❧✐③❡ ❛s ❛ s✉❜s❡t ♦❢ ✉s✐♥❣ t❤❡ ✭❧❡❢t✮ ❲❛❣♥❡r✕Pr❡st♦♥ r❡♣r❡s❡♥t❛t✐♦♥✿

❚❤❡♦r❡♠ ✭❙✐❡❜❡♥✱ ✷✵✵✽✮

❚❤❡ ♠❛♣ ✱ ✇❤❡r❡

✶

❡♠❜❡❞s ✐♥t♦ ✳ ❚❤❡ s✉❜❣r❛♣❤s ❛r✐s✐♥❣ ❛s ❞♦♠❛✐♥s ❛♥❞ ✐♠❛❣❡s ❛r❡ ❡①❛❝t❧② t❤❡ ♦♥❡s ♦❢ t❤❡ ❢♦r♠ ❝❛♥ ❜❡ r❡❛❝❤❡❞ ❜② ❛ ✜♥✐t❡ ♣❛t❤ ❢r♦♠

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-50
SLIDE 50

❚❤❡ ❈❛②❧❡② ❣r❛♣❤

❚❤❡♦r❡♠

■❢ G ✐s ❛ ❣r♦✉♣✱ ❛♥❞ Γ(G) ✐s ✐ts ❈❛②❧❡② ❣r❛♣❤✱ t❤❡♥ t❤❡ ✭❡❞❣❡✲❝♦❧♦r❡❞✱ ❞✐r❡❝t❡❞✮ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ Aut(Γ(G)) ∼ = G✳ Pr♦♦❢✿ Aut(Γ(G)) ≤ SG ✐s ❡①❛❝t❧② ✇❤❛t ♦♥❡ ♦❜t❛✐♥s ❛s t❤❡ ✭❧❡❢t✮ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ G✳ ❈❧❡❛r❧②✱ ✐❢ ✐s ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❛♥❞ ✐s ✐ts ❈❛②❧❡② ❣r❛♣❤✱ ✐♥ ❣❡♥❡r❛❧ ✳ ❍♦✇❡✈❡r✱ ♦♥❡ ❝❛♥ r❡❛❧✐③❡ ❛s ❛ s✉❜s❡t ♦❢ ✉s✐♥❣ t❤❡ ✭❧❡❢t✮ ❲❛❣♥❡r✕Pr❡st♦♥ r❡♣r❡s❡♥t❛t✐♦♥✿

❚❤❡♦r❡♠ ✭❙✐❡❜❡♥✱ ✷✵✵✽✮

❚❤❡ ♠❛♣ ✱ ✇❤❡r❡

✶

❡♠❜❡❞s ✐♥t♦ ✳ ❚❤❡ s✉❜❣r❛♣❤s ❛r✐s✐♥❣ ❛s ❞♦♠❛✐♥s ❛♥❞ ✐♠❛❣❡s ❛r❡ ❡①❛❝t❧② t❤❡ ♦♥❡s ♦❢ t❤❡ ❢♦r♠ ❝❛♥ ❜❡ r❡❛❝❤❡❞ ❜② ❛ ✜♥✐t❡ ♣❛t❤ ❢r♦♠

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-51
SLIDE 51

❚❤❡ ❈❛②❧❡② ❣r❛♣❤

❚❤❡♦r❡♠

■❢ G ✐s ❛ ❣r♦✉♣✱ ❛♥❞ Γ(G) ✐s ✐ts ❈❛②❧❡② ❣r❛♣❤✱ t❤❡♥ t❤❡ ✭❡❞❣❡✲❝♦❧♦r❡❞✱ ❞✐r❡❝t❡❞✮ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ Aut(Γ(G)) ∼ = G✳ Pr♦♦❢✿ Aut(Γ(G)) ≤ SG ✐s ❡①❛❝t❧② ✇❤❛t ♦♥❡ ♦❜t❛✐♥s ❛s t❤❡ ✭❧❡❢t✮ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ G✳ ❈❧❡❛r❧②✱ ✐❢ S ✐s ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❛♥❞ Γ(S) ✐s ✐ts ❈❛②❧❡② ❣r❛♣❤✱ ✐♥ ❣❡♥❡r❛❧ PAut(Γ(G)) ∼ = S✳ ❍♦✇❡✈❡r✱ ♦♥❡ ❝❛♥ r❡❛❧✐③❡ ❛s ❛ s✉❜s❡t ♦❢ ✉s✐♥❣ t❤❡ ✭❧❡❢t✮ ❲❛❣♥❡r✕Pr❡st♦♥ r❡♣r❡s❡♥t❛t✐♦♥✿

❚❤❡♦r❡♠ ✭❙✐❡❜❡♥✱ ✷✵✵✽✮

❚❤❡ ♠❛♣ ✱ ✇❤❡r❡

✶

❡♠❜❡❞s ✐♥t♦ ✳ ❚❤❡ s✉❜❣r❛♣❤s ❛r✐s✐♥❣ ❛s ❞♦♠❛✐♥s ❛♥❞ ✐♠❛❣❡s ❛r❡ ❡①❛❝t❧② t❤❡ ♦♥❡s ♦❢ t❤❡ ❢♦r♠ ❝❛♥ ❜❡ r❡❛❝❤❡❞ ❜② ❛ ✜♥✐t❡ ♣❛t❤ ❢r♦♠

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-52
SLIDE 52

❚❤❡ ❈❛②❧❡② ❣r❛♣❤

❚❤❡♦r❡♠

■❢ G ✐s ❛ ❣r♦✉♣✱ ❛♥❞ Γ(G) ✐s ✐ts ❈❛②❧❡② ❣r❛♣❤✱ t❤❡♥ t❤❡ ✭❡❞❣❡✲❝♦❧♦r❡❞✱ ❞✐r❡❝t❡❞✮ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ Aut(Γ(G)) ∼ = G✳ Pr♦♦❢✿ Aut(Γ(G)) ≤ SG ✐s ❡①❛❝t❧② ✇❤❛t ♦♥❡ ♦❜t❛✐♥s ❛s t❤❡ ✭❧❡❢t✮ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ G✳ ❈❧❡❛r❧②✱ ✐❢ S ✐s ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❛♥❞ Γ(S) ✐s ✐ts ❈❛②❧❡② ❣r❛♣❤✱ ✐♥ ❣❡♥❡r❛❧ PAut(Γ(G)) ∼ = S✳ ❍♦✇❡✈❡r✱ ♦♥❡ ❝❛♥ r❡❛❧✐③❡ S ❛s ❛ s✉❜s❡t ♦❢ PAut(Γ) ✉s✐♥❣ t❤❡ ✭❧❡❢t✮ ❲❛❣♥❡r✕Pr❡st♦♥ r❡♣r❡s❡♥t❛t✐♦♥✿

❚❤❡♦r❡♠ ✭❙✐❡❜❡♥✱ ✷✵✵✽✮

❚❤❡ ♠❛♣ ρ: S → IS, s → ρs✱ ✇❤❡r❡ ρs : s−✶S → sS, t → st ❡♠❜❡❞s S ✐♥t♦ PAut(Γ(G))✳ ❚❤❡ s✉❜❣r❛♣❤s ❛r✐s✐♥❣ ❛s ❞♦♠❛✐♥s ❛♥❞ ✐♠❛❣❡s ❛r❡ ❡①❛❝t❧② t❤❡ ♦♥❡s ♦❢ t❤❡ ❢♦r♠ ❝❛♥ ❜❡ r❡❛❝❤❡❞ ❜② ❛ ✜♥✐t❡ ♣❛t❤ ❢r♦♠

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-53
SLIDE 53

❚❤❡ ❈❛②❧❡② ❣r❛♣❤

❚❤❡♦r❡♠

■❢ G ✐s ❛ ❣r♦✉♣✱ ❛♥❞ Γ(G) ✐s ✐ts ❈❛②❧❡② ❣r❛♣❤✱ t❤❡♥ t❤❡ ✭❡❞❣❡✲❝♦❧♦r❡❞✱ ❞✐r❡❝t❡❞✮ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ Aut(Γ(G)) ∼ = G✳ Pr♦♦❢✿ Aut(Γ(G)) ≤ SG ✐s ❡①❛❝t❧② ✇❤❛t ♦♥❡ ♦❜t❛✐♥s ❛s t❤❡ ✭❧❡❢t✮ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ G✳ ❈❧❡❛r❧②✱ ✐❢ S ✐s ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❛♥❞ Γ(S) ✐s ✐ts ❈❛②❧❡② ❣r❛♣❤✱ ✐♥ ❣❡♥❡r❛❧ PAut(Γ(G)) ∼ = S✳ ❍♦✇❡✈❡r✱ ♦♥❡ ❝❛♥ r❡❛❧✐③❡ S ❛s ❛ s✉❜s❡t ♦❢ PAut(Γ) ✉s✐♥❣ t❤❡ ✭❧❡❢t✮ ❲❛❣♥❡r✕Pr❡st♦♥ r❡♣r❡s❡♥t❛t✐♦♥✿

❚❤❡♦r❡♠ ✭❙✐❡❜❡♥✱ ✷✵✵✽✮

❚❤❡ ♠❛♣ ρ: S → IS, s → ρs✱ ✇❤❡r❡ ρs : s−✶S → sS, t → st ❡♠❜❡❞s S ✐♥t♦ PAut(Γ(G))✳ ❚❤❡ s✉❜❣r❛♣❤s ❛r✐s✐♥❣ ❛s ❞♦♠❛✐♥s ❛♥❞ ✐♠❛❣❡s ❛r❡ ❡①❛❝t❧② t❤❡ ♦♥❡s ♦❢ t❤❡ ❢♦r♠ tail(s) = {t ∈ V : t ❝❛♥ ❜❡ r❡❛❝❤❡❞ ❜② ❛ ✜♥✐t❡ ♣❛t❤ ❢r♦♠ s}.

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

slide-54
SLIDE 54

◗✉❡st✐♦♥✿ ✐s t❤❡r❡ ❛ ♥✐❝❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ρ(S) ❛♥❞ PAut(Γ(S))❄ ▲❡t ❜❡ t❤❡ s♠❛❧❧❡st ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ ✐♥ ❝♦♥t❛✐♥✐♥❣ ✇❤✐❝❤ s❛t✐s✜❡s t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s✿ ✶✳ ✱ ✷✳ ❢♦r ❛♥② ❝♦♠♣❛t✐❜❧❡ s✉❜s❡t ♦❢ r❛♥❦ ✶ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s✱ ✐❢ ❝♦♥t❛✐♥s t❤❡ ❥♦✐♥ ♦❢ ❛♥② t✇♦ ❡❧❡♠❡♥ts ♦❢ ✱ t❤❡♥ ❝♦♥t❛✐♥s t❤❡ ❥♦✐♥ ♦❢ t❤❡ s❡t ✳ ❚❤❛t ✐s✱ ❛ ♠❛♣ ♦❢ r❛♥❦ ✶ ✐s ✐♥ ✐t ✐s ❛ r❡str✐❝t✐♦♥ ♦❢ ❛ ♠❛♣ ✐♥ ❀ ♦❢ r❛♥❦ ✷ ✐s ✐♥ ❛❧❧ ✷✲r❛♥❦ r❡str✐❝t✐♦♥s ♦❢ ❛r❡ r❡str✐❝t✐♦♥s ♦❢ ♠❛♣s ✐♥ ✳ ❚❤❡ ❜❡st ✇❡ ❝❛♥ ❤♦♣❡ ❢♦r✿ ✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 55

◗✉❡st✐♦♥✿ ✐s t❤❡r❡ ❛ ♥✐❝❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ρ(S) ❛♥❞ PAut(Γ(S))❄ ▲❡t ρ(S) ❜❡ t❤❡ s♠❛❧❧❡st ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ ✐♥ IS ❝♦♥t❛✐♥✐♥❣ ρ(S) ✇❤✐❝❤ s❛t✐s✜❡s t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s✿ ✶✳ E(IS) ⊆ ρ(S)✱ ✷✳ ❢♦r ❛♥② ❝♦♠♣❛t✐❜❧❡ s✉❜s❡t A ⊆ S ♦❢ r❛♥❦ ✶ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s✱ ✐❢ ρ(S) ❝♦♥t❛✐♥s t❤❡ ❥♦✐♥ ♦❢ ❛♥② t✇♦ ❡❧❡♠❡♥ts ♦❢ A✱ t❤❡♥ ρ(S) ❝♦♥t❛✐♥s t❤❡ ❥♦✐♥ ♦❢ t❤❡ s❡t A✳ ❚❤❛t ✐s✱ ❛ ♠❛♣ ♦❢ r❛♥❦ ✶ ✐s ✐♥ ✐t ✐s ❛ r❡str✐❝t✐♦♥ ♦❢ ❛ ♠❛♣ ✐♥ ❀ ♦❢ r❛♥❦ ✷ ✐s ✐♥ ❛❧❧ ✷✲r❛♥❦ r❡str✐❝t✐♦♥s ♦❢ ❛r❡ r❡str✐❝t✐♦♥s ♦❢ ♠❛♣s ✐♥ ✳ ❚❤❡ ❜❡st ✇❡ ❝❛♥ ❤♦♣❡ ❢♦r✿ ✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 56

◗✉❡st✐♦♥✿ ✐s t❤❡r❡ ❛ ♥✐❝❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ρ(S) ❛♥❞ PAut(Γ(S))❄ ▲❡t ρ(S) ❜❡ t❤❡ s♠❛❧❧❡st ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ ✐♥ IS ❝♦♥t❛✐♥✐♥❣ ρ(S) ✇❤✐❝❤ s❛t✐s✜❡s t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s✿ ✶✳ E(IS) ⊆ ρ(S)✱ ✷✳ ❢♦r ❛♥② ❝♦♠♣❛t✐❜❧❡ s✉❜s❡t A ⊆ S ♦❢ r❛♥❦ ✶ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s✱ ✐❢ ρ(S) ❝♦♥t❛✐♥s t❤❡ ❥♦✐♥ ♦❢ ❛♥② t✇♦ ❡❧❡♠❡♥ts ♦❢ A✱ t❤❡♥ ρ(S) ❝♦♥t❛✐♥s t❤❡ ❥♦✐♥ ♦❢ t❤❡ s❡t A✳ ❚❤❛t ✐s✱ ❛ ♠❛♣ ϕ

◮ ♦❢ r❛♥❦ ✶ ✐s ✐♥

ρ(S) ⇐ ⇒ ✐t ✐s ❛ r❡str✐❝t✐♦♥ ♦❢ ❛ ♠❛♣ ✐♥ ρS❀

◮ ♦❢ r❛♥❦ ≥ ✷ ✐s ✐♥

ρ(S) ⇐ ⇒ ❛❧❧ ✷✲r❛♥❦ r❡str✐❝t✐♦♥s ♦❢ ϕ ❛r❡ r❡str✐❝t✐♦♥s ♦❢ ♠❛♣s ✐♥ ρS✳ ❚❤❡ ❜❡st ✇❡ ❝❛♥ ❤♦♣❡ ❢♦r✿ ✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 57

◗✉❡st✐♦♥✿ ✐s t❤❡r❡ ❛ ♥✐❝❡ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ ρ(S) ❛♥❞ PAut(Γ(S))❄ ▲❡t ρ(S) ❜❡ t❤❡ s♠❛❧❧❡st ✐♥✈❡rs❡ s❡♠✐❣r♦✉♣ ✐♥ IS ❝♦♥t❛✐♥✐♥❣ ρ(S) ✇❤✐❝❤ s❛t✐s✜❡s t❤❡ ♣r♦♣❡rt✐❡s ♦❢ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s✿ ✶✳ E(IS) ⊆ ρ(S)✱ ✷✳ ❢♦r ❛♥② ❝♦♠♣❛t✐❜❧❡ s✉❜s❡t A ⊆ S ♦❢ r❛♥❦ ✶ ♣❛rt✐❛❧ ♣❡r♠✉t❛t✐♦♥s✱ ✐❢ ρ(S) ❝♦♥t❛✐♥s t❤❡ ❥♦✐♥ ♦❢ ❛♥② t✇♦ ❡❧❡♠❡♥ts ♦❢ A✱ t❤❡♥ ρ(S) ❝♦♥t❛✐♥s t❤❡ ❥♦✐♥ ♦❢ t❤❡ s❡t A✳ ❚❤❛t ✐s✱ ❛ ♠❛♣ ϕ

◮ ♦❢ r❛♥❦ ✶ ✐s ✐♥

ρ(S) ⇐ ⇒ ✐t ✐s ❛ r❡str✐❝t✐♦♥ ♦❢ ❛ ♠❛♣ ✐♥ ρS❀

◮ ♦❢ r❛♥❦ ≥ ✷ ✐s ✐♥

ρ(S) ⇐ ⇒ ❛❧❧ ✷✲r❛♥❦ r❡str✐❝t✐♦♥s ♦❢ ϕ ❛r❡ r❡str✐❝t✐♦♥s ♦❢ ♠❛♣s ✐♥ ρS✳ ❚❤❡ ❜❡st ✇❡ ❝❛♥ ❤♦♣❡ ❢♦r✿ ρ(S) = PAut(Γ(S))✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 58

❆ ♥♦♥✲t❤❡♦r❡♠

❇❛❞ ♥❡✇s✿ PAut(Γ(S)) ❞♦❡s ❞❡♣❡♥❞ ♦♥ t❤❡ s②st❡♠ ♦❢ ❣❡♥❡r❛t♦rs ❝❤♦s❡♥✿ t❤❡ ❜✐❣❣❡r t❤❡ s②st❡♠ ♦❢ ❣❡♥❡r❛t♦rs✱ t❤❡ s♠❛❧❧❡r ✐t ✐s✳ ❚❤❡ ❜❡st ✇❡ ❝❛♥ ❤♦♣❡ ❢♦r ♥♦✇✿ ✐❢ t❤❡ s②st❡♠ ♦❢ ❣❡♥❡r❛t♦rs ✐s ❧❛r❣❡ ❡♥♦✉❣❤✱ t❤❡♥ ✳ ❇❛❞ ♥❡✇s✿ ❡✈❡♥ ✇rt t♦ t❤❡ ❣❡♥❡r❛t✐♥❣ s②st❡♠ ✱ ✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 59

❆ ♥♦♥✲t❤❡♦r❡♠

❇❛❞ ♥❡✇s✿ PAut(Γ(S)) ❞♦❡s ❞❡♣❡♥❞ ♦♥ t❤❡ s②st❡♠ ♦❢ ❣❡♥❡r❛t♦rs ❝❤♦s❡♥✿ t❤❡ ❜✐❣❣❡r t❤❡ s②st❡♠ ♦❢ ❣❡♥❡r❛t♦rs✱ t❤❡ s♠❛❧❧❡r ✐t ✐s✳ ❚❤❡ ❜❡st ✇❡ ❝❛♥ ❤♦♣❡ ❢♦r ♥♦✇✿ ✐❢ t❤❡ s②st❡♠ ♦❢ ❣❡♥❡r❛t♦rs ✐s ❧❛r❣❡ ❡♥♦✉❣❤✱ t❤❡♥ ρ(S) = PAut(Γ(S))✳ ❇❛❞ ♥❡✇s✿ ❡✈❡♥ ✇rt t♦ t❤❡ ❣❡♥❡r❛t✐♥❣ s②st❡♠ ✱ ✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 60

❆ ♥♦♥✲t❤❡♦r❡♠

❇❛❞ ♥❡✇s✿ PAut(Γ(S)) ❞♦❡s ❞❡♣❡♥❞ ♦♥ t❤❡ s②st❡♠ ♦❢ ❣❡♥❡r❛t♦rs ❝❤♦s❡♥✿ t❤❡ ❜✐❣❣❡r t❤❡ s②st❡♠ ♦❢ ❣❡♥❡r❛t♦rs✱ t❤❡ s♠❛❧❧❡r ✐t ✐s✳ ❚❤❡ ❜❡st ✇❡ ❝❛♥ ❤♦♣❡ ❢♦r ♥♦✇✿ ✐❢ t❤❡ s②st❡♠ ♦❢ ❣❡♥❡r❛t♦rs ✐s ❧❛r❣❡ ❡♥♦✉❣❤✱ t❤❡♥ ρ(S) = PAut(Γ(S))✳ ❇❛❞ ♥❡✇s✿ ❡✈❡♥ ✇rt t♦ t❤❡ ❣❡♥❡r❛t✐♥❣ s②st❡♠ S✱

  • ρ(S) = PAut(Γ(S))✳

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 61

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❍❛♣♣② ❜✐rt❤❞❛②✦

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s

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SLIDE 62

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

❍❛♣♣② ❜✐rt❤❞❛②✦

◆ór❛ ❙③❛❦á❝s ■♥✈❡rs❡ ♠♦♥♦✐❞s ♦❢ ♣❛rt✐❛❧ ❣r❛♣❤ ❛✉t♦♠♦r♣❤✐s♠s