O Degree Graph dos grupos alternados e de outros grupos simples

(1)

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛

■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❖ ❉❡❣r❡❡ ●r❛♣❤ ❞♦s ❣r✉♣♦s ❛❧t❡r♥❛❞♦s ❡ ❞❡

♦✉tr♦s ❣r✉♣♦s s✐♠♣❧❡s

❆❧❧❛♥ ❑❛r❞❡❝ ▼❡ss✐❛s ❞❛ ❙✐❧✈❛

(2)

❯♥✐✈❡rs✐❞❛❞❡ ❞❡ ❇r❛sí❧✐❛

■♥st✐t✉t♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s

❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛

❖ ❉❡❣r❡❡ ●r❛♣❤ ❞♦s ❣r✉♣♦s ❛❧t❡r♥❛❞♦s ❡ ❞❡

♦✉tr♦s ❣r✉♣♦s s✐♠♣❧❡s

❆❧❧❛♥ ❑❛r❞❡❝ ▼❡ss✐❛s ❞❛ ❙✐❧✈❛✶

❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ❝♦♥❝❧✉sã♦ ❞♦ ▼❡str❛❞♦ ❡♠ ▼❛t❡♠át✐❝❛

❈♦♠✐ssã♦ ❡①❛♠✐♥❛❞♦r❛

Pr♦❢✳ ❉r✳ ▼❛r❝♦ ❆♥t♦♥✐♦ P❡❧❧❡❣r✐♥✐ ✭❖r✐❡♥t❛❞♦r✮ ▼❆❚✴❯♥❇

Pr♦❢❛✳ ❉r❛✳ ■r✐♥❛ ❙✈✐r✐❞♦✈❛ ▼❆❚✴❯♥❇

Pr♦❢✳ ❉r✳ ■❧✐r ❙♥♦♣❝❤❡ ▼❆❚✴❯❋❘❥

❇r❛sí❧✐❛✱ 05 ❞❡ ❋❡✈❡r❡✐r♦ ❞❡ 2013

(3)

❆❣r❛❞❡❝✐♠❡♥t♦

(4)

❉❡❞✐❝❛tór✐❛

(5)

✺

❘❡s✉♠♦

❖ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ é ✉♠❛ ✐♥tr♦❞✉çã♦ ❛♦ ❡st✉❞♦ ❞❡ ✉♠ ❣r❛❢♦ ❝❤❛♠❛❞♦ ❉❡❣r❡❡ ●r❛♣❤✳ ❊st❡ ❣r❛❢♦ é ❛ss♦❝✐❛❞♦ ❛♦s ❣r❛✉s ❞♦s ❝❛r❛❝t❡r❡s ❞❡ ✉♠ ❣r✉♣♦ ✜♥✐t♦ ♥♦ s❡❣✉✐♥t❡ ♠♦❞♦✿ ♦s ✈ért✐❝❡s sã♦ ♦s ♣r✐♠♦s q✉❡ ❞✐✈✐❞❡♠ ♦s ❣r❛✉s ❞♦s ❝❛r❛❝t❡r❡s ✐rr❡❞✉tí✈❡✐s ❡ ❞♦✐s ✈ért✐❝❡s p, q sã♦ ❝♦♥❡①♦s ❝♦♠ ✉♠❛ ❛r❡st❛ s❡ ♦

❣r✉♣♦ ♣♦ss✉✐ ✉♠ ❝❛rát❡r ✐rr❡❞✉tí✈❡❧ ❝✉❥♦ ❣r❛✉ é ❞✐✈✐sí✈❡❧ ♣❡❧♦ ♣r♦❞✉t♦pq✳ ❖

❉❡❣r❡❡ ●r❛♣❤ ❢♦✐ ❡st✉❞❛❞♦ ✐♥✐❝✐❛❧♠❡♥t❡ ❡♠ ❣r✉♣♦s s♦❧ú✈❡✐s ❡ ❛♣❡♥❛s ❛ ♣♦✉❝♦ t❡✈❡ s❡✉s ❡st✉❞♦s ❛✈❛♥ç❛❞♦s ♣❛r❛ ❣r✉♣♦s ♥ã♦ s♦❧ú✈❡✐s✳ ❉♦♥❛❧❞ ▲✳ ❲❤✐t❡ ❝♦♠♣❧❡t♦✉ ♦ ❡st✉❞♦ ♣❛r❛ ❣r✉♣♦s s✐♠♣❧❡s ❡♠2009❝♦♠ ♦ ❛rt✐❣♦ ❵❉❡❣r❡❡ ●r❛♣❤s

♦❢ ❙✐♠♣❧❡ ●r♦✉♣s✬✱ ♦♥❞❡ ❡❧❡ ❞❡s❝r❡✈❡ ♣❛r❛ t♦❞♦s ♦s ❣r✉♣♦s ✜♥✐t♦s s✐♠♣❧❡s ♦s ❝♦rr❡s♣♦♥❞❡♥t❡s ❉❡❣r❡❡ ●r❛♣❤s✳ ❱❛♠♦s ♥❡st❡ tr❛❜❛❧❤♦ ♠♦str❛r ❡st❡s ❡st✉❞♦s ♣❛r❛ t♦❞♦s ♦s ❣r✉♣♦s ❛❧t❡r♥❛❞♦s✱ ❡ ❛❧❣✉♥s ❣r✉♣♦s s✐♠♣❧❡s ❧✐♥❡❛r❡s✱ s✐♠♣❧ét✐❝♦s ❡ ✉♥✐tár✐♦s✳

❖ ♣r✐♥❝✐♣❛❧ r❡s✉❧t❛❞♦ q✉❡ ✈❛♠♦s ✐❧✉str❛r ❡♠ ❞❡t❛❧❤❡s é ♦ ❢❛t♦ q✉❡✱ s❡

n _≥ 9✱ ♦ ❉❡❣r❡❡ ●r❛♣❤ ❞♦ ❣r✉♣♦ ❛❧t❡r♥❛❞♦ _An é ✉♠ ❣r❛❢♦ ❝♦♠♣❧❡t♦✳ ❊st❡

r❡s✉❧t❛❞♦ ✉s❛ ✉♠❛ ❝♦♥❥❡❝t✉r❛ ❞❡ ❆❧✈✐s✱ ♣r♦✈❛❞❛ ♣♦r ❇❛rr② ❡ ❲❛r❞✳

(6)

✻

❆❜str❛❝t

❚❤❡ ♣r❡s❡♥t ✇♦r❦ ✐s ❛♥ ✐♥tr♦❞✉❝t✐♦♥ t♦ t❤❡ st✉❞② ♦❢ ❛ ❣r❛♣❤ ❝❛❧❧❡❞ ❉❡❣r❡❡ ●r❛♣❤✳ ❚❤✐s ❣r❛♣❤ ✐s ❛ss♦❝✐❛t❡❞ t♦ t❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ ❝❤❛r❛❝t❡rs ♦❢ ❛ ✜♥✐t❡ ❣r♦✉♣ ✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ✇❛②✿ t❤❡ ✈❡rt✐❝❡s ❛r❡ t❤❡ ♣r✐♠❡s t❤❛t ❞✐✈✐❞❡ t❤❡ ❞❡❣r❡❡s ♦❢ t❤❡ ✐rr❡❞✉❝✐❜❧❡ ❝❤❛r❛❝t❡rs ❛♥❞ t✇♦ ✈❡rt✐❝❡sp, q ❛r❡ ❝♦♥♥❡❝t❡❞ ✇✐t❤ ❛♥ ❡❞❣❡

✐❢ t❤❡ ❣r♦✉♣ ❤❛s ❛♥ ✐rr❡❞✉❝✐❜❧❡ ❝❤❛r❛❝t❡r ✇❤♦s❡ ❞❡❣r❡❡ ✐s ❞✐✈✐s✐❜❧❡ t❤❡ ♣r♦❞✉❝t

pq✳ ❖ ❉❡❣r❡❡ ●r❛♣❤ ✇❛s ✐♥✐t✐❛❧❧② st✉❞✐❡❞ ❢♦r s♦❧✉❜❧❡ ❣r♦✉♣s ❛♥❞ ♦♥❧② r❡❝❡♥t❧②

❛❧s♦ ❢♦r ♥♦♥ s♦❧✉❜❧❡ ❣r♦✉♣s✳ ■♥ 2009 ❉♦♥❛❧❞ ▲✳ ❲❤✐t❡ ❝♦♠♣❧❡t❡❞ t❤❡ st✉❞②

❢♦r s✐♠♣❧❡ ❣r♦✉♣s ✐♥ t❤❡ ♣❛♣❡r ❵❉❡❣r❡❡ ●r❛♣❤ ♦❢ ❙✐♠♣❧❡ ●r♦✉♣s✬✱ ✇❤❡r❡ ❤❡ ❞❡s❝r✐❜❡s ❢♦r ❛❧❧ ✜♥✐t❡ s✐♠♣❧❡ ❣r♦✉♣s t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❉❡❣r❡❡ ●r❛♣❤s✳ ■♥ t❤✐s ✇♦r❦✱ ✇❡ ✇✐❧❧ ✐❧❧✉str❛t❡ t❤❡s❡ st✉❞✐❡s ❢♦r ❛❧❧ ❛❧t❡r♥❛t✐♥❣ ❣r♦✉♣s ❛♥❞ s♦♠❡ s✐♠♣❧❡ ❧✐♥❡❛r✱ s②♠♣❧❡❝t✐❝ ❛♥❞ ✉♥✐t❛r② ❣r♦✉♣s✳

❚❤❡ ♠❛✐♥ r❡s✉❧t t❤❛t ✇❡ ✇✐❧❧ ❞❡s❝r✐❜❡ ✐♥ ❞❡t❛✐❧ ✐s t❤❡ ❢❛❝t t❤❛t ✐❢ n _≥ 9✱

t❤❡ ❉❡❣r❡❡ ●r❛♣❤ ♦❢ t❤❡ ❛❧t❡r♥❛t✐♥❣ ❣r♦✉♣ An ✐s ❛ ❝♦♠♣❧❡t❡ ❣r❛♣❤✳ ❚❤✐s

r❡s✉❧t ♠❛❦❡s ✉s❡ ♦❢ ❛ ❝♦♥❥❡❝t✉r❡ ♦❢ ❆❧✈✐s✱ ♣r♦✈❡❞ ❜② ❇❛rr② ❲❛r❞✳

(7)

❙✉♠ár✐♦

■♥tr♦❞✉çã♦ ✾

✶ ❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s ✶✶

✶✳✶ ❘❡♣r❡s❡♥t❛çõ❡s ❡ ❈❛r❛❝t❡r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ●r✉♣♦ ❙✐♠étr✐❝♦ Sn ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷

✷ ❯♠❛ ❈♦♥❥❡❝t✉r❛ ❞❡ ❆❧✈✐s ✹✶

✷✳✶ ❆ ❝♦♥❥❡❝t✉r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶ ✷✳✷ ❊s❝♦❧❤❛ ❞❛ ♣❛rt✐çã♦ ❞❡ n ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✶

✷✳✸ ❖ ❝♦♥❥✉♥t♦ P(αn) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✷✳✸✳✶ Pr✐♠♦s ♠é❞✐♦s q✉❛♥❞♦κ(n)é í♠♣❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼

✷✳✸✳✷ Pr✐♠♦s ♠é❞✐♦s q✉❛♥❞♦κ(n)❢♦r ♣❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✽

✷✳✸✳✸ Pr✐♠♦s P❡q✉❡♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✸

✸ ❉❡❣r❡❡ ●r❛♣❤s ✶✵✶

✸✳✶ ❈♦♥❝❡✐t♦s ❜ás✐❝♦s s♦❜r❡ ❣r❛❢♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✶ ✸✳✷ ❖ ❉❡❣r❡❡ ●r❛♣❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵✸ ✸✳✷✳✶ ❖ ❉❡❣r❡❡ ●r❛♣❤ ❞❡ ❛❧❣✉♥s ●r✉♣♦s ❋✐♥✐t♦s ❙✐♠♣❧❡s ✳ ✳ ✶✵✺

(8)

(9)

■♥tr♦❞✉çã♦

◆❛ t❡♦r✐❛ ❞❡ r❡♣r❡s❡♥t❛çõ❡s✱ ❛ t❛❜❡❧❛ ❞❡ ❝❛r❛❝t❡r❡s ✐rr❡❞✉tí✈❡✐s ❞❡ ✉♠ ❣r✉♣♦ ♣♦❞❡ ❢♦r♥❡❝❡r ♠✉✐t❛s ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦ ❣r✉♣♦✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿ s❡ ♦ ❣r✉♣♦ é ❛❜❡❧✐❛♥♦ ♦✉ ♥ã♦❀ s❡ t❡♠ s✉❜❣r✉♣♦s ♥♦r♠❛✐s ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ s❡ ♦ ❣r✉♣♦ é s✐♠♣❧❡s✱ ♦✉ s❡ ♦ ❣r✉♣♦ é s♦❧ú✈❡❧✳ ▼❛s ❡♠ ♣❛rt✐❝✉❧❛r✱ ♦s ❣r❛✉s ❞♦s ❝❛r❛❝t❡r❡s ✐rr❡❞✉tí✈❡✐s ❞❡ ✉♠ ❣r✉♣♦ t❛♠❜é♠ ❢♦r❛♠ ♠♦t✐✈♦s ❞❡ ❡st✉❞♦s ♣♦r ❞✐✈❡rs♦s ♠❛t❡♠át✐❝♦s ❡♠ r❡❧❛çã♦ ❛ ✉♠ ❣r❛❢♦ ❝❤❛♠❛❞♦ ❉❡❣r❡❡ ●r❛♣❤✳ ❖ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s ❞❡st❡ ❣r❛❢♦ é ♦ ❝♦♥❥✉♥t♦ ❞♦s ♣r✐♠♦s ❞✐✈✐s♦r❡s ❞♦s ❣r❛✉s ❞♦s ❝❛r❛❝t❡r❡s ✐rr❡❞✉tí✈❡✐s ❞❡ ✉♠ ❣r✉♣♦ ✜♥✐t♦G✳ ❯♠❛ ❛r❡st❛ ❧✐❣❛ ❞♦✐s

✈ért✐❝❡s p, q s❡ ❡①✐st✐r ❛❧❣✉♠ ❣r❛✉ ❞❡ ✉♠ ❝❛rát❡r ✐rr❡❞✉tí✈❡❧ ❞❡ G t❛❧ q✉❡ pq

♦ ❞✐✈✐❞❛✳ ■♥✐❝✐❛❧♠❡♥t❡ ❡st❡ ❣r❛❢♦ ❢♦✐ ❡st✉❞❛❞♦ s♦♠❡♥t❡ ♣❛r❛ ❣r✉♣♦s s♦❧ú✈❡✐s✱ ❡ só ❞❡♣♦✐s ❞❡s❡♥✈♦❧✈✐❞♦ ♣❛r❛ ♦✉tr♦s t✐♣♦s ❞❡ ❣r✉♣♦s✳ ❱❛♠♦s ♠♦str❛r ✉♠ ♣♦✉❝♦ ❞❡ss❡s ❡st✉❞♦s ❢❡✐t♦s ❡♠ ❣r✉♣♦s ♥ã♦ ❛❜❡❧✐❛♥♦s ✜♥✐t♦s s✐♠♣❧❡s✱ ♣♦✐s ❡♠ ❣r✉♣♦s ❛❜❡❧✐❛♥♦s ✜♥✐t♦s s✐♠♣❧❡s ✭♦s ❣r✉♣♦s ❞❡ ♦r❞❡♥s ♣r✐♠❛s✮ ❡ss❡s ❡st✉❞♦s ♥ã♦ sã♦ ✐♥t❡r❡ss❛♥t❡s ❞❡✈✐❞♦ ♦ ❝♦♥❥✉♥t♦ ❞❡ ✈ért✐❝❡s s❡r ✈❛③✐♦✳

❆q✉✐ ❞❡st❛❝❛♠♦s ❛❧❣✉♥s ❛rt✐❣♦s ♥♦ ❡st✉❞♦ ❞♦s ❉❡❣r❡❡ ●r❛♣❤s✱ q✉❡ sã♦ ❡❧❡s✿

• ❚❤❡ ❉✐❛♠❡t❡r ♦❢ t❤❡ ❈❤❛r❛❝t❡r ❉❡❣r❡❡ ●r❛♣❤[✶✽]✱ ❡s❝r✐t♦ ♣♦r ❖✳ ▼❛♥③✱

❲✳ ❲✐❧❧❡♠s ❡ ❚✳ ❘✳ ❲♦❧❢ ❡♠ 1989❀

• ❉✐❛♠❡t❡rs ♦❢ ❉❡❣r❡❡ ●r❛♣❤s ♦❢ ◆♦♥s♦❧✈❛❜❧❡ ●r♦✉♣s [✶✺]✱ ❡s❝r✐t♦ ♣♦r

▼❛r❦ ▲✳ ▲❡✇✐s ❡ ❉♦♥❛❧❞ ▲✳ ❲❤✐t❡ ❡♠2005❀ ❡

• ❉❡❣r❡❡ ●r❛♣❤ ♦❢ ❙✐♠♣❧❡ ●r♦✉♣s [✷✶]✱ ❡s❝r✐t♦ ♣♦r ❉♦♥❛❧❞ ▲✳ ❲❤✐t❡ ❡♠ 2009✳

❊st❡ ú❧t✐♠♦ ❛rt✐❣♦ é ✉♠ r❡s✉♠♦ ❝♦♠♣❧❡t♦ ❞♦s ❡st✉❞♦s ❢❡✐t♦s ❞✉r❛♥t❡ ❛♥♦s ♣♦r ❲❤✐t❡✳ P❛r❛ ♦s ❣r✉♣♦s ❛❧t❡r♥❛❞♦s ❡❧❡ ❝♦♥t♦✉ ❝♦♠ ❛ ❛❥✉❞❛ ❞❡ ✉♠❛ ❝♦♥❥❡❝t✉r❛ ❢❡✐t❛ ♣♦r ❉✳ ▲✳ ❆❧✈✐s ❡♠1991♥♦ ❛rt✐❣♦ ❵❈❤❛r❛❝t❡r ❉❡❣r❡❡s ♦❢ ❙✐♠♣❧❡ ●r♦✉♣s✬ [✶]✱ ♣r♦❞✉③✐❞♦ ♣♦r ❡❧❡ ❆❧✈✐s ❡ ❇❛rr②✳ ❊st❛ ❝♦♥❥❡❝t✉r❛ ❞✐③ ❡①✐st✐r ✉♠ ❝❛rát❡r

✐rr❡❞✉tí✈❡❧ χ ❞♦ ❣r✉♣♦ ❛❧t❡r♥❛❞♦ An ❝♦♠ n ≥ 15✱ t❛❧ q✉❡ t♦❞♦ ♣r✐♠♦p ≤n

❞✐✈✐❞❡ χ(1)✳ ❊st❛ ❝♦♥❥❡❝t✉r❛ ❢♦✐ ♠♦str❛❞❛ ♣♦r ❇❛rr② ❡ ❲❛r❞ ♥♦ ❛rt✐❣♦ ❵❖♥

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✶✵ ❙❯▼➪❘■❖

❝♦♥❥❡❝t✉r❡ ♦❢ ❆❧✈✐s✬ ❡♠ 2005✳ ❲❤✐t❡ t❛♠❜é♠ ❝♦♥t♦✉ ❝♦♠ ❛ ❛❥✉❞❛ ❞❡ ♦✉tr♦s

❛rt✐❣♦s✱ ❝♦♠♦ ♣♦r ❡①❡♠♣❧♦✿

• ❚❤❡ ❈❤❛r❛❝t❡r ♦❢ t❤❡ ❋✐♥✐t❡ ❙✐♠♣❧❡❝t✐❝ ●r✉♣♦sSp(4, q) [✷✵]✱ ❡s❝r✐t♦ ♣♦r

❇✳ ❙r✐♥✐✈❛s❛♥ ❡♠ 1968❀

• ❚❤❡ ❈❤❛r❛❝t❡r ♦❢ t❤❡ ❋✐♥✐t❡ ❙✐♠♣❧❡❝t✐❝ ●r♦✉♣s Sp(4, q)✱ q = 2f _[_✼_]_✱

❡s❝r✐t♦ ♣♦r ❍✳ ❊♥♦♠♦t♦ ❡♠1972❀

• ❚❤❡ ❈❤❛r❛❝t❡r ❚❛❜❧❡ ❢♦r SL(3, q)✱ SU(3, q2₎_✱ _{P SL}₍₃_{, q}₎ _❡ _{P SU}₍₃_{, q}2₎

[✽]✱ ❡s❝r✐t♦ ♣♦r ❏✳ ❙✳ ❋r❛♠❡ ❡ ❲✳ ❆✳ ❙✐♠♣s♦♥ ❡♠ 1973✳

❊st❡s ❛rt✐❣♦s ♦ ❛❥✉r❛♠ ♥♦ ❡st✉❞♦ ❞♦s ❉❡❣r❡❡ ●r❛♣❤s ❞❛s ❢❛♠í❧✐❛s ❞❡ ❣r✉♣♦s s✐♠♣❧❡s✿ Pr♦❥❡t✐✈♦ ❊s♣❡❝✐❛❧ ❙✐♠♣❧ét✐❝♦ P Sp(4, q)✱ Pr♦❥❡t✐✈♦ ❊s♣❡❝✐❛❧ ▲✐♥❡❛r

P SL(3, q)✱ Pr♦❥❡t✐✈♦ ❊s♣❡❝✐❛❧ ❯♥✐tár✐♦P SU(3, q2₎_{✳ ❖s ♦✉tr♦s ❣r✉♣♦s ✜♥✐t♦s}

s✐♠♣❧❡s q✉❡ ✈❡r❡♠♦s ♦s ❡st✉❞♦s ❞♦s s❡✉s ❉❡❣r❡❡ ●r❛♣❤s é ❛ ❢❛♠í❧✐❛ ❞❡ ❣r✉♣♦s s✐♠♣❧❡s Pr♦❥❡t✐✈♦ ❊s♣❡❝✐❛❧ ▲✐♥❡❛rP SL(2, q)✱ q✉❡ t❡♠ s✉❛ t❛❜❡❧❛ ❞❡ ❝❛r❛❝t❡r❡s

✐rr❡❞✉tí✈❡✐s✱ ♣♦r ❡①❡♠♣❧♦✱ ♥♦ ❧✐✈r♦ ❵●r♦✉♣ ❘❡♣r❡s❡♥t❛t✐♦♥ ❚❤❡♦r②✱ ♣❛rt ❆✬[✻]✱

❡s❝r✐t♦ ♣♦r ❉♦r♥❤♦✛ ❡♠ 1971✱ ❡ ♦s 26 ❣r✉♣♦s s✐♠♣❧❡s ❡s♣♦rá❞✐❝♦s q✉❡ tê♠

t♦❞❛s ❛s t❛❜❡❧❛s ❞❡ ❝❛r❛❝t❡r❡s ✐rr❡❞✉tí✈❡✐s ♠♦str❛❞❛s ♥♦ ❧✐✈r♦ ❵❆t❧❛s ♦❢ ❋✐♥✐t❡ ●r♦✉♣s✬ [✹]✱ ❡s❝r✐t♦ ♣♦r ❏✳ ❍✳ ❈♦♥✇❛②✱ ❘✳ ❚✳ ❈✉rt✐s✱ ❙✳ P✳ ◆♦rt♦♥✱ ❘✳ ❆✳

P❛r❦❡r✱ ❘✳ ❆✳ ❲✐❧s♦♥ ❡♠ 1985✳

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❈❛♣ít✉❧♦ ✶

❘❡s✉❧t❛❞♦s ♣r❡❧✐♠✐♥❛r❡s

❊st❡ ♣r✐♠❡✐r♦ ❝❛♣✐t✉❧♦ t❡♠ ❝♦♠♦ ♦❜❥❡t✐✈♦ ✐♥tr♦❞✉③✐r t♦❞♦s ♦s r❡s✉❧t❛❞♦s ✐♠✲ ♣♦rt❛♥t❡s q✉❡ ✈❡♥❤❛♠♦s ❛ ♣r❡❝✐s❛r ♥♦ ❞❡❝♦rr❡r ❞♦ ♥♦ss♦ tr❛❜❛❧❤♦ s♦❜r❡ ❛ ❚❡♦r✐❛ ❞❡ ❘❡♣r❡s❡♥t❛çõ❡s ❞❡ ❣r✉♣♦s s✐♠étr✐❝♦s✳

✶✳✶ ❘❡♣r❡s❡♥t❛çõ❡s ❡ ❈❛r❛❝t❡r❡s

❙❡❥❛ F ✉♠ ❝♦r♣♦ ❡ s❡❥❛ V ✉♠ ❡s♣❛ç♦ ✈❡t♦r✐❛❧ ❞❡ ❞✐♠❡♥sã♦ ✜♥✐t❛ s♦❜r❡ F✱

❞✐r❡♠♦s ♥❡st❡ ❝❛s♦ q✉❡ V é ✉♠ F✲❡s♣❛ç♦✳

❉❡✜♥✐çã♦ ✶✳✶✳✶ ✭❘❡♣r❡s❡♥t❛çã♦✮✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ s❡❥❛ V ✉♠ F✲❡s♣❛ç♦✳

❯♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ G ❡♠ V é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s ϕ : G −→

GL(V)✱ s❡♥❞♦ GL(V) ♦ ❣r✉♣♦ ❞♦s F✲❛✉t♦♠♦r✜s♠♦s ❞❡ V✳ ◆❡st❡ ❝❛s♦✱ V é

❝❤❛♠❛❞♦ ❞❡ ✉♠ F✲❡s♣❛ç♦ ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ G ❡ s✉❛ ❞✐♠❡♥sã♦ s❡rá ❝❤❛✲

♠❛❞❛ ❞❡ ❣r❛✉ ❞❛ r❡♣r❡s❡♥t❛çã♦ ϕ✳

◗✉❛♥❞♦ V ❢♦r ✉♠ F✲❡s♣❛ç♦ ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ G✱ ❞❡♥♦t❡♠♦s ♣♦r ϕg :

V _−→V ❛ ✐♠❛❣❡♠ ϕ(g)_∈GL(V)✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✷✳ ❚♦♠❡♠♦s ♣♦r V ✉♠ F✲❡s♣❛ç♦ ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ G

❝♦♠ ❞✐♠❡♥sã♦n✳ ❉❡♥♦t❡♠♦s ♣♦rGL(n,F)♦ ❣r✉♣♦ ❞❛s ♠❛tr✐③❡sn×n ✐♥✈❡r✲

sí✈❡✐s ❝♦♠ ❡♥tr❛❞❛s ❡♠ F✳ ❙❡ ✜①❛r♠♦s ✉♠❛ ❜❛s❡ ❞❡ V ♣♦❞❡♠♦s ❞❡✜♥✐r ✉♠

✐s♦♠♦r✜s♠♦

ψ :GL(V)_−→GL(n,F),

❛ss♦❝✐❛♥❞♦ ❛ ❝❛❞❛ ❛✉t♦♠♦r✜s♠♦ ϕ ∈ GL(V) ❛ s✉❛ r❡s♣❡❝t✐✈❛ ♠❛tr✐③ ❡♠

r❡❧❛çã♦ à ❜❛s❡ ❝♦♥s✐❞❡r❛❞❛✳

❊♥tã♦✱ s✐♠✐❧❛r♠❡♥t❡ ❞❡✜♥✐♠♦s r❡♣r❡s❡♥t❛çã♦ ♠❛tr✐❝✐❛❧ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳

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✶✷ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙

❉❡✜♥✐çã♦ ✶✳✶✳✸ ✭❘❡♣r❡s❡♥t❛çã♦ ▼❛tr✐❝✐❛❧✮✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ s❡❥❛ F ✉♠

❝♦r♣♦✳ ❯♠❛ r❡♣r❡s❡♥t❛çã♦ ♠❛tr✐❝✐❛❧ ❞❡ G ❝♦♠ ❣r❛✉ n s♦❜r❡ F é ✉♠ ❤♦♠♦✲

♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s ψ :G−→GL(n,F)✳

❆ss✉♠❛ q✉❡ V, W sã♦ F✲❡s♣❛ç♦s ❞❡ r❡♣r❡s❡♥t❛çõ❡s ❞❡ G ♣❡❧❛s s❡❣✉✐♥t❡s

r❡♣r❡s❡♥t❛çõ❡s

ϕ :G_−→GL(V) ❡ ϑ:G_−→GL(W).

❉✐r❡♠♦s q✉❡ϕ ❡ϑ sã♦ r❡♣r❡s❡♥t❛çõ❡s ❡q✉✐✈❛❧❡♥t❡s s❡ ❡①✐st✐r ✉♠ ✐s♦♠♦r✜s♠♦ τ :V −→W t❛❧ q✉❡

ϑg =τ◦ϕg◦τ−1, ∀ g ∈G.

❉❡st❛ ❢♦r♠❛✱ ❞✉❛s r❡♣r❡s❡♥t❛çõ❡s ♠❛tr✐❝✐❛✐s ❞❡ Gs❡❥❛♠ ❡❧❛s

ψ :G−→GL(n,F) ❡ξ :G−→GL(n,F),

sã♦ r❡♣r❡s❡♥t❛çõ❡s ❡q✉✐✈❛❧❡♥t❡s s❡ ❡①✐st✐r ✉♠❛ ♠❛tr✐③ ✐♥✈❡rsí✈❡❧U _∈GL(n,F)

t❛❧ q✉❡

ψg =U ξgU−1, ∀ g ∈G.

❋❛❝✐❧♠❡♥t❡ s❡ ♦❜s❡r✈❛ q✉❡ ♣❛r❛ ❞✉❛s r❡♣r❡s❡♥t❛çõ❡s ♠❛tr✐❝✐❛✐s ♦❜t✐❞❛s ❞♦ ♠❡s♠♦ F✲❡s♣❛ç♦ ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ G✱ ♣♦r s✐♠♣❧❡s♠❡♥t❡ ♠✉❞❛♥❞♦ ❛ ❜❛s❡

❝♦♥s✐❞❡r❛❞❛ s♦❜r❡F✱ t❡♠♦s ❡ss❛s r❡♣r❡s❡♥t❛çõ❡s ❡q✉✐✈❛❧❡♥t❡s✳

❉❡✜♥✐çã♦ ✶✳✶✳✹✳ ❙❡❥❛ ϕ : G _−→ GL(V) ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞♦ ❣r✉♣♦ G✳

❯♠ s✉❜❡s♣❛ç♦ W ⊆V é ❞✐t♦ G✲s✉❜❡s♣❛ç♦ ❞❡V s❡ ϕgW ⊆W ♣❛r❛ ∀ g ∈G✳

❚♦♠❡♠♦s W ❝♦♠♦G✲s✉❜❡s♣❛ç♦ ❞❡ V✱ ❡♥tã♦ ϕ ✐♥❞✉③✐rá ✉♠❛ r❡♣r❡s❡♥t❛✲

çã♦ ❞❡G ❡♠ W✳ ❉❡♥♦t❡♠♦s ❡ss❛ r❡♣r❡s❡♥t❛çã♦ ♣♦r ϕ _↓W✳

❙❡❥❛ V ✉♠ F✲❡s♣❛ç♦ ❡ s❡❥❛♠ U1, . . . , Ut s✉❜❡s♣❛ç♦s ✈❡t♦r✐❛✐s ❞❡ V ❝♦♠

❞✐♠❡♥sõ❡sa1, . . . , at✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❊♥tã♦ ❛ s♦♠❛ ❞❡st❡s s✉❜❡s♣❛ç♦s s❡rá

❞❡✜♥✐❞❛ ♣♦r

U1+· · ·+Ut={u1+· · ·+ut, ui ∈Ui ❝♦♠ 1≤i≤t}.

■r❡♠♦s ❞✐③❡r q✉❡ ❛ s♦♠❛ U1 +· · ·+Ut s❡rá ❞✐r❡t❛✱ s❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡st❛

s♦♠❛ ♣♦❞❡rá s❡r ❡s❝r✐t♦ ❞❡ ♠❛♥❡✐r❛ ú♥✐❝❛

u1+· · ·+ut, ui ∈Ui ❝♦♠ 1≤i≤t.

❈❛s♦ ✐st♦ ❛❝♦♥t❡ç❛✱ ❞❡♥♦t❛r❡♠♦s ♣♦rU1⊕ · · · ⊕Uts❡♥❞♦ ❛ s♦♠❛ ❞✐r❡t❛ ✭♥❡st❡

(13)

✶✳✶✳ ❘❊P❘❊❙❊◆❚❆➬Õ❊❙ ❊ ❈❆❘❆❈❚❊❘❊❙ ✶✸

❉❡✜♥✐çã♦ ✶✳✶✳✺✳ ❙❡❥❛ V ✉♠ F✲❡s♣❛ç♦ ❞❡ r❡♣r❡s❡♥t❛çã♦ ❞❡ G ♦❜t✐❞♦ ♣♦r ϕ:G_→GL(V)✳ ❉✐r❡♠♦s q✉❡ ϕ é:

(✐) ■rr❡❞✉tí✈❡❧✱ s❡ ♦s ú♥✐❝♦s G✲s✉❜❡s♣❛ç♦s ❞❡ V ❢♦r❡♠ _{0_} ❡ V;

(✐✐) ❘❡❞✉tí✈❡❧✱ s❡ ♥ã♦ ❢♦r ✐rr❡❞✉tí✈❡❧;

(✐✐✐) ❈♦♠♣❧❡t❛♠❡♥t❡ r❡❞✉tí✈❡❧✱ s❡ V =V1⊕ · · · ⊕Vt ♦♥❞❡ ❝❛❞❛ Vi ♣❛r❛ 1 ≤

i ≤ t✱ é ✉♠ G✲s✉❜❡s♣❛ç♦ ✐rr❡❞✉tí✈❡❧✳ ◆❡st❡ ❝❛s♦✱ ❡s❝r❡✈❡r❡♠♦s ϕ =

ϕ1⊕· · ·⊕ϕt s❡♥❞♦ϕi =ϕ ↓_Vi ♣❛r❛1≤i≤tr❡♣r❡s❡♥t❛çõ❡s ✐rr❡❞✉tí✈❡✐s

❞❡ G✳

❙❡❥❛ ψ :G_→GL(n,F) ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♠❛tr✐❝✐❛❧ ❞♦ ❣r✉♣♦ G s♦❜r❡ ♦ F✲❡s♣❛ç♦V ❡ s❡❥❛ W ✉♠ G✲s✉❜❡s♣❛ç♦ ❞❡ V ❞❡ ❞✐♠❡♥sã♦m✳ ❉✐r❡♠♦s q✉❡ ψ

é ✉♠❛ r❡♣r❡s❡♥t❛çã♦ r❡❞✉tí✈❡❧✱ s❡ ❡①✐st✐r ✉♠❛ ♠❛tr✐③ n_×n ✐♥✈❡rsí✈❡❧C ❝♦♠

❝♦❡✜❝✐❡♥t❡s ❡♠ F t❛❧ q✉❡ _∀g _∈G

C−1_·ψg ·C= 

   

φg ωg

0 · · · 0

✳✳✳ ✳✳✳ ✳✳✳

0 _{· · ·} 0

θg 

   

,

♦♥❞❡φ:G_→GL(m,F)❡θ:G_→GL(n₋m,F)sã♦ r❡♣r❡s❡♥t❛çõ❡s ♠❛tr✐❝✐❛✐s

❞❡G✱ ❡ ♦ ✐♥t❡✐r♦ m ♥ã♦ ❞❡♣❡♥❞❡ ❞❛ ❡s❝♦❧❤❛ ❞❡ g✳

❆ r❡♣r❡s❡♥t❛çã♦ ψ é ❞✐t❛ s❡r ✐rr❡❞✉tí✈❡❧✱ s❡ ♥ã♦ ❢♦r r❡❞✉tí✈❡❧ ❡ s❡rá ❝♦♠✲

♣❧❡t❛♠❡♥t❡ r❡❞✉tí✈❡❧ ♦✉ ❞❡❝♦♠♣♦♥í✈❡❧✱ s❡ ❡①✐st✐r ✉♠❛ ♠❛tr✐③n_×n ✐♥✈❡rsí✈❡❧ C ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ F t❛❧ q✉❡ _∀g ∈G

C−1_·ψ(g)_·C =



     

φ1(g) 0 0 · · · 0

0 φ2(g) 0 · · · 0

✳✳✳ 0 ✳✳✳ ✳✳✳ ✳✳✳

✳✳✳ ✳✳✳ ✳✳✳ ✳✳✳ 0 0 0 · · · 0 φm(g)



     

=φ1(g)⊕ · · · ⊕φm(g),

♦♥❞❡ φi sã♦ t♦❞❛s r❡♣r❡s❡♥t❛çõ❡s ♠❛tr✐❝✐❛✐s ✐rr❡❞✉tí✈❡✐s ❞❡ G✳

❯♠ ❞♦s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s s♦❜r❡ ❛ r❡❞✉t✐❜✐❧✐❞❛❞❡ ❞❡ r❡♣r❡s❡♥t❛çõ❡s é ❞❡✈✐❞♦ ❛ ❍❡✐♥r✐❝❦ ▼❛s❝❤❦❡ q✉❡ ♥♦s ❞✐③ s♦❜r❡ ❞❡t❡r♠✐♥❛❞❛s ❝✐r❝✉♥stâ♥❝✐❛s q✉❡ t♦❞❛ r❡♣r❡s❡♥t❛çã♦ é ❝♦♠♣❧❡t❛♠❡♥t❡ r❡❞✉tí✈❡❧✳

❚❡♦r❡♠❛ ✶✳✶✳✻ ✭❚❡♦r❡♠❛ ❞❡ ▼❛s❝❤❦❡✮✳ ❙❡❥❛ G✉♠ ❣r✉♣♦ ✜♥✐t♦ ❡ s❡❥❛F ✉♠

❝♦r♣♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ q✉❡ ♥ã♦ ❞✐✈✐❞❡ ❛ ♦r❞❡♠ ❞❡ G✳ ❊♥tã♦✱ t♦❞❛ r❡♣r❡s❡♥✲

(14)

✶✹ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙

❉❡♠♦♥str❛çã♦✳ ❱❡r [✶✶] ♣✳ 4✳

❏✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❛ r❡❞✉t✐❜✐❧✐❞❛❞❡ ❞❛s r❡♣r❡s❡♥t❛çõ❡s✱ ❡stá ♦ ❡st✉❞♦ ❞♦s tr❛ç♦s ❞❡st❛s r❡♣r❡s❡♥t❛çõ❡s✱ ♣♦✐s ♥❡❧❡s é ♣♦ssí✈❡❧ ❞❡s❝♦❜r✐r ♠✉✐✲ t❛s ✐♥❢♦r♠❛çõ❡s s♦❜r❡ ♦ ♥♦ss♦ ❣r✉♣♦✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ q✉❛♥❞♦ ❛s r❡♣r❡s❡♥t❛✲ çõ❡s sã♦ ✐rr❡❞✉tí✈❡✐s✳

❉❡✜♥✐çã♦ ✶✳✶✳✼ ✭❈❛rát❡r✮✳ ❙❡❥❛ ϕ : G _−→ GL(n,F) ✉♠❛ r❡♣r❡s❡♥t❛çã♦

♠❛tr✐❝✐❛❧ ❞♦ ❣r✉♣♦ G s♦❜r❡ ♦ ❝♦r♣♦ F✳ ❉❡✜♥❛ θ : G _−→ F ❛ ❢✉♥çã♦ θ(g) =

tr(ϕ(g))✱ g ∈ G✳ ❈❤❛♠❡♠♦s θ ❞❡ ❝❛rát❡r ❞❡ G ❛ss♦❝✐❛❞♦ ❛ r❡♣r❡s❡♥t❛çã♦ ϕ✳ ◗✉❛♥❞♦ ϕ ❢♦r ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ✐rr❡❞✉tí✈❡❧✱ ❡♥tã♦ ❞✐r❡♠♦s q✉❡ θ é ✉♠

❝❛rát❡r ✐rr❡❞✉tí✈❡❧✳

❆ss✉♠❛ ϕ : G _→ GL(n,F) ❝♦♠♦ ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♠❛tr✐❝✐❛❧ ❞♦ ❣r✉♣♦

G✳ ❙❡ψ é ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ❞❡G ❡q✉✐✈❛❧❡♥t❡ ❛ϕ✱ ❡♥tã♦ ♦s ❝❛r❛❝t❡r❡s ❞❡ G

❛ss♦❝✐❛❞♦s ❛s r❡♣r❡s❡♥t❛çõ❡sϕ❡ψsã♦ ✐❣✉❛✐s✳ ■st♦ ♣♦rq✉❡ ♦ tr❛ç♦ ❞❡ ♠❛tr✐③❡s

é ✐♥✈❛r✐❛♥t❡ ♣♦r ❝♦♥❥✉❣❛çã♦✳

❉❡♥♦t❡♠♦s ♣♦r C ♦ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s✳ ❆ ♣r♦♣♦s✐çã♦ ❛❜❛✐①♦

♥♦s ♠♦str❛ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s s♦❜r❡ r❡♣r❡s❡♥t❛çõ❡s r❡❛❧✐③❛❞❛s s♦❜r❡ ♦ ❝♦r♣♦ C✳ ❆❧é♠ ❞✐ss♦✱ ❛ ♣❛rt✐r ❞❡ ❛❣♦r❛ ♥♦ss♦s ❣r✉♣♦s s❡rã♦ s❡♠♣r❡ ✜♥✐t♦s✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✽✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦✳ ❊♥tã♦✱ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s sã♦

s❛t✐s❢❡✐t❛s:

(✐) ❊①✐st❡♠r ❞✐st✐♥t❛s r❡♣r❡s❡♥t❛çõ❡s ✐rr❡❞✉tí✈❡✐s ϕ1, . . . , ϕr ❞❡ G s♦❜r❡ C

r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛ ♠❡♥♦s ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❝♦♠ ❣r❛✉sn1 = 1, n2, . . . , nr✱

♦♥❞❡ r é ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛çã♦ ❞❡ G✳ ❊♠ ♣❛rt✐❝✉✲

❧❛r✱ ✜①❛♠♦s ♣♦rϕ1 ❛ r❡♣r❡s❡♥t❛çã♦ ♦❜t✐❞❛ ❡♥✈✐❛♥❞♦ t♦❞♦s ♦s ❡❧❡♠❡♥t♦s

❞♦ ❣r✉♣♦ G ♣❛r❛ ✐❞❡♥t✐❞❛❞❡ 1C❀ ❡ss❛ r❡♣r❡s❡♥t❛çã♦ s❡rá ❝❤❛♠❛❞❛ r❡✲ ♣r❡s❡♥t❛çã♦ tr✐✈✐❛❧ ❞❡G;

(✐✐) _|G_|= 1 +n2

2+· · ·+n2r;

(✐✐✐) ❈❛❞❛ ni ❞✐✈✐❞❡ |G|✳

❉❡♠♦♥str❛çã♦✳ ❱❡r [✶✶] ♣✳ 5−17✳

▲❡♠❛ ✶✳✶✳✾✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦✳ ❆ss✉♠❛ ♣♦r ϕi :G−→GL(ni,C) ❛s r❡♣r❡✲

s❡♥t❛çõ❡s ♠❛tr✐❝✐❛✐s ✐rr❡❞✉tí✈❡✐s ❞❡G❡θi s❡✉s ❝❛r❛❝t❡r❡s ❛ss♦❝✐❛❞♦s✳ ❊♥tã♦✱

❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s sã♦ s❛t✐s❢❡✐t❛s:

(✐) θi(1) é ♦ ❣r❛✉ ❞❡ ϕi ❡ θi(1) ❞✐✈✐❞❡ |G|;

(✐✐) ∀g ∈ G✱ θi(g) é s♦♠❛ ❞❡ r❛í③❡s d✲és✐♠❛ ❞❛ ✉♥✐❞❛❞❡ ♦♥❞❡ d é ❛ ♦r❞❡♠

(15)

✶✳✶✳ ❘❊P❘❊❙❊◆❚❆➬Õ❊❙ ❊ ❈❆❘❆❈❚❊❘❊❙ ✶✺

(✐✐✐) _∀g _∈G✱ θi(g−1) = θi(g);

(✐✈) θi é ✉♠❛ ❢✉♥çã♦ ❝♦♥st❛♥t❡ s♦❜r❡ ❛s ❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛çã♦ ❞❡ G;

(✈) ❙❡ϕ(g) =ϕ1(g)⊕· · ·⊕ϕt(g)✱ ∀g ∈G✱ ❡♥tã♦θ(g) =θ1(g) +· · ·+θt(g);

(✈✐) ❙❡❥❛♠ ϕi = [ars] ❡ ϕj r❡♣r❡s❡♥t❛çõ❡s ♠❛tr✐❝✐❛✐s ❞❡ ❣r❛✉s ni ❡ nj r❡s✲

♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡✜♥✐♠♦s ♣♦r

ϕi⊗ϕj = 

 

a11ϕj · · · a1nϕj

✳✳✳ ✳✳✳ ✳✳✳

an1ϕj · · · annϕj 

 

❛ r❡♣r❡s❡♥t❛çã♦ ❞❡ ❣r❛✉ (ni·nj) ♦❜t✐❞♦ ♣❡❧♦ ♣r♦❞✉t♦ ❞❡ ❑r♦♥❡❝❦❡r ❞❡

ϕi ❡ ϕj✳ ❙❡ ♣❛r❛ t♦❞♦ g ∈ G✱ ϕ(g) = ϕ1(g) ⊗ · · · ⊗ ϕt(g)✱ ❡♥tã♦

θ(g) = θ1(g)· · · θt(g);

(✈✐✐) ❉✉❛s r❡♣r❡s❡♥t❛çõ❡s tê♠ ♦ ♠❡s♠♦ ❝❛rát❡r s❡ ❡ s♦♠❡♥t❡ sã♦ ❡q✉✐✈❛❧❡♥✲

t❡s✳

❉❡♠♦♥str❛çã♦✳ ❱❡r [✶✶]♣✳ 13₋35✳

❆ ♣❛rt✐r ❞❡ ❛❣♦r❛ ❛ss✉♠❛ q✉❡ ♥♦ss❛s r❡♣r❡s❡♥t❛çõ❡s s❡rã♦ r❡❛❧✐③❛❞❛s s♦❜r❡

C✳ ❉❡✜♥❛ ♣♦r_Irr(G)♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ❝❛r❛❝t❡r❡s ✐rr❡❞✉tí✈❡✐s ❝♦♠♣❧❡①♦s

❞♦ ❣r✉♣♦ G✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✵✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ s❡❥❛ f : G −→ C ✉♠❛ ❢✉♥çã♦✳ ❉✐✲

r❡♠♦s q✉❡ f é ✉♠❛ ❢✉♥çã♦ ❞❡ ❝❧❛ss❡ ❞❡ G q✉❛♥❞♦ f ❢♦r ❝♦♥st❛♥t❡ s♦❜r❡ ❛s

❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛çã♦ ❞❡ G✳ ❉❡♥♦t❡♠♦s ♣♦r _FC(G) ♦ ❝♦♥❥✉♥t♦ ❞❛s ❢✉♥çõ❡s

❞❡ ❝❧❛ss❡s ❞♦ ❣r✉♣♦ G✳

◆♦t❡ q✉❡FC(G)é ✉♠C✲❡s♣❛ç♦ ❝♦♠ ❞✐♠❡♥sã♦ ✐❣✉❛❧ ❛♦ ♥ú♠❡r♦ ❞❡ ❝❧❛ss❡s

❞❡ ❝♦♥❥✉❣❛çã♦ ❞❡ G✳ ◆♦t❡ ❛✐♥❞❛ q✉❡ t♦❞♦s ♦s ❝❛r❛❝t❡r❡s ❞❡ G sã♦ ❞❡ ❢❛t♦

❢✉♥çõ❡s ❞❡ ❝❧❛ss❡s ❞❡ G✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ♦ ❝♦♥❥✉♥t♦ _Irr(G)_{⊆ FC}(G)✳

❚❡♦r❡♠❛ ✶✳✶✳✶✶✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦✳ ❚♦❞♦ ❡❧❡♠❡♥t♦ ϕ ❡♠ _FC(G) ♣♦❞❡ s❡r

❡①♣r❡ss❛❞♦ ✉♥✐❝❛♠❡♥t❡ ♥❛ ❢♦r♠❛

ϕ = X

χ∈Irr(G)

aχχ, aχ ∈C.

■st♦ é✱ Irr(G) é ✉♠❛ ❜❛s❡ ♣❛r❛ FC(G)✳

(16)

✶✻ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙

❉❡✜♥✐çã♦ ✶✳✶✳✶✷✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ s❡❥❛♠ ϕ, ψ ❡❧❡♠❡♥t♦s ❡♠ _FC(G)✳

❉❡✜♥✐♠♦s ❡♠ FC(G) ♦ s❡❣✉✐♥t❡ ♣r♦❞✉t♦

(ϕ, ψ)_G = 1

|G|

X

g∈G

ϕ(g)ψ(g).

❉❡st❛q✉❡♠♦s ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡st❡ ♣r♦❞✉t♦✿ ✭✐✮ (ϕ, ψ)_G= (ψ, ϕ)_G;

✭✐✐✮ (ϕ, ϕ)_G >0 ❛ ♠❡♥♦s q✉❡ ϕ(g) = 0, _∀g _∈G;

✭✐✐✐✮ (a1ϕ1+a2ϕ2, ψ)G=a1(ϕ1, ψ)G+a2(ϕ2, ψ)G;

✭✐✈✮ (ϕ, b1ψ1+b2ψ2)_G =b1(ϕ, ψ1)_G+b2(ϕ, ψ2)_G.

❊♥tã♦✱ ❝♦♠ ❡st❛s ♣r♦♣r✐❡❞❛❞❡s (, )_G é ✉♠ ♣r♦❞✉t♦ ❍❡r♠✐t✐❛♥♦ ♥♦ C✲

❡s♣❛ç♦FC(G)✳

❚❡♦r❡♠❛ ✶✳✶✳✶✸✳ ❙❡❥❛♠ χ ❡ θ ❝❛r❛❝t❡r❡s ❞♦ ❣r✉♣♦ G✳ ❊♥tã♦✱ ❛s s❡❣✉✐♥t❡s

♣r♦♣r✐❡❞❛❞❡s sã♦ s❛t✐s❢❡✐t❛s:

(✐) χ_{∈ I}rr(G)_⇐⇒(χ, χ)_G = 1;

(✐✐) ❙❡ χ, θ∈ Irr(G) ❡ θ 6=χ✱ ❡♥tã♦ (χ, θ)_G = 0; (✐✐✐) (θ, χ)_G é ✉♠ ✐♥t❡✐r♦ ♥ã♦ ♥❡❣❛t✐✈♦;

(✐✈) (θ, χ)_G= (χ, θ)_G✳

❉❡♠♦♥str❛çã♦✳ ❱❡r [✶✶] ♣✳ 20₋21✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✶✹✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ _Irr(G) = _{χ1 = 1G, χ2, . . . , χr}✳

❊♥tã♦✱ Irr(G) é ✉♠❛ ❜❛s❡ ♦rt♦♥♦r♠❛❧ ❞❡ _FC(G) ❝♦♠ r❡s♣❡✐t♦ ❛♦ ♣r♦❞✉t♦ ( , )_G✳

❉❡♠♦♥str❛çã♦✳ ❆ ❞❡♠♦♥str❛çã♦ s❡❣✉✐ ❞♦s t❡♦r❡♠❛s 1.1.11❡ 1.1.13✳

❆ss✉♠❛ θ ✉♠ ❝❛rát❡r ❞♦ ❣r✉♣♦ G ❡ _Irr(G) ❞❡✜♥✐❞♦ ❝♦♠♦ ♥❛ ♣r♦♣♦s✐çã♦

❛❝✐♠❛✳ ❊♥tã♦✱ θ=a1χ1+· · ·+atχr ♦♥❞❡ a1, . . . , ar ∈N∪ {0} ❡

(θ, χj)_G = r X

i=1

ai(χi, χj)_G=aj.

❙❡ ♦ ♣r♦❞✉t♦ ❢♦r aj 6= 0✱ ✐st♦ ♠♦str❛ q✉❡ χj é ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❞❡ θ ❝♦♠

♠✉❧t✐♣❧✐❝✐❞❛❞❡ aj✳ ❉❡ ❝❡rt❛ ❢♦r♠❛✱ ♦ ♣r♦❞✉t♦ ( , )_G ♥♦s ❢♦r♥❡❝❡ ✐♥❢♦r♠❛✲

(17)

✶✳✶✳ ❘❊P❘❊❙❊◆❚❆➬Õ❊❙ ❊ ❈❆❘❆❈❚❊❘❊❙ ✶✼

❉❡✜♥✐çã♦ ✶✳✶✳✶✺✳ ❙❡❥❛ Irr(G) = _{χ1 = 1G, χ2, . . . , χr}❡ s❡❥❛♠ θ=a1χ1+

· · ·+arχr ❡ ψ =b1χ1 +· · ·+brχr✱ ❝♦♠ aibj ∈N∪ {0}✱ ❝❛r❛❝t❡r❡s ❞♦ ❣r✉♣♦

G✳ ❉❡✜♥❛ ♣♦r

θ\ψ =c1χ1+c2χ2+· · ·+crχr,

♦♥❞❡ ci = min{ai.bi} ✳

❊♠ ♣❛❧❛✈r❛s✱ ❛ ✐♥t❡rs❡❝çã♦ ❛❝✐♠❛ é ❛ s♦♠❛ ❞❛s ❝♦♠♣♦♥❡♥t❡s ❝♦♠✉♠ ❡♥tr❡

θ ❡ ψ ♠✉❧t✐♣❧✐❝❛❞❛s ♣♦r ci✳

❖❜s❡r✈❛çã♦ ✶✳✶✳✶✻✳ ◆♦t❡ q✉❡

(θ, ψ)_G=

r X

i,j=1

aibj(χi, χj)_G = X

i

aibi.

❆♦ s✉♣♦r♠♦s (θ, ψ)_G _≥ 1✱ ❡♥tã♦ θ ❡ ψ ♣♦ss✉❡♠ ❝♦♠♣♦♥❡♥t❡s ❡♠ ❝♦♠✉♠✳

❆❣♦r❛ s❡ (θ, ψ)G = 1✱ ❡♥tã♦ θ ❡ ψ tê♠ ✉♠❛ ú♥✐❝❛ ❝♦♠♣♦♥❡♥t❡ ✐rr❡❞✉tí✈❡❧ ❡♠

❝♦♠✉♠❀ ❝❤❛♠❡♠♦s ❡st❛ ❝♦♠♣♦♥❡♥t❡ ♣♦r χs✳ ◆❡st❡ ❝❛s♦✱ ♦ ✈❛❧♦r ❞❡ cs s❡rá

❡①❛t❛♠❡♥t❡ 1✱ ♦✉ s❡❥❛✱ ♦ r❡s✉❧t❛❞♦ ❞❡ θTψ =χs é ✉♠ ❝❛rát❡r ✐rr❡❞✉tí✈❡❧✳

❆ss✉♠❛ q✉❡ ϕ :G_−→GL(n,C)é ✉♠❛ r❡♣r❡s❡♥t❛çã♦ ♠❛tr✐❝✐❛❧ ❞♦ ❣r✉♣♦

G ❡ H _≤ G✳ ❉❡♥♦t❛♠♦s ♣♦r ϕ _↓H ❛ ❛♣❧✐❝❛çã♦ ❞❛ r❡♣r❡s❡♥t❛çã♦ ϕ r❡str✐t❛

❛♦s ❡❧❡♠❡♥t♦s h∈H✳ ❈❤❛♠❡♠♦s ❛ ❢✉♥çã♦ ϕ↓H ♣♦r ϕ r❡str✐t❛ ❛ H✳

❉❡✜♥✐çã♦ ✶✳✶✳✶✼✳ ❙❡❥❛G✉♠ ❣r✉♣♦ ❡H _≤G✳ ❙✉♣♦♥❤❛ϕ_{∈ FC}(H)✳ ❉❡♥♦t❡

♣♦r ϕG _{❛ ❢✉♥çã♦ ❞❡ ❝❧❛ss❡ ✐♥❞✉③✐❞❛ ♣❛r❛} _G _{❞❡✜♥✐❞❛ ♣♦r}

ϕG₍_g_{) =} 1

|H|

X

x∈G

ϕ◦₍_xgx−1₎_, _♦♥❞❡_ϕ◦₍_xgx−1_{) =}  



ϕ(xgx−1₎_, _s❡ _xgx−1 _∈_H

0, s❡ xgx−1 _∈_/_H

.

▲❡♠❛ ✶✳✶✳✶✽ ✭❘❡❝✐♣r♦❝✐❞❛❞❡ ❞❡ ❋r♦❜❡♥✐✉s✮✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ H _≤ G✳

❙✉♣♦♥❤❛ q✉❡ θ ∈ FC(H) ❡ ϕ ∈ FC(G)✳ ❊♥tã♦✱

(θG, ϕ)_G = (θ, ϕ_↓H)H

❉❡♠♦♥str❛çã♦✳ ❱❡r [✶✶]♣✳ 62✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✶✾✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ H ≤G✳ ❙✉♣♦♥❤❛ ϕ ✉♠ ❝❛rát❡r ❞❡ H✳ ❊♥tã♦✱ ϕG é ✉♠ ❝❛rát❡r ❞❡ G ❝♦♠ ❣r❛✉ |G/H|ϕ(1)✳

❉❡♠♦♥str❛çã♦✳ ❚♦♠❡χ_{∈ I}rr(G)✱ ❡♥tã♦χ_↓H é ✉♠ ❝❛rát❡r ❞❡H ❡ ♣❡❧♦ ❧❡♠❛

1.1.18

0_≤(ϕ, χ_↓H)H = (ϕ G_{, χ}₎

(18)

✶✽ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙

❙❡❣✉❡ ❞♦ t❡♦r❡♠❛ 1.1.13q✉❡ ϕG _{é ✉♠ ❝❛rát❡r ❞❡} _G_{✱ ♣♦✐s ♦ ♣r♦❞✉t♦ ❛❝✐♠❛ é}

✉♠ ✐♥t❡✐r♦ ♥ã♦ ♥❡❣❛t✐✈♦✳ ❏á ♦ ❣r❛✉ ❞❡ ϕG _{é ♦❜t✐❞♦ ♣♦r}

ϕG(1) = 1

|H_|

X

x∈G

ϕ◦(x1x−1) = 1

|H_|

X

x∈G

ϕ(1) =|G/H|ϕ(1)

❝♦♠♦ q✉❡rí❛♠♦s✳

❚❡♦r❡♠❛ ✶✳✶✳✷✵✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ H ≤ K ≤ G✳ ❊♥tã♦✱ ❛s s❡❣✉✐♥t❡s

♣r♦♣r✐❡❞❛❞❡s sã♦ s❛t✐s❢❡✐t❛s:

(✐) ❙❡ ϕ1, ϕ2 ∈ FC(H)✱ ❡♥tã♦ (ϕ1 +ϕ2)G=ϕG1 +ϕG2;

(✐✐) ❙❡ ϕ_{∈ FC}(H)✱ ❡♥tã♦ (ϕK₎G ₌_ϕG_;

(✐✐✐) ❙❡ ϕ_{∈ FC}(H) ❡ χ_{∈ FC}(G)✱ ❡♥tã♦ χ_·ϕG _{= (}_χ_↓

H ϕ)

G

;

(✐✈) ❙❡ χ_{∈ FC}(G)✱ ❡♥tã♦ (χ_↓H)

G

=χ_·1G H;

(✈) ❙❡ χ_{∈ FC}(G)✱ ❡♥tã♦ (χ_↓H, χ↓H)H = (χ, χ·1GH)G.

❉❡♠♦♥str❛çã♦✳ ◆♦ ✐t❡♠ ✭✐✮ t❡♠♦s

(ϕ1+ϕ2)G(g) =

1

|H_|

X

x∈G

(ϕ1+ϕ2)◦(xgx−1)

= 1

|H|

X

x∈G

ϕ◦₁(xgx−1) +ϕ◦₂(xgx−1)

= ϕG

1(g) +ϕG2(g).

◆♦ ✐t❡♠ ✭✐✐✮✱ ❝♦♥s✐❞❡r❡ χ ✉♠ q✉❛❧q✉❡r ❝❛rát❡r ✐rr❡❞✉tí✈❡❧ ❞❡G✳ ❙❡❣✉❡ ❞❛

r❡❝✐♣r♦❝✐❞❛❞❡ ❞❡ ❋r♦❜❡♥✐✉s q✉❡

((ϕK)G, χ)_G = (ϕK, χ_↓K)K = (ϕ,(χ↓K)↓H)H = (ϕ, χ↓H)H = (ϕ G_{, χ}₎

G.

❙❡❣✉❡ ❞❛ ♣r♦♣♦s✐çã♦1.1.14q✉❡ (ϕK₎G ₌_ϕG_✳

◆♦ ✐t❡♠ ✭✐✐✐✮

χϕG(g) = χ(g) 1

|H_|

X

x∈G

ϕ◦(xgx−1)

= 1

|H_|

X

x∈G

χ◦(xgx−1)ϕ◦(xgx−1)

= (χ_↓H ϕ)

G

.

◆♦ ✐t❡♠ ✭✐✈✮✱ χ1G

H = (χ↓H 1H)

G

= (χ↓H)

G_✳

◆♦ ✐t❡♠ ✭✈✮✱ (χ, χ1G H)G

(✐✈)

= χ,(χ_↓H)

G

(19)

✶✳✶✳ ❘❊P❘❊❙❊◆❚❆➬Õ❊❙ ❊ ❈❆❘❆❈❚❊❘❊❙ ✶✾

❚❡♦r❡♠❛ ✶✳✶✳✷✶✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ H _≤G✳ ❙✉♣♦♥❤❛χ ✉♠ ❝❛rát❡r ❞❡ G✳

❊♥tã♦

(χ_↓H, χ↓H)H ≤ |G/H|(χ, χ)G,

❝♦♠ ❛ ✐❣✉❛❧❞❛❞❡ s❡ ❡ s♦♠❡♥t❡ s❡ χ(g) = 0 ♣❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦ g ∈G\H✳

❉❡♠♦♥str❛çã♦✳ ❱❡r [✶✶]♣✳ 28✳

❱❛♠♦s ❛❣♦r❛ ✈♦❧t❛r ❛s ♥♦ss❛s ❛t❡♥çõ❡s ♣❛r❛ q✉❛♥❞♦ ♦ ♥♦ss♦ s✉❜❣r✉♣♦

H ❢♦r ♥♦r♠❛❧ ❡♠ G✳ ❯s❛r❡♠♦s ❛ ♥♦t❛çã♦ H ⊳ G ♣❛r❛ ♠♦str❛r q✉❡ t❛❧ ❢❛t♦

❛❝♦♥t❡❝❡✳

❉❡✜♥✐çã♦ ✶✳✶✳✷✷✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ H ⊳ G✳ ❙✉♣♦♥❤❛♠♦s q✉❡ ϕ_{∈ FC}(H)

❡ ❞❡✜♥❛♠♦s ❛ ❢✉♥çã♦ ϕg _:_H _−→_C _♣♦r

ϕg₍_h_{) =}_ϕ₍_ghg−1₎_, _♦♥❞❡ _g _∈_G _❡ _h_∈_H.

❈❤❛♠❡♠♦s ❝❛❞❛ ϕg _{♣♦r ❝♦♥❥✉❣❛❞♦ ❞❡} _ϕ _{s♦❜ ❛ ❛çã♦ ❞❡} _G_✳

▲❡♠❛ ✶✳✶✳✷✸✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ H ⊳ G✳ ❆ss✉♠❛ q✉❡ ϕ, ϑ ∈ FC(H) ❡

χ∈ FC(G)✳ P❛r❛ x, y ∈G✱ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s sã♦ s❛t✐s❢❡✐t❛s:

(✐) ϕx _{∈ FC}₍_H_);

(✐✐) (ϕx₎y ₌_ϕxy_;

(✐✐✐) (ϕx_{, ϑ}x₎

H = (ϕ, ϑ)H;

(✐✈) (χ↓H, ϕ

x₎

H = (χ↓H, ϕ)_H;

(✈✐) ϕx _{é ✉♠ ❝❛rát❡r s❡ t❛♠❜é♠} _ϕ _{♦ ❢♦r✳}

❉❡♠♦♥str❛çã♦✳ ❱❡r [✶✶]♣✳ 78✳

◗✉❛♥❞♦ t✐✈❡r♠♦s ϕ∈ Irr(H)✱ ❡♥tã♦ t♦❞♦s s❡✉s ❝♦♥❥✉❣❛❞♦s ❞✐st✐♥t♦s s♦❜

❛ ❛çã♦ ❞❡ G sã♦ ✐rr❡❞✉tí✈❡✐s ♣❡❧♦ ✐t❡♠(✐✐✐)❞♦ ❧❡♠❛ ❛❝✐♠❛✳

❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ H ⊳ G✳ ❉❡✜♥❛ ♣❛r❛ ϕ _{∈ I}rr(H)

IG(ϕ) ={g ∈G|ϕg(h) =ϕ(h), ∀h∈H},

♦ ❣r✉♣♦ ❞❡ ✐♥ér❝✐❛ ❞❡ϕ s♦❜ ❛ ❛çã♦ ❞❡G✳ ❖❜s❡r✈❡ q✉❡ H ≤ I(G)≤G❡ ❝❛s♦

(20)

✷✵ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙

❚❡♦r❡♠❛ ✶✳✶✳✷✹ ✭❈❧✐✛♦r❞✮✳ ❙❡❥❛G✉♠ ❣r✉♣♦ ❡H ⊳ G✳ ❙✉♣♦♥❤❛χ_{∈ I}rr(G)

❝♦♠θ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ✐rr❡❞✉tí✈❡❧ ❞❡χ_↓H ❡θ1 =θ, θ2, . . . , θts❡✉s ❝♦♥❥✉❣❛❞♦s

❞✐st✐♥t♦s s♦❜ ❛ ❛çã♦ ❞❡ G✳ ❊♥tã♦✱

χ_↓H=e

t X

i=1

θi,

♦♥❞❡ e= (χ_↓H, θ)H✳

❉❡♠♦♥str❛çã♦✳ ❱❡r [✶✶] ♣✳ 79✳

❚❡♦r❡♠❛ ✶✳✶✳✷✺✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ H ⊳ G✳ ❙✉♣♦♥❤❛ q✉❡ ϑ _{∈ I}rr(H) ❡

T =_IG(ϑ)✳ ❆❣♦r❛ ❞❡✜♥❛ ♣♦r

A =_{ψ _{∈ I}rr(T)_|(ψ _↓H, ϑ)H 6= 0} ❡ B ={χ∈ Irr(G)|(χ↓H, ϑ)H 6= 0}.

❊♥tã♦✱

(✐) ❙❡ ψ ∈ A✱ ❡♥tã♦ ψG _{é ✐rr❡❞✉tí✈❡❧}_;

(✐✐) ❆ ❢✉♥çã♦ ψ _7→ψG _{é ✉♠❛ ❜✐❥❡çã♦ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦s} _A _❡ _B_;

(✐✐✐) ❙❡ ψG ₌ _χ _❝♦♠ _ψ _{∈ A✱ ❡♥tã♦} _ψ _{é ❛ ú♥✐❝❛ ❝♦♠♣♦♥❡♥t❡ ✐rr❡❞✉tí✈❡❧ ❞❡}

χ↓T q✉❡ ❡stá ❡♠ A;

(✐✈) ❙❡ ψG₌_χ _❝♦♠ _ψ _{∈ A✱ ❡♥tã♦} ₍_ψ _↓

H, ϑ)H = (χ↓H, ϑ)H✳

❉❡♠♦♥str❛çã♦✳ ❱❡r [✶✶] ♣✳ 82₋83✳

❚❡♦r❡♠❛ ✶✳✶✳✷✻✳ ❙❡❥❛G✉♠ ❣r✉♣♦ ❡ H ⊳G❝♦♠ _|G/H_|=p♣r✐♠♦✳ ❙✉♣♦♥❤❛ χ ∈ Irr(G) ❡ θ ✉♠❛ ❝♦♠♣♦♥❡♥t❡ ✐rr❡❞✉tí✈❡❧ ❞❡ χ ↓H ❝♦♠ θ1 = θ, θ2, . . . θt

s❡✉s ❝♦♥❥✉❣❛❞♦s ❞✐st✐♥t♦s s♦❜ ❛ ❛çã♦ ❞❡ G✳ ❊♥tã♦✱

(✐) χ_↓H é ✐rr❡❞✉tí✈❡❧ ♦✉;

(✐✐) χ_↓H=

t X

i=1

θi✱ ♦♥❞❡ t =p✳

❉❡♠♦♥str❛çã♦✳ ❱❡r [✶✶] ♣✳ 85−86✳

❙❡❥❛ G ✉♠ ❣r✉♣♦ ✜♥✐t♦ ❡ s❡❥❛ H _≤ G ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ♥♦r♠❛❧✳ ❯♠

❝❛rát❡rϑ∈ Irr(H)é ❞✐t♦ s❡r ❡st❡♥❞í✈❡❧ ❛Gs❡ ❡①✐st✐r ✉♠ ❝❛rát❡rχ∈ Irr(G)

t❛❧ q✉❡ χ_↓H=ϑ✳

❈♦r♦❧ár✐♦ ✶✳✶✳✷✼✳ ❙❡❥❛G ✉♠ ❣r✉♣♦ ❡H ⊳ G✳ ❙✉♣♦♥❤❛ q✉❡_|G/H_|=pé ✉♠

(21)

✶✳✶✳ ❘❊P❘❊❙❊◆❚❆➬Õ❊❙ ❊ ❈❆❘❆❈❚❊❘❊❙ ✷✶

❉❡♠♦♥str❛çã♦✳ ❙❡❥❛χ_{∈ I}rr(G)✉♠❛ ❝♦♠♣♦♥❡♥t❡ ❞❡ϑG_{✱ ❡♥tã♦ ♣❡❧♦ t❡♦r❡♠❛}

1.1.24✱ χ_↓H= eϑ ♣❛r❛ ❛❧❣✉♠ e✱ ♣❡❧♦ t❡♦r❡♠❛ 1.1.26 e = 1✳ ❊♥tã♦✱ χ↓H=ϑ

❡ ♣♦rt❛♥t♦ ϑ é ❡st❡♥❞í✈❡❧ ♣❛r❛ G✳

❈♦r♦❧ár✐♦ ✶✳✶✳✷✽✳ ❙♦❜ ❛s ❤✐♣ót❡s❡s ❞♦ t❡♦r❡♠❛ 1.1.26✳ ❉❡♥♦t❡ ♣♦r Iθ =

{χ_{∈ I}rr(G)_|(χ_↓H, θ)H ≥1}✳ ❊♥tã♦✱ s❡❣✉❡ q✉❡:

(1) ❙❡ ❛❝♦♥t❡❝❡r ♦ ✐t❡♠ (✐) ♥♦ t❡♦r❡♠❛ 1.1.26✱ ❡♥tã♦ _|Iθ|=p❀

(2) ❙❡ ❛❝♦♥t❡❝❡r ♦ ✐t❡♠ (✐✐) ♥♦ t❡♦r❡♠❛ 1.1.26✱ ❡♥tã♦ θG _{∈ I}_rr₍_G₎ _❡

θG₌_θG

i ♣❛r❛ q✉❛❧q✉❡r ❝♦♥❥✉❣❛❞♦ ❞✐st✐♥t♦ ❞❡ θ s♦❜ ❛ ❛çã♦ ❞❡ G✳

❉❡♠♦♥str❛çã♦✳ ❙❡❣✉❡ ❞♦ t❡♦r❡♠❛ 1.1.26✱ ♣❛r❛ t♦❞♦ χ _{∈ I}θ ❛❝♦♥t❡❝❡ ♦ ❝❛s♦

(✐)♦✉ (✐✐)✳

❙✉♣♦♥❤❛ q✉❡ ❛❝♦♥t❡❝❡ ♦ ✐t❡♠ ✭✐✮ ❞♦ t❡♦r❡♠❛ 1.1.26✳ ❊♥tã♦ θ é ❡st❡♥❞í✈❡❧

♣❛r❛G♣❡❧♦ ❝♦r♦❧ár✐♦1.1.27✳ ❆ss✐♠✱θG ₌ X χ∈Iθ

χ❡ ❛✐♥❞❛θG_{(1) =}_|_G/H_|·_θ₍₁₎_✳

❆❣♦r❛θG_{(1) =} X χ∈Iθ

χ(1) ❡θ(1) =χ(1)✱ ❧♦❣♦ θG_{(1) =} X χ∈Iθ

χ(1) =_|G/H_{| ·}χ(1)

✐♠♣❧✐❝❛♥❞♦ |Iθ|=|G/H|=p✱ ♣r♦✈❛♥❞♦ ♦ ✐t❡♠(1)✳

❆❣♦r❛ s✉♣♦♥❤❛ ❛❝♦♥t❡❝❡r ♦ ✐t❡♠ ✭✐✐✮ ❞♦ t❡♦r❡♠❛ 1.1.26✳ ❊♥tã♦ χ _↓H=

p X

i=1

θi ❡ ❝♦♥s❡q✉❡♥t❡♠❡♥t❡ IG(θ) = H✱ ♣♦✐s H ≤ IG(θ) ❡ |G/IG(θ)| = p✳

❆❣♦r❛ ❝♦♥s✐❞❡r❡ T = H ♥♦ t❡♦r❡♠❛ 1.1.25✱ ❡♥tã♦ θG _{é ✐rr❡❞✉tí✈❡❧ ❡ ❛❧é♠}

❞✐ss♦✱

θ_iG(y) = (θg)G(y) = 1

|H_|

X

x∈G

(θg)◦(xyx−1)

= 1

|H_|

X

x∈G

θ◦(gxyx−1g−1), (gx) =z _∈G

= 1

|H|

X

z∈G

θ◦(zyz−1)

= θG

♣❛r❛ q✉❛❧q✉❡r ❝♦♥❥✉❣❛❞♦θi ❞❡θ✳ ■st♦ ❝♦♥❝❧✉✐ ❛ ❞❡♠♦♥str❛çã♦✳

❉❡✜♥✐çã♦ ✶✳✶✳✷✾✳ ❙❡ H ⊳ G ❡ χ˜ é ✉♠ ❝❛rát❡r ❞❡ G/H✱ ❡♥tã♦ ♦ ❝❛rát❡r χ

❞❡ G q✉❡ é ❞❛❞♦ ♣♦r

χ(g) = ˜χ(Hg) (g ∈G)

(22)

✷✷ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙

❚❡♦r❡♠❛ ✶✳✶✳✸✵✳ ❆ss✉♠❛ q✉❡H ⊳ G✳ ❆ss♦❝✐❛♥❞♦ ❝❛❞❛ ❝❛rát❡r ❞❡G/H ❝♦♠

s❡✉ ❡❧❡✈❛❞♦ ❛ G✱ ♦❜t❡♠♦s ✉♠❛ ❜✐❥❡çã♦ ❝♦rr❡s♣♦♥❞❡♥t❡ ❡♥tr❡ ♦ ❝♦♥❥✉♥t♦ ❞❡

❝❛r❛❝t❡r❡s ❞❡ G/H ❡ ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❛r❛❝t❡r❡s χ ❞❡ G q✉❡ s❛t✐s❢❛③❡♠ H ≤

Ker(χ)✱ ♦♥❞❡Ker(χ) = _{g _∈G_|χ(g) = χ(1)_}é ♦ ♥ú❝❧❡♦ ❞❡χ✳ ❉❡st❡ ♠♦❞♦✱

❝❛r❛❝t❡r❡s ✐rr❡❞✉tí✈❡✐s ❞❡G/H ❝♦rr❡s♣♦♥❞❡♠ ❛♦s ❝❛r❛❝t❡r❡s ✐rr❡❞✉tí✈❡✐s ❞❡G

q✉❡ tê♠ H ♥♦ s❡✉ ♥ú❝❧❡♦✳

❉❡♠♦♥str❛çã♦✳ ❱❡r [✶✹] ♣✳ 169✳

❚❡♦r❡♠❛ ✶✳✶✳✸✶ ✭▼❛❝❦❡②✮✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❡ H, K _≤G✳ ❈♦♥s✐❞❡r❡ D ♦

❝♦♥❥✉♥t♦ ❝♦♠♣❧❡t♦ ❞❡ r❡♣r❡s❡♥t❛♥t❡s ❞❛s ❝❧❛ss❡s ❧❛t❡r❛✐s ❞✉♣❧❛sHxK s❛t✐s❢❛✲

③❡♥❞♦ G= [˙

x∈D

HxK ✉♥✐ã♦ ❞✐s❥✉♥t❛✳ ❙✉♣♦♥❤❛ ϕ s❡♥❞♦ ✉♠ ❝❛rát❡r ❞❡ H ❡ ϑ

✉♠ ❝❛rát❡r ❞❡ K✳ ❉❡✜♥❛ ♦ ❝❛rát❡r ϑx _❡♠ _Kx _♣♦r _ϑx₍_k_{) =} _ϑ₍_xkx−1₎_{✳ ❊♥tã♦✱}

(1) ϑG _↓

H=

X

x∈D

(ϑx ↓H∩Kx)

H

;

(2) (ϕG_{, ϑ}G₎

G =

X

x∈D

(ϕ_↓H∩Kx, ϑ

x

↓H∩Kx)_H∩Kx✳

❉❡♠♦♥str❛çã♦✳ ❆ ❞❡♠♦♥str❛çã♦ ❞♦ ✐t❡♠ (1) é ❡♥❝♦♥tr❛❞❛ ❡♠ [✶✵] ♣✳ 218₋ 219✳

◆♦ ✐t❡♠ (2) ✉t✐❧✐③❡♠♦s ❛ r❡❝✐♣r♦❝✐❞❛❞❡ ❞❡ ❋r♦❜❡♥✐✉s ❡ t❡♠♦s

(ϕG, ϑG)_G = (ϕ, ϑG_↓H)H = 

ϕ, X

x∈D

ϑx _↓H∩Kx

!H



H

=

= ϕ _↓H∩Ky,

X

x∈D

ϑx_↓H∩Kx

!

H∩Ky

=

= X

x∈D

(ϕ _↓H∩Kx, ϑ

x

↓H∩Kx)_H_∩_Kx.

■st♦ ❝♦♥❝❧✉✐ ♦ t❡♦r❡♠❛✳

✶✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ●r✉♣♦ ❙✐♠étr✐❝♦

S

n

❙❡❥❛Ω✉♠ ❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ ❡❧❡♠❡♥t♦s✳ ❯♠❛ ♣❡r♠✉t❛çã♦πs♦❜r❡ ♦ ❝♦♥❥✉♥t♦

Ωé ✉♠❛ ❛♣❧✐❝❛çã♦ ❜✐❥❡t✐✈❛ π : Ω_−→Ω q✉❡ ❞❡♥♦t❛♠♦s ♣♦r

. . . a . . . . . . π(a) . . .

(23)

✶✳✷✳ P❘❖P❘■❊❉❆❉❊❙ ❉❖ ●❘❯P❖ ❙■▼➱❚❘■❈❖ SN ✷✸

❖ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ♣❡r♠✉t❛çõ❡s ❡♠ Ω ❥✉♥t❛♠❡♥t❡ ❝♦♠ ❛ ♦♣❡r❛çã♦ ❞❡

❝♦♠♣♦s✐çã♦ ❞❡ ❛♣❧✐❝❛çõ❡s ❢♦r♠❛♠ ✉♠ ❣r✉♣♦✳ ❊st❡ ❣r✉♣♦ é ❝♦♥❤❡❝✐❞♦ ❝♦♠♦ ❣r✉♣♦ s✐♠étr✐❝♦ ❡ ❞❡♥♦t❡♠♦s ♣♦r SΩ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ s❡ Ω = {1,2, . . . , n}

❞❡♥♦t❛r❡♠♦sSΩ ♣♦r Sn✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❙❡❥❛♠ i1, i2, . . . , ik✱ k ≥1 ✐♥t❡✐r♦s ❞✐st✐♥t♦s ❡♠ {1, . . . , n}✳

❯♠❛ ♣❡r♠✉t❛çã♦ σ ∈ Sn é ✉♠ k✲❝✐❝❧♦ s❡

σ(i1) = i2, σ(i2) =i3, . . . , σ(ij) =ij+1, j < k ❡ σ(ik) =i1

❡ ✜①❛ t♦❞♦s ♦s ✐♥t❡✐r♦s ❞✐st✐♥t♦s ❞❡i1, i2, . . . , ik✳ ❊s❝r❡✈❡♠♦sσ= (i1, i2, . . . , ik)✱

♠❛s t❛♠❜é♠ ♣♦❞❡♠♦s ❡s❝r❡✈❡r σ s❡♥❞♦ (i2, . . . , ik, i1)✱ . . . , (ik, i1, . . . , ik−1)

♣♦ré♠ t♦❞❛s ❡ss❛s ❡s❝r✐t✉r❛s ❞❡♥♦t❛♠ ❛ ♠❡s♠❛ ♣❡r♠✉t❛çã♦ σ✳

❯♠ ❡❧❡♠❡♥t♦ τ _{∈ S}n é ✉♠ ❝✐❝❧♦ s❡ ❢♦r ✉♠ k✲❝✐❝❧♦ ♣❛r❛ ❛❧❣✉♠ k✳ ❉♦✐s

❝✐❝❧♦s σ ❡ τ sã♦ ❞✐t♦s ❞✐s❥✉♥t♦s s❡ σ ✜①❛ t♦❞♦s ♦s ♣♦♥t♦s ♠♦✈✐❞♦s ♣♦r τ ❡

✈✐❝❡✲✈❡rs❛✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❝✐❝❧♦s ❞✐s❥✉♥t♦s s❡♠♣r❡ ❝♦♠✉t❛♠✳

❊①❡♠♣❧♦ ✶✳✷✳✷✳ ❚♦♠❡ ❞♦✐s ❝✐❝❧♦s ❡♠ Sn✱ σ =

1 2 3 4 5 6 5 1 3 4 2 6

❡ τ =

1 2 3 4 5 6 1 2 4 3 5 6

✳ ❊s❝r❡✈❛ σ = (152)(3)(4)(6) ❡ τ = (34)(1)(2)(5)(6)✳

❖❜s❡r✈❡♠♦s q✉❡ σ ❡ τ sã♦ ❞❡ ❢❛t♦ ❝✐❝❧♦s ❞✐s❥✉♥t♦s ❡

στ = (152)(34)(6) = (34)(152)(6) =τ σ,

✈❡r✐✜❝❛♥❞♦ ❛ss✐♠ q✉❡ σ ❡ τ ❝♦♠✉t❛♠✳

❚❡♦r❡♠❛ ✶✳✷✳✸✳ ❈❛❞❛ ❡❧❡♠❡♥t♦ π _{∈ S}n ♣♦ss✉✐ ✉♠❛ ❡s❝r✐t✉r❛ ❝♦♠♦ ♣r♦❞✉t♦

❞❡ ❝✐❝❧♦s ❞✐s❥✉♥t♦s π = τ1τ2. . . τt✱ ♦♥❞❡ τk sã♦ ❝✐❝❧♦s ❡♠ Sn ❝♦♠ τi ❡ τj

❞✐s❥✉♥t♦s ♣❛r❛ i6=j✳ ❊st❛ ❡s❝r✐t✉r❛ s❡rá ú♥✐❝❛ ❛ ♠❡♥♦s ❞❛ ♦r❞❡♠ ❞♦s ❝✐❝❧♦s τk✳

❉❡♠♦♥str❛çã♦✳ ❱❡r [✶✸]♣✳ 5✳

❆♣❧✐❝❛♠♦s s♦❜r❡ ❝❛❞❛k✲❝✐❝❧♦σ _{∈ S}n✉♠❛ ❢✉♥çã♦ q✉❡ ❝❤❛♠❡♠♦s ♣♦r s✐❣♥♦✱

❞❡✜♥✐❞❛ ♣♦r

sgn(σ) =

1, s❡k é í♠♣❛r

−1, s❡k é ♣❛r .

❆❣♦r❛ ❝♦♥s✐❞❡r❛♥❞♦π =τ1. . . τt❛ ❡s❝r✐t✉r❛ ❞❡ π ♣♦r ❝✐❝❧♦s ❞✐s❥✉♥t♦s✳ ❉❡✜♥❛

♣♦r

sgn(π) =

t Y

k=1

(24)

✷✹ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙

❙❡❣✉❡ ❞♦ t❡♦r❡♠❛1.2.3q✉❡ ♦ ✈❛❧♦r ❞❡ sgn(π)é ❜❡♠ ❞❡✜♥✐❞♦✱ ♣♦✐s ♦ ❝♦♠♣r✐✲

♠❡♥t♦ ❞♦s ❝✐❝❧♦s q✉❡ ❛♣❛r❡❝❡♠ ♥❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ ❝✐❝❧♦s ❞✐s❥✉♥t♦s ❞❡π sã♦

✉♥✐❝❛♠❡♥t❡ ❞❡t❡r♠✐♥❛❞♦s ♣♦rπ✳ ❉✐r❡♠♦s q✉❡ ❛ ♣❡r♠✉t❛çã♦π é ♣❛r q✉❛♥❞♦ sgn(π) = 1 ❡ í♠♣❛r q✉❛♥❞♦sgn(π) =₋1✳

❊①❡♠♣❧♦ ✶✳✷✳✹✳ ❱❡❥❛♠♦s ❡①❡♠♣❧♦s ❞❡ ♣❡r♠✉t❛çõ❡s ♣❛r❡s ❡ í♠♣❛r❡s✳ ❚♦♠❡ ❛s s❡❣✉✐♥t❡s ♣❡r♠✉t❛çõ❡s ❥á ❡s❝r✐t❛s ❡♠ ❝✐❝❧♦s ❞✐s❥✉♥t♦s a= (134)(25)_{∈ S}5✱

b= (1)(236)(4)(578)∈ S8✱ c= (47)(125)(63) ∈ S7 ❡ d = (12)∈ S2✳ ❈❛❧❝✉❧❡✲

♠♦s ♦ s✐❣♥♦ ❡♠ a, b, c ❡ d

sgn(a) =−1, sgn(b) = 1, sgn(c) = 1 ❡ sgn(d) = −1.

❊♥tã♦✱ ❛s ♣❡r♠✉t❛çõ❡s b ❡ c sã♦ ♣❛r❡s ❡ ❛s ♣❡r♠✉t❛çõ❡s a ❡ d sã♦ í♠♣❛r❡s✳

❙❡❥❛ π = τ1. . . τt ❛ ❞❡❝♦♠♣♦s✐çã♦ ❞❡ π ❡♠ ❝✐❝❧♦s ❞✐s❥✉♥t♦s✳ ❉❡♥♦t❡♠♦s

♣♦r ls, 1 ≤ s ≤ t ♦ ❝♦♠♣r✐♠❡♥t♦ ❞♦ ❝✐❝❧♦ τs✳ ❈❤❛♠❛♠♦s ♦ ❝♦♥❥✉♥t♦ ✭❝♦♠

❡✈❡♥t✉❛✐s r❡♣❡t✐çõ❡s✮{l1, l2, . . . , lt}♣♦r ❡str✉t✉r❛ ❝í❝❧✐❝❛ ❞❡ π✳

❊①❡♠♣❧♦ ✶✳✷✳✺✳ ❙❡❥❛ π = (2,5,11)(1,8,9,12,15,16)(3,4,10,7)(6,13)(14) ❛

❞❡❝♦♠♣♦s✐çã♦ ❞❡ π ❡♠ ❝✐❝❧♦s ❞✐s❥✉♥t♦s ❡♠ _S16✳ ❆ ❡str✉t✉r❛ ❝í❝❧✐❝❛ ❞❡ π é

{3,6,4,2,1} ❡ t❡♠♦s sgn(π) = 1·(−1)·(−1)·(−1)·1 =−1✳

❉✉❛s ♣❡r♠✉t❛çõ❡s π ❡ σ ❡♠ Sn sã♦ ❞✐t❛s ❞♦ ♠❡s♠♦ t✐♣♦ s❡ ❡❧❛s t✐✈❡r❡♠

❛ ♠❡s♠❛ ❡str✉t✉r❛ ❝í❝❧✐❝❛✳

❊①❡♠♣❧♦ ✶✳✷✳✻✳ ❚♦♠❡ (123)❡ (132) ♣❡r♠✉t❛çõ❡s ❡♠ _S3✳ ❋❛❝✐❧♠❡♥t❡ ♦❜s❡r✲

✈❡♠♦s q✉❡ (123) ❡ (132) sã♦ ❞♦ ♠❡s♠♦ t✐♣♦✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ (123) ❡ (132)

sã♦ ❝♦♥❥✉❣❛❞♦s ❡♠ S3✳ ❊st❛ ❝♦♥❝❧✉sã♦ t❛♠❜é♠ s❡❣✉❡ ❞♦ ♣ró①✐♠♦ t❡♦r❡♠❛✳

❚❡♦r❡♠❛ ✶✳✷✳✼✳ ❉♦✐s ❡❧❡♠❡♥t♦s α ❡ β ❡♠ Sn sã♦ ❝♦♥❥✉❣❛❞♦s s❡ ❡ s♦♠❡♥t❡

s❡ ❢♦r❡♠ ❞♦ ♠❡s♠♦ t✐♣♦ ❡♠ Sn✳

❉❡♠♦♥str❛çã♦✳ ❱❡r [✶✸] ♣✳ 8−9✳

❉❡✜♥✐çã♦ ✶✳✷✳✽✳ ❙❡❥❛ C = {C1 = 1,C2, . . . ,Cr} ♦ ❝♦♥❥✉♥t♦ ❞❛s ❝❧❛ss❡s ❞❡

❝♦♥❥✉❣❛çã♦ ❞❡ Sn✳ ❊s❝♦❧❤❛♠♦s ❝♦♠♦ ♥♦ss♦ r❡♣r❡s❡♥t❛♥t❡ ❞❛ ❝❧❛ss❡ ❞❡ ❝♦♥✲

❥✉❣❛çã♦ Ck✱ ✉♠ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ ❝✉❥❛ ❡str✉t✉r❛ ❝í❝❧✐❝❛ s❛t✐s❢❛③❡r l1 ≥ l2 ≥

· · · ≥ lt✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❞❡♥♦t❡♠♦s ♣♦r C(π) ❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉❣❛çã♦ ❞❡ ✉♠

q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ π∈ Sn✳

■♠♣♦rt❛♥t❡ ♦❜s❡r✈❛r q✉❡ ❛ s♦♠❛ ❞♦s ❡❧❡♠❡♥t♦s ❞❛ ❡str✉t✉r❛ ❝í❝❧✐❝❛ ❞❡ ✉♠ q✉❛❧q✉❡r ❡❧❡♠❡♥t♦ ❡♠ Sn ❞❡✈❡ s❡rn✳

❉❡✜♥✐çã♦ ✶✳✷✳✾✳ ❙❡❥❛n ✉♠ ✐♥t❡✐r♦ ♣♦s✐t✐✈♦ ❡_{λ1, λ2, . . . , λt}✉♠❛ s❡q✉ê♥❝✐❛

❞❡ ✐♥t❡✐r♦s ♣♦s✐t✐✈♦s t❛✐s q✉❡ λ1 + λ2 + · · · + λt = n✳ ❉❡✜♥❛ ♣♦r

λ = (λ1, λ2, . . . , λt) ❡ ❞✐r❡♠♦s q✉❡ é ✉♠❛ ♣❛rt✐çã♦ ❞❡ n✱ s❡ λi ≤ λi−1 ❝♦♠

(25)

✶✳✷✳ P❘❖P❘■❊❉❆❉❊❙ ❉❖ ●❘❯P❖ ❙■▼➱❚❘■❈❖ SN ✷✺

❉✉❛s ♣❛rt✐çõ❡s λ _⊢ n ❡ α _⊢ n sã♦ ❞✐t❛ ❞✐st✐♥t❛s✱ s❡ ❡①✐st❡ ❛❧❣✉♠ i q✉❡

s❛t✐s❢❛ç❛ λi 6=αi✳

❚❡♦r❡♠❛ ✶✳✷✳✶✵✳ ❉❡♥♦t❡ ♣♦r Par(n) ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ♣❛rt✐çõ❡s ❞♦

✐♥t❡✐r♦ n ≥1✳ ❊①✐st❡ ✉♠❛ ❜✐❥❡çã♦ f ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦sPar(n) ❡ C✳

❉❡♠♦♥str❛çã♦✳ ❆❞♦t❡ ♣♦rτj ♦ r❡♣r❡s❡♥t❛♥t❡ ❞❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉❣❛çã♦Cj ❝♦♠♦

♥❛ ❞❡✜♥✐çã♦ 1.2.8 ❡ ❝♦♥s✐❞❡r❡ s✉❛ ❡str✉t✉r❛ ❝í❝❧✐❝❛ s❡♥❞♦ {l1, . . . , lr}✳ ❊♥tã♦

❞❡✜♥✐♠♦s ❛ ❢✉♥çã♦f :_{C −→ P}ar(n)♣♦rf(_C(τj)) = (l1, l2, . . . , lr)✳ ❱❡r✐✜q✉❡✲

♠♦s s❡ f é ❛ ❢✉♥çã♦ ♣r♦❝✉r❛❞❛✳

❆ ❢✉♥çã♦f é ❜❡♠ ❞❡✜♥✐❞❛✳ ❚♦♠❡♠♦sτj ❡τ˜j❞♦✐s r❡♣r❡s❡♥t❛♥t❡s ❞❛ ♠❡s♠❛

❝❧❛ss❡ ❞❡ ❝♦♥❥✉❣❛çã♦ Cj✳ ❙❡❣✉❡ ❞♦ t❡♦r❡♠❛ 1.2.7 q✉❡ τj ❡ τ˜j sã♦ ❞♦ ♠❡s♠♦

t✐♣♦ ❡♠ Sn ❡ ♣♦rt❛♥t♦ f(C(τj)) =f(C( ˜τj))✳

❆ ❢✉♥çã♦f é ✐♥❥❡t✐✈❛✳ ❙❡❥❛♠τi ❡τj r❡♣r❡s❡♥t❛♥t❡s ❞❛s ❝❧❛ss❡s ❞❡ ❝♦♥❥✉❣❛✲

çã♦ C(τi)❡ C(τj)❡♠ Sn✳ ❙✉♣♦♥❤❛ q✉❡f(C(τi)) = f(C(τj)) =λ ✉♠❛ ♣❛rt✐çã♦

❡♠Par(n)✳ ❊♥tã♦✱τi ❡τj sã♦ ❞♦ ♠❡s♠♦ t✐♣♦ ❡♠Sn❡ s❡❣✉❡ ❞♦ t❡♦r❡♠❛1.2.7

q✉❡ τi ❡ τj sã♦ ❝♦♥❥✉❣❛❞♦s✳ P♦rt❛♥t♦ C(τi) =C(τj)✳

❆ ❢✉♥çã♦ f é s♦❜r❡❥❡t✐✈❛✳ ❙✉♣♦♥❤❛ α = (l1, l2, . . . , lr) ✉♠❛ ♣❛rt✐çã♦ ❡♠

Par(n)✳ ❋♦r♠❛r❡♠♦s ✉♠ ❡❧❡♠❡♥t♦ γ ∈ Sn t❛❧ q✉❡ f(C(γ)) = α✳ ❱❛♠♦s ❛

❡st❛ ❢♦r♠❛çã♦✿

• ❈♦♠♣♦♥❞♦ ♦ ♣r✐♠❡✐r♦ l1✲❝✐❝❧♦✱ ♦s ✐♥t❡✐r♦s ❞❡ 1 ❛ l1✳ ❉❡♥♦t❡♠♦s ❡st❡ ❝✐❝❧♦

♣♦r γ1 = (1,2, . . . , l1)❀

• ❈♦♠♣♦♥❞♦ ♦ s❡❣✉♥❞♦ l2✲❝✐❝❧♦✱ ♦s ✐♥t❡✐r♦s ❞❡ l1 + 1 ❛ (l1 +l2)✳ ❉❡♥♦t❡♠♦s

❡st❡ ❝✐❝❧♦ ♣♦r γ2 = (l1+ 1, l1+ 2, . . . , l1+l2)❀

• ❈♦♠♣♦♥❞♦ ♥❡st❛ ❢♦r♠❛ ❛té ♦ r✲és✐♠♦ lr✲❝✐❝❧♦✱ ❢♦r♠❛❞♦ ♣♦r ✐♥t❡✐r♦s ❞❡

(l1+l2+· · ·+lr−1+ 1) ❛(l1+l2+· · ·+lr−1+lr)✳ ❉❡♥♦t❡♠♦s ❡st❡ ❝✐❝❧♦ ♣♦r

γr = (l1+l2+· · ·+lr−1+ 1, l1+l2+· · ·+lr−1+ 2, . . . , l1+l2+· · ·+lr−1+lr)✳

❋á❝✐❧ ✈❡r q✉❡ γ = γ1γ2· · ·γr é ♣r♦❞✉t♦ ❞❡ ❝✐❝❧♦s ❞✐s❥✉♥t♦s ❡♠ Sn ❡

f(_C(γ)) =α✳ ❈♦♠♦ γ s❛t✐s❢❛③ ❛ ❞❡✜♥✐çã♦ 1.2.8 ❝♦♥❝❧✉í♠♦s q✉❡ f é s♦❜r❡❥❡✲

t✐✈❛✳ ■st♦ ♠♦str❛ q✉❡ ❛ ❢✉♥çã♦f é ❛ ❜✐❥❡çã♦ ♣r♦❝✉r❛❞❛ ❡♥tr❡C ❡Par(n)✳

❙❡❣✉❡ ❞♦ t❡♦r❡♠❛ ❛❝✐♠❛ q✉❡ ❝❛❞❛ ❝❧❛ss❡ ❞❡ ❝♦♥❥✉❣❛çã♦ Ci ❡♠ Sn ❝♦rr❡s✲

♣♦♥❞❡ ❛ ✉♠❛ ú♥✐❝❛ ♣❛rt✐çã♦ λ ❡♠ _Par(n)✳ ◆❡st❡ ❝❛s♦✱ ❞✐r❡♠♦s q✉❡ λ é ❛

♣❛rt✐çã♦ ❛ss♦❝✐❛❞❛ ❛ Ci✳

❆ ❝❛❞❛ ♣❛rt✐çã♦λ = (λ1,· · · , λr)⊢n ✐r❡♠♦s ❛ss♦❝✐❛r ✉♠❛ t❛❜❡❧❛ q✉❡ ❞❡✲

♥♦t❡♠♦s ♣♦r Tλ✱ ❝♦♥st✐t✉í❞❛ ♣♦r ❝❛✐①❛s ❛❧✐♥❤❛❞❛s ❤♦r✐③♦♥t❛❧♠❡♥t❡✱ ✈❡rt✐❝❛❧✲

♠❡♥t❡ s❡♥❞♦ ❥✉st✐✜❝❛❞❛s à ❡sq✉❡r❞❛✱ s❛t✐s❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

• ❆ ❧✐♥❤❛ 1✱ t❡ráλ1 ❝❛✐①❛s❀

(26)

✷✻ ❈❆P❮❚❯▲❖ ✶✳ ❘❊❙❯▲❚❆❉❖❙ P❘❊▲■▼■◆❆❘❊❙

• . . .

• ❆ ❧✐♥❤❛ r✱ t❡rá λr ❝❛✐①❛s✳

❊①❡♠♣❧♦ ✶✳✷✳✶✶✳ ❙❡❥❛ λ = (4,3,3,2,2,1) ✉♠❛ ♣❛rt✐çã♦ ❞♦ ✐♥t❡✐r♦ 15✳ ❙✉❛

t❛❜❡❧❛ ❛ss♦❝✐❛❞❛ é

T(4,3,3,2,2,1) =

−→4 caixas

−→3 caixas

−→2 caixas

−→1 caixa

P❛r❛ λ _⊢ n ❡ Tλ s✉❛ t❛❜❡❧❛ ❛ss♦❝✐❛❞❛✱ ✐r❡♠♦s ❞❡♥♦t❛r ♣♦r (Tλ)t ❛ s✉❛

t❛❜❡❧❛ ♦r✐❣✐♥❛❞❛ ❞❛ tr❛♥s♣♦s✐çã♦ ❞❡ ❧✐♥❤❛s ❡♠ Tλ ♣♦r ❝♦❧✉♥❛s✱ ❞❛ s❡❣✉✐♥t❡

♠❛♥❡✐r❛✿

• ❧✐♥❤❛1 ❞❡ Tλ s❡rá ❝♦❧✉♥❛ 1❞❡ (Tλ)t❀

• ❧✐♥❤❛2 ❞❡ Tλ s❡rá ❝♦❧✉♥❛ 2❞❡ (Tλ)t❀

• . . .

• ❧✐♥❤❛r ❞❡Tλ s❡rá ❝♦❧✉♥❛ r ❞❡ (Tλ)t✳

❊♥tã♦ é ♣♦ssí✈❡❧ ❛ss♦❝✐❛r ❛ t❛❜❡❧❛ (Tλ)t ✉♠❛ s❡q✉ê♥❝✐❛ λt = (λt1, . . . , λts)✱

♦♥❞❡ λt

i é ✐❣✉❛❧ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❛✐①❛s ♥❛ ❝♦❧✉♥❛ i✳ ❱❡❥❛ q✉❡ λt é ❞❡ ❢❛t♦

✉♠❛ ♣❛rt✐çã♦ ❞❡n ❡ ❝❤❛♠❡♠♦s ♣♦r ♣❛rt✐çã♦ tr❛♥s♣♦st❛ ❞❡λ✳ ❆❧é♠ ❞✐ss♦✱ ♦s

✈❛❧♦r❡s ❞❛s s✉❛s ❝♦♠♣♦♥❡♥t❡s λt

i ♣♦❞❡♠ s❡r ❝❛❧❝✉❧❛❞♦s ❢❛❝✐❧♠❡♥t❡ s❡♥❞♦

λt_i =|{j |λj ≥i}|.

❯♠❛ ♣❛rt✐çã♦λ é ❞✐t❛ ❛✉t♦✲❛ss♦❝✐❛❞❛ s❡ ❡ s♦♠❡♥t❡ s❡λ=λt_{✳ ◆❡st❡ ❝❛s♦✱}

♦❜✈✐❛♠❡♥t❡ t❡♠♦s Tλ = (Tλ)t✳

❊①❡♠♣❧♦ ✶✳✷✳✶✷✳ ❙❡❥❛ ❛ ♣❛rt✐çã♦λ = (3,1,1)❞♦ ✐♥t❡✐r♦5✳ ❊♥tã♦ s✉❛ t❛❜❡❧❛

❛ss♦❝✐❛❞❛ é

T(3,1,1) = ❛❧é♠ ❞✐ss♦ (Tλ)t= .

(27)

✶✳✷✳ P❘❖P❘■❊❉❆❉❊❙ ❉❖ ●❘❯P❖ ❙■▼➱❚❘■❈❖ SN ✷✼

❊①❡♠♣❧♦ ✶✳✷✳✶✸✳ ◆❛ t❛❜❡❧❛ ❞♦ ❡①❡♠♣❧♦ 1.2.11

(Tλ)t =

❡ ❛ ♣❛rt✐çã♦ tr❛♥s♣♦st❛ ❞❡ λ é λt _{= (6}_,₅_,₃_,₁₎_✳

❖❜s❡r✈❛çã♦ ✶✳✷✳✶✹✳ ❆ss✉♠❛ q✉❡ λ _⊢ n é ❛✉t♦✲❛ss♦❝✐❛❞❛✳ ❙✉♣♦♥❤❛ q✉❡ λ= (λ1, λ2, . . . , λr)❡λt= (λt1, λt2, . . . , λts)✳ ❖❜s❡r✈❡ q✉❡ ♦ ♥ú♠❡r♦ ❞❡ ❝♦❧✉♥❛s

❡♠ Tλ é λt1 =|{j |λj ≥1}|=r✱ ♦✉ s❡❥❛✱ ❡①❛t❛♠❡♥t❡ ♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❞❡

Tλ✳ ❊♥tã♦✱ ♦ ♥ú♠❡r♦ ❞❡ ❧✐♥❤❛s ❡ ❝♦❧✉♥❛s ♥❛ t❛❜❡❧❛ ❛ss♦❝✐❛❞❛ Tλ sã♦ ✐❣✉❛✐s✳

❆❣♦r❛ ❡♠ ❝❛❞❛ t❛❜❡❧❛ ❛ss♦❝✐❛❞❛ Tλ ❞❡ ✉♠❛ q✉❛❧q✉❡r ♣❛rt✐çã♦ λ ⊢ n✱

❝♦❧♦q✉❡♠♦s ♥♦ ✐♥t❡r✐♦r ❞❡ ❝❛❞❛ ❝❛✐①❛ ✉♠ ✐♥t❡✐r♦ ❞❡ 1 ❛ n s❡♠ r❡♣❡t✐çõ❡s✱

s❛t✐s❢❛③❡♥❞♦ ❛s s❡❣✉✐♥t❡s ♣r♦♣r✐❡❞❛❞❡s✿

• ❆ ♦r❞❡♠ ❞❡ss❡s ✐♥t❡✐r♦s ❞❡✈❡♠ s❡r ❝r❡s❝❡♥t❡ ♥❛s ❧✐♥❤❛s ❞❡ ❡sq✉❡r❞❛ ♣❛r❛ ❞✐r❡✐t❛❀

• ❆ ♦r❞❡♠ ❞❡ss❡s ✐♥t❡✐r♦s ❞❡✈❡♠ s❡r ❝r❡s❝❡♥t❡ ♥❛s ❝♦❧✉♥❛s ❞❡ ❝✐♠❛ ♣❛r❛ ❜❛✐①♦✳

❉❡♥♦t❛r❡♠♦s ❡ss❛ ♥♦✈❛ t❛❜❡❧❛ ♣♦r TS

λ ❡ ❝❤❛♠❛r❡♠♦s ❞❡ t❛❜❡❧❛ st❛♥❞❛r❞ ❞❡

λ✳ ❖❜s❡r✈❡ q✉❡ ♣♦ss✐✈❡❧♠❡♥t❡ ❡①✐st❡♠ ♣❛r❛ ❝❛❞❛ ♣❛rt✐çã♦ ❞❡ n✱ ❞✐st✐♥t❛s

t❛❜❡❧❛s st❛♥❞❛r❞s✳

❊①❡♠♣❧♦ ✶✳✷✳✶✺✳ ➚ t❛❜❡❧❛ ❞♦ ❡①❡♠♣❧♦ 1.2.11 ♣♦❞❡♠♦s ❛ss♦❝✐❛r ❞✐✈❡rs❛s t❛✲

❜❡❧❛s st❛♥❞❛r❞✱ ♣♦r ❡①❡♠♣❧♦ ❞✉❛s ❞❡❧❛s sã♦

1 3 4 6 2 5 7 8 9 10 11 13 12 15 14

❡

1 2 3 5 4 6 8 7 9 11 10 12 13 14 15

❆ ♣❛rt✐r ❞❡ ❝❛❞❛ t❛❜❡❧❛ st❛♥❞❛r❞✱ ✐r❡♠♦s ❞❛r ♦r✐❣❡♠ ❛ ✉♠ s✉❜❣r✉♣♦ ❞❡ Sn ❝♦♥str✉í❞♦ ❞❛ s❡❣✉✐♥t❡ ❢♦r♠❛✳ ❈♦♥s✐❞❡r❡ λ = (λ1, λ2, . . . , λr) ⊢ n ❡ TλS

✉♠❛ t❛❜❡❧❛ st❛♥❞❛r❞ ❡ ❞❡✜♥❛ ♦ s❡❣✉✐♥t❡ s✉❜❣r✉♣♦

Sλ =SN1 ×SN2 × · · · ×SNr,

♦♥❞❡ Ni = {✐♥t❡✐r♦s ❞❛ ❧✐♥❤❛i} ❡ SNi ♦ ❣r✉♣♦ ❞❡ ♣❡r♠✉t❛çõ❡s r❡❛❧✐③❛❞❛s