Elementos Simétricos e Antissimétricos sob Involuções Orientadas Generalizadas em Anéis de Grupo

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛ ✲ ❯❋❇❆

■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ✲ ■▼

Pr♦❣r❛♠❛ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ✲ P●▼❆❚ ❚❡s❡ ❞❡ ❉♦✉t♦r❛❞♦

❊❧❡♠❡♥t♦s ❙✐♠étr✐❝♦s ❡ ❆♥t✐ss✐♠étr✐❝♦s s♦❜

■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s ●❡♥❡r❛❧✐③❛❞❛s ❡♠ ❆♥é✐s ❞❡

●r✉♣♦

❊❞✇❛r❞ ▲❛♥❞✐ ❚♦♥✉❝❝✐

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❊❧❡♠❡♥t♦s ❙✐♠étr✐❝♦s ❡ ❆♥t✐ss✐♠étr✐❝♦s s♦❜

■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s ●❡♥❡r❛❧✐③❛❞❛s ❡♠ ❆♥é✐s ❞❡

●r✉♣♦

❊❞✇❛r❞ ▲❛♥❞✐ ❚♦♥✉❝❝✐

❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦❧❡❣✐❛❞♦ ❞❛ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❯❋❇❆✴❯❋❆▲ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ▼❛t❡♠át✐❝❛✱ ❛♣r♦✈❛❞❛ ❡♠ ✷✺ ❞❡ ❆❣♦st♦ ❞❡ ✷✵✶✼✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❚❤✐❡rr② ❈♦rrê❛ P❡t✐t ▲♦❜ã♦✳

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❚♦♥✉❝❝✐✱ ❊❞✇❛r❞ ▲❛♥❞✐✳

❊❧❡♠❡♥t♦s ❙✐♠étr✐❝♦s ❡ ❆♥t✐ss✐♠étr✐❝♦s s♦❜ ■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s ●❡♥❡r❛❧✐③❛❞❛s ❡♠ ❆♥é✐s ❞❡ ●r✉♣♦✴ ❊❞✇❛r❞ ▲❛♥❞✐ ❚♦♥✉❝❝✐✳ ✕ ❙❛❧✈❛❞♦r✿ ❯❋❇❆✱ ✷✵✶✼✳

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❚❤✐❡rr② ❈♦rrê❛ P❡t✐t ▲♦❜ã♦✳

❉✐ss❡rt❛çã♦ ✭❞♦✉t♦r❛❞♦✮ ✕ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✱ Pr♦❣r❛♠❛ ❞❡ Pós✲❣r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛✱ ✷✵✶✼✳

❘❡❢❡rê♥❝✐❛s ❜✐❜❧✐♦❣rá✜❝❛s✳

✶✳ ❆♥é✐s ✭➪❧❣❡❜r❛✮✳ ✷✳ ❆♥é✐s ❞❡ ●r✉♣♦✳ ✸✳ ❚❡♦r✐❛ ❞❡ ●r✉♣♦s✳ ■✳ P❡t✐t ▲♦❜ã♦✱ ❚❤✐❡rr② ❈♦rrê❛✳ ■■✳ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ ❇❛❤✐❛✱ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛✳ ■■■✳ ❚ít✉❧♦✳

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❊❧❡♠❡♥t♦s ❙✐♠étr✐❝♦s ❡ ❆♥t✐ss✐♠étr✐❝♦s s♦❜

■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s ●❡♥❡r❛❧✐③❛❞❛s ❡♠ ❆♥é✐s ❞❡

●r✉♣♦

❊❞✇❛r❞ ▲❛♥❞✐ ❚♦♥✉❝❝✐

❚❡s❡ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈♦❧❡❣✐❛❞♦ ❞❛ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ ❯❋❇❆✴❯❋❆▲ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ❉♦✉t♦r ❡♠ ▼❛t❡♠át✐❝❛✱ ❛♣r♦✈❛❞❛ ❡♠ ✷✺ ❞❡ ❆❣♦st♦ ❞❡ ✷✵✶✼✳

❇❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛✿

Pr♦❢✳ ❉r✳ ❚❤✐❡rr② ❈♦rrê❛ P❡t✐t ▲♦❜ã♦ ✭❖r✐❡♥t❛❞♦r✮ ❯❋❇❆

Pr♦❢❛✳ ❉r❛✳ ▼❛♥✉❡❧❛ ❞❛ ❙✐❧✈❛ ❙♦✉③❛

❯❋❇❆

Pr♦❢❛✳ ❉r❛✳ ❈❛r♠❡❧❛ ❙✐❝❛

❯❋❇❆

Pr♦❢❛✳ ❉r❛✳ ❆♥❛ ❈r✐st✐♥❛ ❱✐❡✐r❛

❯❋▼●

Pr♦❢❛✳ ❉r✳ ❖s♥❡❧ ❇r♦❝❤❡ ❈r✐st♦

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❆❣r❛❞❡❝✐♠❡♥t♦s

Pr✐♠❡✐r♦✱ ❛❣r❛❞❡ç♦ ✐♠❡♥s❛♠❡♥t❡ à ♠✐♥❤❛ ❢❛♠í❧✐❛ ♣♦r t♦❞♦ ❛♣♦✐♦ ♠♦r❛❧ ❡ ✜♥❛♥✲ ❝❡✐r♦✱ ♣♦✐s s❡♠ ❡st❡ t❡♥❤♦ ❝❡rt❡③❛ q✉❡ ♥✉♥❝❛ ❛❧❝❛♥ç❛r✐❛ ♦ ♥í✈❡❧ ✐♥t❡❧❡❝t✉❛❧✱ ❛❝❛❞ê♠✐❝♦ ❡ ♣r♦✜ss✐♦♥❛❧ ♥♦ q✉❛❧ ♠❡ ❡♥❝♦♥tr♦✳ ❊♠ ❡s♣❡❝✐❛❧✱ ❛❣r❛❞❡ç♦ ♠✉✐t♦ ❛♦s ♠❡✉s ♣❛✐s ❈❧á✉❞✐♦ ❡ ❘✐t❛ ♣♦r t♦❞♦ ❛♠♦r✱ ❝❛r✐♥❤♦✱ ✐♥❝❡♥t✐✈♦ ❡ ❡❞✉❝❛çã♦ ♠♦r❛❧✱ ♣♦✐s ❝❛❞❛ ✉♠ ❞❡ss❡s ❞❡t❛❧❤❡s ❛❥✉❞♦✉ ❛ ❝♦♥str✉✐r ♠✐♥❤❛ ♣❡rs♦♥❛❧✐❞❛❞❡ ❡ ❛ ♣❡ss♦❛ q✉❡ s♦✉✳ ❆❣r❛❞❡ç♦ t❛♠❜é♠ ❛♦s ♠❡✉s ✐r♠ã♦s✱ ❏♦ã♦ P❛✉❧♦ ❡ ❈❛r♦❧✐♥❡✱ ❡ ❛♦s ♠❡✉s ♣r✐♠♦s ♣❡❧♦s q✉❛✐s ♣♦ss✉♦ t❛♥t♦ ❛❢❡t♦✱ ♣♦r t♦❞♦ ♦ ❝❛r✐♥❤♦✱ ❛♠✐③❛❞❡ ❡ ❞✐✈❡rsã♦ q✉❡ ♠❡ ♣r♦♣♦r❝✐♦♥❛r❛♠ ❞❡s❞❡ ❛ ♠✐♥❤❛ ✐♥❢â♥❝✐❛ ❛té ♦ ♣r❡s❡♥t❡ ♠♦♠❡♥t♦✳

❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❝♦♠♣❛♥❤❡✐r❛ ❏❛❝q✉❡❧✐♥❡ ♣♦r t♦❞♦ ❝❛r✐♥❤♦✱ ❛♠♦r✱ ❝♦♥✜❛♥ç❛✱ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❡ ♣♦r ❡st❛r s❡♠♣r❡ ❛♦ ♠❡✉ ❧❛❞♦ ♠❡ ❛❥✉❞❛♥❞♦ ❛ ♣❛ss❛r ♣♦r t♦❞♦s ♦s ♠♦♠❡♥t♦s ❞✐❢í❝❡✐s q✉❡ ❛♣❛r❡❝❡r❛♠ ♥❡st❛ ❥♦r♥❛❞❛✱ ✐♥❝❡♥t✐✈❛♥❞♦ ❡ ❛♣♦✐❛♥❞♦ ❛s ❞❡❝✐sõ❡s ✐♠♣♦rt❛♥t❡s q✉❡ t✐✈❡ q✉❡ t♦♠❛r ❛♣ós ✐♥✐❝✐❛r ♠✐♥❤❛ ❝❛rr❡✐r❛ ❛❝❛❞ê♠✐❝❛✱ ❛❧é♠ ❞❡ s❡r ✉♠❛ ♣❡ss♦❛ ❡①❝❡♣❝✐♦♥❛❧ ❝♦♠ q✉❡♠ ♣♦ss♦ ❞✐✈✐❞✐r ♠❡✉s ❞❡s❡❥♦s ❡ ❣♦st♦s✳

❆❣r❛❞❡ç♦ ❛♦s ♠❡✉s ❛♠✐❣♦s✱ ❡①✲❝♦❧❡❣❛s ❞❡ ❣r❛❞✉❛çã♦✱ ♠❡str❛❞♦ ❡ ❞♦✉t♦r❛❞♦ ♣❡❧♦s ❜♦♥s ♠♦♠❡♥t♦s q✉❡ ♣❛ss❛♠♦s ❡st✉❞❛♥❞♦ ❡ ♣♦r ❛❥✉❞❛r❡♠ ❛ ♠❡ ❞✐✈❡rt✐r ❡ ❞✐str❛✐r ♥♦s ♠♦♠❡♥t♦s ❞❡ ❞❡s❝❛♥s♦✳ ❆❣r❛❞❡ç♦ ❡♠ ❡s♣❡❝✐❛❧ às ♠✐♥❤❛s ✐r♠ãs ❆❧❡❥❛♥❞r❛ ❡ ❊❧❡♥ ♣♦r t♦❞♦ ♦ ❝♦♠♣❛♥❤❡✐r✐s♠♦ ❞✉r❛♥t❡ t♦❞❛ ❛ ❥♦r♥❛❞❛✳

❆❣r❛❞❡ç♦ ♠✉✐t♦ t❛♠❜é♠ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ ❚❤✐❡rr② P❡t✐t ▲♦❜ã♦✱ t❛♥t♦ ♣❡❧❛ s✉❛ ❞✐s♣♦s✐çã♦✱ ❞❡❞✐❝❛çã♦ ❡ ♣r❡st❛t✐✈✐❞❛❞❡ q✉❛♥t♦ ♣❡❧♦ s❡✉ ❝♦♥st❛♥t❡ ✐♥❝❡♥t✐✈♦ ❡ ♣r♦✜ss✐♦♥❛✲ ❧✐s♠♦ ❞✉r❛♥t❡ ❛ ♠✐♥❤❛ ❥♦r♥❛❞❛ ❞❛ ♣ós✲❣r❛❞✉❛çã♦✳

❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ▼❛♥✉❡❧❛ ❞❛ ❙✐❧✈❛ ❙♦✉③❛✱ ❈❛r♠❡❧❛ ❙✐❝❛✱ ❆♥❛ ❈r✐st✐♥❛ ❱✐❡✐r❛ ❡ ❖s♥❡❧ ❇r♦❝❤❡ ❈r✐st♦ ♣♦r ♣❛rt✐❝✐♣❛r❡♠ ❞❛ ❝♦♠✐ssã♦ ❥✉❧❣❛❞♦r❛ ❞❛ ♠✐♥❤❛ t❡s❡ ❡ ♠❡ ❞❛r❡♠ ❛ ❣r❛♥❞❡ ❤♦♥r❛ ❞❡ tê✲❧♦s ❝♦♠♦ ♠❡♠❜r♦s ❞❛ ❜❛♥❝❛ ❡①❛♠✐♥❛❞♦r❛ ❞❡ ♠✐♥❤❛ ❞❡❢❡s❛✳

❆❣r❛❞❡ç♦ ❛ t♦❞♦s ♦s ♣r♦❢❡ss♦r❡s ❞♦ ❉❈❊✲❯❊❙❇ ❡ ■▼✲❯❋❇❆ q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❡❢❡t✐✈❛♠❡♥t❡ ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ❝♦♠♦ ♠❛t❡♠át✐❝♦✳ ❆❣r❛❞❡ç♦ ❛✐♥❞❛ ♠❛✐s ❛♦s q✉❡ ❝♦♥✲ tr✐❜✉ír❛♠ ♣❛r❛ ♠✐♥❤❛ ❢♦r♠❛çã♦ ♥ã♦ s♦♠❡♥t❡ ❝♦♠♦ ♠❛t❡♠át✐❝♦✱ ♠❛s t❛♠❜é♠ ❝♦♠♦ s❡r ❤✉♠❛♥♦✳ ❆❣r❛❞❡❝✐♠❡♥t♦s ❡s♣❡❝✐❛✐s ❛♦s ♣r♦❢❡ss♦r❡s ❆❞❡❧③✐t♦✱ ❆❝✐♦❧②✱ ❈❧❛✉❞✐♥❡✐✱ ❈❧ê♥✐❛✱ ❉é❜♦r❛✱ ❊r✐❞❛♥✱ ❋❧❛✉❧❧❡s✱ ❏ú❧✐♦✱ ▼ár❝✐♦✱ ❘❡❣✐♥❛❧❞♦✱ ❚â♥✐❛✱ ❉✐❡❣♦ ❡ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ❆✉❣✉st♦✱ ♣♦r s❡✉s ❡①❡♠♣❧♦s ❝♦♠♦ ♣r♦✜ss✐♦♥❛✐s ❡ ót✐♠♦s ♣r♦❢❡ss♦r❡s✳

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✏❙♦♥❤♦ q✉❡ s❡ s♦♥❤❛ só é só ✉♠ s♦♥❤♦ q✉❡ s❡ s♦♥❤❛ só✳ ▼❛s s♦♥❤♦ q✉❡ s❡ s♦♥❤❛ ❥✉♥t♦ é r❡❛❧✐❞❛❞❡✳✑

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❘❡s✉♠♦

❙❡❥❛RG ✉♠ ❛♥❡❧ ❞❡ ❣r✉♣♦ ❞❡ ✉♠ ❣r✉♣♦Gs♦❜r❡ ✉♠ ❛♥❡❧ Rt❛❧ q✉❡char(R)6= 2✳

❆ ♣❛rt✐r ❞❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦ σ : G → {±1}✱ ❝❤❛♠❛❞♦ ❞❡ ♦r✐❡♥t❛çã♦ ❡♠ G✱ ❡ ✉♠❛

✐♥✈♦❧✉çã♦∗✱ ♣♦❞❡♠♦s ♠✉♥✐r RG ❝♦♠ ✉♠❛ ✐♥✈♦❧✉çã♦ ❞❡ ❛♥é✐s σ∗✱ ❝❤❛♠❛❞❛ ❞❡ ✐♥✈♦❧✉çã♦

♦r✐❡♥t❛❞❛✱ ❛ ♣❛rt✐r ❞❛ ❡①t❡♥sã♦ ❧✐♥❡❛r ❞❡ ∗t♦r❝✐❞❛ ♣❡❧❛ ♦r✐❡♥t❛çã♦ σ✳

➱ ♥♦tá✈❡❧ ❛ ❝♦♥❝❡♥tr❛çã♦ ❞❡ ♣❡sq✉✐s❛ r❡❧❛❝✐♦♥❛♥❞♦ ✐❞❡♥t✐❞❛❞❡s ♣♦❧✐♥♦♠✐❛✐s ❡ ✐♥✲ ✈♦❧✉çõ❡s✱ s❡♥❞♦ q✉❡✱ ♥♦ ❝❛s♦ ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦∗ ❡♠ RG✱ ♣❛rt❡ ❞❡ss❛ é ❢♦❝❛❞❛ ♥♦ ❡st✉❞♦

❞❛ r❡❧❛çã♦ ❡①✐st❡♥t❡ ❡♥tr❡ ✉♠❛ ✐❞❡♥t✐❞❛❞❡ ✈❡r✐✜❝❛❞❛ ❡♠ RG+

= {αRG:α∗ =α} ♦✉ ❡♠ RG−{αRG:α=α} ❡ ❛s ❡str✉t✉r❛s ❜❛s❡ ❞❡st❡ ❛♥❡❧✱ ♥♦ ❝❛s♦ R G

❖s ♣r✐♥❝✐♣❛✐s r❡s✉❧t❛❞♦s q✉❡ ❡①✐❜❡♠ ❡st❛s r❡❧❛çõ❡s sã♦ ❛❧❝❛♥ç❛❞♦s q✉❛♥❞♦ RG+

♦✉RG−✱ s♦❜ ✐♥✈♦❧✉çõ❡s ❞❛❞❛s ♣❡❧❛ ❡①t❡♥sã♦ ❧✐♥❡❛r ❞❡ ❡♠ RG ♦✉ ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s

σ✱ ✈❡r✐✜❝❛♠ ❝♦♠✉t❛t✐✈✐❞❛❞❡✱ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ♦✉ ❛❧❣✉♠❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ▲✐❡✳ ◆❡st❡

s❡♥t✐❞♦✱ ❜✉s❝❛♠♦s ♥♦ ♣r❡s❡♥t❡ tr❛❜❛❧❤♦ ❡st✉❞❛r ♦s ❝❛s♦s ♥ã♦ ❡①♣❧♦r❛❞♦s ♥❛ ❧✐t❡r❛t✉r❛✱ ❛s✲ s✐♠ ❝♦♠♦ ❣❡♥❡r❛❧✐③❛r ♦s r❡s✉❧t❛❞♦s ❝♦♥❤❡❝✐❞♦s ♣❛r❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ ❣❡♥❡r❛❧✐③❛❞❛✱ ♦♥❞❡ s✉♣♦♠♦s ❛ ♦r✐❡♥t❛çã♦σ ❛❣♦r❛ s❡♥❞♦ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡G ❡♠ U(R)✳

❈❧❛ss✐✜❝❛r❡♠♦s ❞❡ ❢♦r♠❛ ♣❛r❝✐❛❧ q✉❛♥❞♦ RG+✱ s♦❜ ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛✱ é

❝♦♠✉t❛t✐✈♦✱ s❡♥❞♦ ❡st❡ ♦ ú❧t✐♠♦ ❝❛s♦ ❛ s❡r tr❛t❛❞♦ ❞♦ ♣♦♥t♦ ❞❡ ✈✐st❛ ❡①♣♦st♦ ❛❝✐♠❛✱ ❡✱ ❛❧é♠ ❞✐st♦✱ ❞❡s❝r❡✈❡r❡♠♦s ❞❡ ❢♦r♠❛ ✐♥t❡❣r❛❧ q✉❛♥❞♦ ❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ é ✈❡r✐✜❝❛❞❛ ❡♠

RG+ ♦✉

RG− ❛❣♦r❛ s♦❜ ❛ ót✐❝❛ ❞❛s ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s ❣❡♥❡r❛❧✐③❛❞❛s✳

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❆❜str❛❝t

▲❡t RG ❜❡ ❛ ❣r♦✉♣ r✐♥❣ ♦❢ ❛ ❣r♦✉♣ G ♦✈❡r ❛ r✐♥❣ R ✇✐t❤ charR 6= 2✳ ❋r♦♠ ❛

❤♦♠♦♠♦r✜s♠σ :G→ {±1}✱ ❝❛❧❧❡❞ ♦r✐❡♥t❛t✐♦♥ ✐♥ G✱ ❛♥❞ ❛♥ ✐♥✈♦❧✉t✐♦♥ ✱ ✇❡ ❝❛♥ ❛ss✐❣♥ RG ✇✐t❤ ❛ r✐♥❣ ✐♥✈♦❧✉t✐♦♥ σ∗✱ ❝❛❧❧❡❞ ♦r✐❡♥t❡❞ ✐♥✈♦❧✉t✐♦♥✱ ❜② t❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦❢ ∗

t✇✐st❡❞ ❜② t❤❡ ♦r✐❡♥t❛t✐♦♥σ✳

❆ r❡♠❛r❦❛❜❧❡ ❛♠♦✉♥t ♦❢ r❡s❡❛r❝❤ ❤❛s ❜❡ ❞♦♥❡ ❝♦♥♥❡❝t✐♥❣ ♣♦❧②♥♦♠✐❛❧ ✐❞❡♥t✐t✐❡s ✇✐t❤ ✐♥✈♦❧✉t✐♦♥s✳ ■♥ t❤✐s s❡♥s❡✱ ♣❛rt ♦❢ t❤❡ r❡s❡❛r❝❤✱ ✐♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ✐♥✈♦❧✉t✐♦♥ ♦♥ ❘●✱ ✐s ❢♦❝✉s❡❞ ♦♥ t❤❡ st✉❞② ♦❢ t❤❡ r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ ❛♥ ✐❞❡♥t✐t② ✐♥RG+

={αRG :α∗ =α} ♦r ✐♥RG−{α RG:α=α} ❛♥❞ t❤❡ ❜❛s✐❝ str✉❝t✉r❡s ♦❢ t❤❡ r✐♥❣✱ ♥❛♠❡❧②✱ ❘ ❛♥❞ ●✳

❚❤❡ ♠❛✐♥ r❡s✉❧ts ❝♦♥❝❡r♥✐♥❣ t❤❡s❡ r❡❧❛t✐♦♥s ❛r❡ ♦❜t❛✐♥❡❞ ✇❤❡♥ RG+ ♦r

RG− ✉♥❞❡r ✐♥✈♦❧✉t✐♦♥s ❣✐✈❡♥ ❜② t❤❡ ❧✐♥❡❛r ❡①t❡♥s✐♦♥ ♦❢∗✐♥RG♦✉ t❤❡ ♦r✐❡♥t❡❞ ✐♥✈♦❧✉t✐♦♥sσ

s❛t✐✜❡s ❝♦♠♠✉t❛t✐✈✐t② ♦r ❛♥t✐❝♦♠♠✉t❛t✐✈✐t②✱ ♦r ❡✈❡♥ s♦♠❡ ▲✐❡ ♣r♦♣❡rt②✳

■♥ t❤✐s ✇❛②✱ ✇❡ ✐♥t❡♥❞ t♦ ❛❞❞r❡ss ❝❛s❡s ♥♦t ②❡t ❡①♣❧♦r❡❞ ✐♥ t❤❡ ❧✐t❡r❛t✉r❡✱ ❛s ✇❡❧❧ ❛s ❡①t❡♥❞✐♥❣ ❦♥♦✇♥ r❡s✉❧ts t♦ ❛ ❣❡♥❡r❛❧✐③❡❞ ♦r✐❡♥t❡❞ ✐♥✈♦❧✉t✐♦♥✱ ❝♦♥s✐❞❡r✐♥❣ t❤❡ ♦r✐❡♥t❛t✐♦♥

σ ❛s ❛ ❤♦♠♦♠♦r✜s♠ ♦❢ G✐♥t♦ U(R)✳

❲❡ s❤❛❧ ♣❛rt✐❛❧❧② ❝❧❛ss✐❢② t❤❡ ❝❛s❡ ✐♥ ✇❤✐❝❤ RG+✱ ✉♥❞❡r ❛♥ ♦r✐❡♥t❡❞ ✐♥✈♦❧✉t✐♦♥✱

✐s ❝♦♠♠✉t❛t✐✈❡✱ s✉❝❤ ✐s t❤❡ ❧❛st ❝❛s❡ t♦ ❜❡ tr❡❛t❡❞ ✐♥ t❤❡ s❡♥s❡ ♠❡♥t✐♦♥❡❞ ❛❜♦✈❡✳ ❇❡s✐❞❡s✱ ✇❡ s❤❛❧❧ ❢✉❧❧② ❞❡s❝r✐❜❡ t❤❡ ❝❛s❡ ✐♥ ✇❤✐❝❤ ❛♥t✐❝♦♠♠✉t❛t✐✈✐t② ✐s ✈❡r✐✜❡❞ ✐♥ RG+ ♦r

RG− ✉♥❞❡r t❤❡ ❧✐❣❤t ♦❢ ❣❡♥❡r❛❧✐③❡❞ ♦r✐❡♥t❡❞ ✐♥✈♦❧✉t✐♦♥s✳

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❮♥❞✐❝❡

■♥tr♦❞✉çã♦ ✶

✶ Pr❡❧✐♠✐♥❛r❡s ✼

✶✳✶ ❆♥é✐s ❞❡ ●r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼ ✶✳✷ ■♥✈♦❧✉çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✶ ■♥✈♦❧✉çõ❡s ❡♠ ●r✉♣♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✷ ■♥✈♦❧✉çõ❡s ❡♠ ❆♥é✐s ❞❡ ●r✉♣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✵ ✶✳✸ ■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✹ ■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s ●❡♥❡r❛❧✐③❛❞❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻

✷ ❆♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❆♥t✐ss✐♠étr✐❝♦s ✶✾

✸ ❆♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❙✐♠étr✐❝♦s ✸✷

✹ ❈♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❆♥t✐ss✐♠étr✐❝♦s ✺✶

✹✳✶ R2 6={0} ♦✉(G∗∩ Z(G))\N 6=∅ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✹✳✶✳✶ G2

6⊂G∗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✻ ✹✳✶✳✷ G2

⊂G∗ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✾

✺ ❈♦♥❝❧✉sõ❡s ✻✻

✺✳✶ ❈♦♥s✐❞❡r❛çõ❡s ❋✐♥❛✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✺✳✷ ❚r❛❜❛❧❤♦s ❋✉t✉r♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✼

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■♥tr♦❞✉çã♦

❆ ❚❡♦r✐❛ ❞♦s ❆♥é✐s ❞❡ ●r✉♣♦s ❝✉♠♣r❡ ✉♠ ♣❛♣❡❧ ❞❡ ❞❡st❛q✉❡ ❞❡♥tr♦ ❞❛ ♠❛t❡♠á✲ t✐❝❛✱ ✈✐st♦ q✉❡ ❡ss❛ ❢✉♥❝✐♦♥❛ ❝♦♠♦ ✉♠ ♣♦♥t♦ ❡♠ q✉❡ t❡♠♦s ✉♠❛ ✈✐❛ ❞❡ ♠ã♦ ❞✉♣❧❛ ❧✐❣❛♥❞♦ ❛s ♠❛✐s ❞✐✈❡rs❛s s✉❜ár❡❛s ❞❛ á❧❣❡❜r❛✱ ♥❛ q✉❛❧ ❝♦♥❤❡❝❡♥❞♦ r❡s✉❧t❛❞♦s ❛❝❡r❝❛ ❞❡ ❣r✉♣♦s✱ ❛♥é✐s✱ ♠ó❞✉❧♦s ❡ á❧❣❡❜r❛s✱ ♣♦❞❡♠♦s ❛✈❛♥ç❛r ❝♦♠ ❛ t❡♦r✐❛ ❡♠ q✉❡stã♦ ❡✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛❞♦ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ✐♥❢♦r♠❛çõ❡s q✉❡ ♣♦ss✉í♠♦s ❞❡ ✉♠ ❞❛❞♦ ❛♥❡❧ ❞❡ ❣r✉♣♦✱ ♣♦❞❡♠♦s ❡①tr❛✐r ♣r♦♣r✐❡❞❛❞❡s ❜❛st❛♥t❡ ❢♦rt❡s ❞❡ss❛s ❡str✉t✉r❛s✱ ❛❧é♠ ❞❡ t♦♠á✲❧❛s ❝♦♠♦ ✉♠ ♣♦♥t♦ ❞❡ ♣❛rt✐❞❛ ❡ ♠♦t✐✈❛çã♦ ♣❛r❛ ❛ ❜✉s❝❛ ❞❡ ❛♥á❧♦❣♦s ♣❛r❛ ❛s ❚❡♦r✐❛s ❞❡ ❆♥é✐s ❡ ▼ó❞✉❧♦s✳

❯♠ ❡①❡♠♣❧♦ ❝❧❛r♦ ❞❡ ❝♦♠♦ ❢✉♥❝✐♦♥❛ ❡ss❛ tr♦❝❛ ❞❡ ✐♥❢♦r♠❛çõ❡s ♣♦❞❡ s❡r ✈✐st❛ ♥♦s r❡s✉❧t❛❞♦s ❛❜♦r❞❛❞♦s ♥❡ss❡ tr❛❜❛❧❤♦✳

❉❛❞♦ ✉♠ ❛♥❡❧R❝♦♠ ✐♥✈♦❧✉çã♦✱ ❝❤❛♠❛♠♦s ❞❡ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s s✐♠étr✐❝♦s

❡ ❛♥t✐ss✐♠étr✐❝♦s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ ❛♦s ❝♦♥❥✉♥t♦s

R+

:={rR:r∗ =r} ❡ R− :={rR:r∗ =r};

✉t✐❧✐③❛♠♦s t❛♠❜é♠ ❛ ♥♦t❛çã♦ R∗ ❡ R−∗ q✉❛♥❞♦ é ❝♦♥✈❡♥✐❡♥t❡ ❡s♣❡❝✐✜❝❛r ❛ ✐♥✈♦❧✉çã♦ ❡♠ t❡❧❛✳ ❖ ✐♥t❡r❡ss❡ ❡♠ ❡st✉❞❛r t❛✐s ❝♦♥❥✉♥t♦s t♦r♥♦✉✲s❡ ❡✈✐❞❡♥t❡ ❛♣ós ❆♠✐ts✉r t❡r ♣r♦✈❛❞♦ ❡♠ ✉♠ ❛rt✐❣♦ ❝❧áss✐❝♦✱ ❬❆✻✾❪✱ q✉❡ s❡ R+ ♦✉

R− s❛t✐s❢❛③ ❛❧❣✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧✱ ❡♥tã♦ R t❛♠❜é♠ s❛t✐s❢❛③ ❛❧❣✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧✳

■♥❢❡❧✐③♠❡♥t❡✱ ♥♦ ❝♦♥t❡①t♦ ❞♦ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r✱ ❛ ✐❞❡♥t✐❞❛❞❡ q✉❡R✈❡r✐✜❝❛✱ ♥ã♦ é✱

❡♠ ❣❡r❛❧✱ ❛ ♠❡s♠❛ ✈❡r✐✜❝❛❞❛ ♣❡❧❛ ✐❞❡♥t✐❞❛❞❡ ♣r❡✈✐❛♠❡♥t❡ ❝♦♥❤❡❝✐❞❛ ❡♠R+♦✉

R−✳ ❈❛s♦

Rs❛t✐s❢❛ç❛ ❛ ♠❡s♠❛ ✐❞❡♥t✐❞❛❞❡ ❡♥❝♦♥tr❛❞❛ ❡♠ R+ ✭♦✉

R−✮✱ ❞✐③❡♠♦s q✉❡ ❛ ✐❞❡♥t✐❞❛❞❡ ❢♦✐ ❧❡✈❛♥t❛❞❛ ❞♦s s✐♠étr✐❝♦s ✭♦✉ ❞♦s ❛♥t✐ss✐♠étr✐❝♦s✮ ♣❛r❛R✳ ❉❡ss❛ ❢♦r♠❛✱ s✉r❣❡ ❛ ♣❡r❣✉♥t❛

♣❛r❛ q✉❛✐s ✐❞❡♥t✐❞❛❞❡s ♦ ❧❡✈❛♥t❛♠❡♥t♦ r❡❢❡r✐❞♦ ❛❝✐♠❛ é ❣❛r❛♥t✐❞♦✳

❚♦♠❛♥❞♦R ✉♠ ❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❝♦♠ ✉♥✐❞❛❞❡ ❡ G✉♠ ❣r✉♣♦ q✉❛❧q✉❡r✱ ❞❡✜♥✐♠♦s RG❝♦♠♦ ♦R✲♠ó❞✉❧♦ ❧✐✈r❡♠❡♥t❡ ❣❡r❛❞♦ ♣♦rG✳ ▼✉♥✐♥❞♦RG ❞❡ ✉♠ ♣r♦❞✉t♦ ❛ ♣❛rt✐r ❞❛s

♦♣❡r❛çõ❡s ❞❡R ❡ G✱ r❡s♣❡✐t❛♥❞♦ ❛ ❞✐str✐❜✉t✐✈✐❞❛❞❡✱ t❡♠♦s q✉❡RG ♣♦ss✉✐ ✉♠❛ ❡str✉t✉r❛

♥❛t✉r❛❧ ❞❡ ❛♥❡❧✱ ♦ q✉❛❧ ❝❤❛♠❛♠♦s ❞❡ ❛♥❡❧ ❞❡ ❣r✉♣♦✳

❆ ♣❛rt✐r ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗ ❞❡ ❣r✉♣♦s ❡♠ G✱ é ♣♦ssí✈❡❧ ❡st❡♥❞❡rR✲❧✐♥❡❛r♠❡♥t❡

❡ss❛ ✐♥✈♦❧✉çã♦ ♣❛r❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ❞❡ ❛♥é✐s ♥♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG✱ ♥❡ss❡ ❝❛s♦ t❛♠❜é♠

(13)

❞❡♥♦t❛❞❛ ♣♦r∗✱ ♦✉ s❡❥❛✱ s❡ α=Px∈Gαxx✱ ❡♥tã♦

α∗ = X

x∈G

αxx !∗

=X

x∈G

αxx∗,

❝❤❛♠❛❞❛ ❞❡ ✐♥✈♦❧✉çã♦ ✐♥❞✉③✐❞❛ ❞❡G❡♠RG✳ ◆♦t❡ q✉❡ ❛ ❛♣❧✐❝❛çã♦ q✉❡ ❧❡✈❛ ✉♠ ❡❧❡♠❡♥t♦

❞♦ ❣r✉♣♦ ❡♠ s❡✉ ✐♥✈❡rs♦ é ✉♠❛ ✐♥✈♦❧✉çã♦ ❞❡ ❣r✉♣♦s❀ ❝❤❛♠❛♠♦s ❞❡ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛ ❞♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG ❛ ✐♥✈♦❧✉çã♦ ✐♥❞✉③✐❞❛ ♣♦r ❡ss❛ ❛♥t❡r✐♦r✳ ❖❜✈✐❛♠❡♥t❡✱ ❞❛❞❛ ✉♠❛

✐♥✈♦❧✉çã♦ ∗ ✐♥❞✉③✐❞❛ ❡♠ ✉♠ ❛♥❡❧ ❞❡ ❣r✉♣♦✱ ♣♦❞❡♠♦s ❝♦♥s✐❞❡r❛r ♦s s✉❜❝♦♥❥✉♥t♦s RG+

RG−

❉❛❞♦ ✉♠ ❛♥❡❧R✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡[r, s] =rssr❡✱ ♣♦r ✐♥❞✉çã♦✱

[r1, r2, . . . , rn] = [[r1, r2], r3, . . . , rn],

♣❛r❛ t♦❞♦ r, s, ri ∈ R✳ ❚r✐✈✐❛❧♠❡♥t❡ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ A ⊂ R é ❝♦♠✉t❛t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ [a, b] = 0, a, b A❀ ♥❡ss❡ s❡♥t✐❞♦✱ ❢❛③❡♥❞♦ ✉♠❛ ❡①t❡♥sã♦

❞❡ss❡ ❝♦♥❝❡✐t♦✱ ♣♦❞❡♠♦s ❡st✉❞❛r ♦s s✉❜❝♦♥❥✉♥t♦sAR t❛✐s q✉❡

[a1, . . . , an] = 0,

♦✉

[a, b, . . . , b | {z } n✈❡③❡s

] = 0,

♣❛r❛ t♦❞♦ a, ai, b ∈ A✳ ❉✐③❡♠♦s q✉❡ ✉♠ s✉❜❝♦❥✉♥t♦ q✉❡ ✈❡r✐✜❝❛ ❛ ♣r✐♠❡✐r❛ ✐❞❡♥t✐❞❛❞❡✱ ♣❛r❛ ❛❧❣✉♠n✱ é ▲✐❡ ♥✐❧♣♦t❡♥t❡✱ ❡ ▲✐❡n✲❊♥❣❡❧✱ s❡ ❡st❡ ✈❡r✐✜❝❛ ❛ s❡❣✉♥❞❛✳ ➱ ❢á❝✐❧ ✈❡r q✉❡

❈♦♠✉t❛t✐✈✐❞❛❞❡⇒▲✐❡ ♥✐❧♣♦tê♥❝✐❛⇒▲✐❡ n✲❊♥❣❡❧.

❊♠ ❬●❙✾✸❪✱ ●✐❛♠❜r✉♥♦ ❡ ❙❡❤❣❛❧ ♠♦str❛r❛♠ q✉❡ s❡Ké ✉♠ ❝♦r♣♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛

❞✐❢❡r❡♥t❡ ❞❡ 2 ❡ G ✉♠ ❣r✉♣♦ q✉❡ ♥ã♦ ♣♦ss✉✐ 2✲❡❧❡♠❡♥t♦s✱ ❡♥tã♦ ❛ ♣r♦♣r✐❡❞❛❞❡ ❞❡ ▲✐❡

♥✐❧♣♦tê♥❝✐❛ ♣♦❞❡ s❡r ❧❡✈❛♥t❛❞❛ ❞❡KG+

KG−✱ ♥♦ ❝❛s♦ ❞❡s❡r ❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛✳ ❊♠ ❬▲✾✾✱ ▲✵✵❪✱ ▲❡❡ ❡st❡♥❞❡✉ ❡ss❡ r❡s✉❧t❛❞♦ ♣❛r❛ ❣r✉♣♦s ♠❛✐♦r❡s ❡ ♣❛r❛ ♦ ❝❛s♦ ▲✐❡n✲❊♥❣❡❧✱ ❛❧é♠

❞❡ ❡st✉❞❛r q✉❛♥❞♦ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ♥ã♦ é ♣♦ssí✈❡❧✳ P♦st❡r✐♦r♠❡♥t❡✱ ❡♠ ❬●P❙✵✾✱ ▲❙❙✵✾❪✱ ♦s r❡s✉❧t❛❞♦s ❛❝✐♠❛ ❢♦r❛♠ tr❛❜❛❧❤❛❞♦s ♥♦ ❝❛s♦ ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ✐♥❞✉③✐❞❛ q✉❛❧q✉❡r✳

◆♦t❡ q✉❡ ♥♦ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❡♠ q✉❡RG+ ❡

RG−sã♦ ❝♦♠✉t❛t✐✈♦s✱ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ❞❡ss❛ ✐❞❡♥t✐❞❛❞❡ é ❡q✉✐✈❛❧❡♥t❡ ❛♦ ❣r✉♣♦ G s❡r ❛❜❡❧✐❛♥♦✳ ❉❡ss❛ ❢♦r♠❛ ❡st✉❞❛r ❛ ❝♦♠✉t❛✲

t✐✈✐❞❛❞❡ ❞♦s s✐♠étr✐❝♦s s❡ r❡s✉♠❡ ❛♦ ❡st✉❞♦ ❞❛ ❡str✉t✉r❛ ❞♦ ❣r✉♣♦ G✳ ❆ ❝❛r❛❝t❡r✐③❛çã♦

♥❡ss❡s ❝❛s♦s s❡ ❡♥❝♦♥tr❛ ❡♠ ❬❏❘✵✻✱ ❇❏P❘✵✾❪✳

(14)

❛♥t✐❝♦♠✉t❛t✐✈♦ s❡ ♦ Pr♦❞✉t♦ ❞❡ ❏♦r❞❛♥ é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ s♦❜r❡ ❡ss❡ ❝♦♥❥✉♥t♦✱ ♦✉ s❡❥❛✱

rs = 0, r, sA✳ ◆ã♦ é ❞✐✜❝í❧ ✈❡r✐✜❝❛r q✉❡ ♥♦ ❝❛s♦ ❞❡RG+♦✉

RG−s❡r ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ❞❡ss❛ ✐❞❡♥t✐❞❛❞❡ é ❡q✉✐✈❛❧❡♥t❡ ❛ G s❡r ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ❡ charR = 2✱

♣♦rt❛♥t♦✱ ❛ss✐♠ ❝♦♠♦ ♥♦ ❝❛s♦ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡✱ ♦ ❡st✉❞♦ é ❢❡✐t♦ s♦❜r❡ ❛ ❡str✉t✉r❛ ❞♦ ❣r✉♣♦ G✱ ❡ ❡st❡ ❢♦✐ ❝♦♠♣❧❡t❛♠❡♥t❡ ❝❛r❛❝t❡r✐③❛❞♦ ❡♠ ❬●P✶✸❛❪✳

❉❛❞♦ ❛s r❡❢❡rê♥❝✐❛s ❛❝✐♠❛✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ ♣❛r❛ ♦ ❡st✉❞♦ ❞❛ r❡❧❛çã♦ ❡♥tr❡ ♦s ❝♦♥❥✉♥t♦sRG+

RG− ❡ ❛s ✐❞❡♥t✐❞❛❞❡s ❝✐t❛❞❛s ❛♥t❡r✐♦r♠❡♥t❡✱ ♣♦✉❝♦ t❡♠✲s❡ ❛ ❛❝r❡s❝❡♥t❛r✳ ❉✐t♦ ✐st♦✱ s✉r❣❡♠ ❛s ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s σ ❡♠ ❛♥é✐s ❞❡ ❣r✉♣♦s✱ q✉❡ sã♦ ❞❛❞❛s ♣❡❧❛

❡①t❡♥sã♦ ❧✐♥❡❛r ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗ ❡♠ G ❛❝r❡s❝✐❞❛ ❞❡ ✉♠❛ t♦rsã♦ ♣r♦✈❡♥✐❡♥t❡ ❞❡ ✉♠

❤♦♠♦♠♦r✜s♠♦ σ:G→ {±1}✱ ♠❛✐s ♣r❡❝✐s❛♠❡♥t❡σ:RGRG ❝♦♠

ασ∗ = X

x∈G

αxx !σ∗

=X

x∈G

σ(x)αxx∗.

❚❛✐s ✐♥✈♦❧✉çõ❡s ❢♦r❛♠ ✐♥tr♦❞✉③✐❞❛s ♣♦r ◆♦✈✐❦♦✈✱ ❡♠ ❬◆✼✵❪✱ ♥♦ ❝♦♥t❡①t♦ ❞❛ K✲t❡♦r✐❛✳

◆❛t✉r❛❧♠❡♥t❡ ♣❛r❛ q✉❡ ❡st❛ ✐♥✈♦❧✉çã♦ s❡❥❛ ❞✐st✐♥t❛ ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ♥ã♦ ♦r✐❡♥t❛❞❛✱ é ♥❡❝❡ssár✐♦ q✉❡char(R)6= 2✱ ♦ q✉❛❧ s❡rá ❤✐♣ót❡s❡ ❛ss✉♠✐❞❛ ❡♠ t♦❞♦ ♦ tr❛❜❛❧❤♦✳

❚♦♠❛♥❞♦ N := kerσ✱ ♣♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ s❡ N∗ = {n∗ :n∈N} ⊂ N✱ ❛ r❡str✐çã♦ ❞❡ σ∗ ❛♦ s✉❜❛♥❡❧ RN ⊂ RG ❞❡✜♥❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ RN✳ P♦r ♦✉tr♦ ❧❛❞♦✱

❛❞♠✐t✐♥❞♦ q✉❡RG+ ♦✉

RG− s❛t✐s❢❛ç❛ ❛❧❣✉♠❛ ❞❛s ✐❞❡♥t✐❞❛❞❡s r❡❢❡r✐❞❛s ❛❝✐♠❛✱ t❡♠♦s q✉❡

RN+ ♦✉

RN− t❛♠❜é♠ ♦ ❢❛③✱ ❞❡ss❛ ❢♦r♠❛✱ ♦s r❡s✉❧t❛❞♦s ❥á ❝✐t❛❞♦s ❢♦r♥❡❝❡♠ ✐♥❢♦r♠❛çõ❡s ✐♠♣♦rt❛♥t❡s s♦❜r❡ ❛ ❡str✉t✉r❛ ❞❡ N ❡✱ ❛ ♣❛rt✐r ❞✐ss♦✱ ♣♦❞❡♠♦s ❞❡s❝r❡✈❡r ♦ ❣r✉♣♦ G✳

❚❛✐s té❝♥✐❝❛s ✜❝❛rã♦ ♠❛✐s ❝❧❛r❛s ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦✳ ❙✉r♣r❡❡♥❞❡♥t❡♠❡♥t❡✱ ♥❛ ♠❛✐♦r✐❛ ❞♦s ❝❛s♦s✱ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ❞❛ ✐❞❡♥t✐❞❛❞❡ ❝♦♥t✐♥✉❛ s❡♥❞♦ ❣❛r❛♥t✐❞♦ ❝♦♠ ❛s ♠❡s♠❛s ❤✐♣ót❡s❡s ❞♦s ❝❛s♦s ♥ã♦ ♦r✐❡♥t❛❞♦s❀ ♥♦ ❝❛s♦ ❞♦ ♥ã♦ ❧❡✈❛♥t❛♠❡♥t♦✱ t❛♠❜é♠ é ♣♦ssí✈❡❧ ✈❡r✐✜❝❛r q✉❡✱ ❡①❝❡t♦ ❡♠ ❛❧❣✉♠❛s ✐♥❞❡♥t✐❞❛❞❡s✱ ♦s ❣r✉♣♦s t❛✐s q✉❡(RG)σ∗ ♦✉(RG)−σ∗ s❛t✐s❢❛③❡♠ ❛❧❣✉♠❛ ❞❛s ✐❞❡♥t✐❞❛❞❡s ❝✐t❛❞❛s ♣❡rt❡♥❝❡♠ ❛ ♠❡s♠❛ ❝❧❛ss❡ ❞♦ ❝❛s♦ ♥ã♦ ♦r✐❡♥t❛❞♦✳ ❚❛✐s r❡s✉❧t❛❞♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬❈P✶✷✱ ●P✶✸❜✱ ●P✶✹❪✱ ♣♦r ❡①❡♠♣❧♦✳

❊st❡♥❞❡♥❞♦ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛✱ ❝❤❡❣❛♠♦s ✜♥❛❧♠❡♥t❡ às ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s ❣❡♥❡r❛❧✐③❛❞❛s✱ ♦❜t✐❞❛s ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛ ❛♦ s✉❜st✐t✉✐r ♦ ❝♦♥tr❛❞♦♠í♥✐♦ ❞❛ ♦r✐✲ ❡♥t❛çã♦ σ ❞❡ {±1} ♣♦r U(R)✱ ♦ ❣r✉♣♦ ❞❛s ✉♥✐❞❛❞❡s ❞♦ ❛♥❡❧ R✳ ◆❡ss❡ ❝❛s♦✱ ♣❛r❛ q✉❡ ❛

❛♣❧✐❝❛çã♦ σ s❡❥❛ ❞❡ ❢❛t♦ ✉♠❛ ✐♥✈♦❧✉çã♦✱ ❛ ❝♦♥❞✐çã♦ N∗ N s❡ tr❛❞✉③ ♥❡ss❡ ❝❛s♦ ♣♦r

xx∗ N

❊♠ ❬❱✶✸❪✱ ❱✐❧❧❛ ❡st✉❞♦✉ ❛s ♣r♦♣r✐❡❞❛❞❡s ❞❡ ▲✐❡ s♦❜ ❛ ót✐❝❛ ❞❡ss❛ ♥♦✈❛ ✐♥✈♦❧✉çã♦✱ ❣❡♥❡r❛❧✐③❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ❡♠ ❬❈P✶✷❪✱ ♠♦str❛♥❞♦✱ ❞❡ ❢♦r♠❛ s✐♠✐❧✐❛r ❛♦s ❝❛s♦s ❛♥t❡r✐♦r❡s✱ q✉❡ ♦ ❣r✉♣♦G♥ã♦ ♣♦ss✉✐ ❡❧❡♠❡♥t♦s ❞❡ ♦r❞❡♠2é ✉♠❛ ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ♣❛r❛

♦ ❧❡✈❛♥t❛♠❡♥t♦ ❞❛s ✐❞❡♥t✐❞❛❞❡s ❞❡ ▲✐❡ ✈❡r✐✜❝❛❞❛s ❡♠ KG+ ❡

KG−✱ ♥♦ ❝❛s♦ ❞❡ K s❡r ✉♠ ❝♦r♣♦ t❛❧ q✉❡ charK 6= 2✳

(15)

❞❛♥❞♦ ❛❧❣✉♥s ❝❛s♦s ❛✐♥❞❛ ♥ã♦ ❛❜♦r❞❛❞♦s✱ r❡❧❛❝✐♦♥❛❞♦s à ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛♥t✐❝♦♠✉t❛t✐✲ ✈✐❞❛❞❡ ❞♦s ❝♦♥❥✉♥t♦sRG+ ❡

RG−

◆♦ ❈❛♣ít✉❧♦ ✷✱ ❝❛r❛❝t❡r✐③❛r❡♠♦s ♦ ❣r✉♣♦ G✱ ❛ss✐♠ ❝♦♠♦ ✐r❡♠♦s ❢♦r♥❡❝❡r ❛❧❣✉♠❛s

♣r♦♣r✐❡❞❛❞❡s r❡❧❛t✐✈❛s ❛♦ ❛♥❡❧R ♣❛r❛ q✉❡ ♦ ❝♦♥❥✉♥t♦RG−s❡❥❛ ❛♥t✐❝♦♠✉t❛t✐✈♦✱ ♥♦ ❝❛s♦ ❞❡

σ s❡r ✉♠❛ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛✱ ❡st❡♥❞❡♥❞♦ ❛ss✐♠ ♦s r❡s✉❧t❛❞♦s ❝♦♥t✐❞♦s ❡♠ ❬●P✶✸❜❪✳

◆❡ss❡ ❝❛s♦✱ ♦❜s❡r✈❛r❡♠♦s q✉❡ ❛ ❝❧❛ss❡ ❞❡ ❣r✉♣♦s ♣❛r❛ q✉❡RG− s❡❥❛ ❛♥t✐❝♦♠✉t❛t✐✈♦ ♣❡r✲ ♠❛♥❡❝❡ ❛ ♠❡s♠❛ ❞♦s ❝❛s♦s ❡♠ q✉❡σé ❛ ♦r✐❡♥t❛çã♦ ❝❧áss✐❝❛ ✭σ ≡ {±1}✮ ♦✉ ✉♠❛ ♦r✐❡♥t❛çã♦

tr✐✈✐❛❧✳

◆♦ ❈❛♣ít✉❧♦ ✸✱ ❢❛r❡♠♦s ✉♠ ❡st✉❞♦ ❛♥á❧♦❣♦ ❛♦ ❞♦ ❈❛♣ít✉❧♦ ✷✱ ♣♦ré♠ ❛❞♠✐t✐♥❞♦ ❛❣♦r❛ q✉❡ ❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ é ✈❡r✐✜❝❛❞❛ ❡♠ RG+✳ ❊ss❡ ❝❛s♦ ✈❡♠ ❛ t❡r ✉♠ ❞❡st❛q✉❡

♥❡ss❡ tr❛❜❛❧❤♦✱ ✈✐st♦ q✉❡✱ ❛s ❝❧❛ss❡s ❞❡ ❣r✉♣♦s q✉❡ ❝❛r❛❝t❡r✐③❛♠ ❡ss❛ ✐❞❡♥t✐❞❛❞❡ ♥❡ss❡ ❝♦♥❥✉♥t♦ sã♦ ❛s ♠❡s♠❛s ♥♦ ❝❛s♦ ❞❡ ✉♠❛ ♦r✐❡♥t❛çã♦ σ ❝❧áss✐❝❛ ♦✉ tr✐✈✐❛❧✱ ❛♦ ♣❛ss♦ q✉❡ ❛♦

❢❛③❡r♠♦s ❛ ❡①t❡♥sã♦ ♣❛r❛ ❛ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛❧✐③❛❞❛✱ ✉♠❛ ♥♦✈❛ ❝❧❛ss❡ ❞❡ ❣r✉♣♦s ❛♣❛r❡❝❡ ❝♦♠♦ ♣♦ss✐❜✐❧✐❞❛❞❡✳ ❖✉tr♦ ❢❛t♦ ✐♥t❡r❡ss❛♥t❡ é q✉❡✱ ♣❛r❛ ❝♦♥❝❧✉✐r ♦s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ♥♦ ❈❛♣ít✉❧♦ ✷ ❡ ❡♠ ❬❱✶✸❪✱ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦ s✉❜❣r✉♣♦N✱ ❡st✉❞❛❞♦s q✉❛♥❞♦ ❛ ♦r✐❡♥t❛çã♦ é

tr✐✈✐❛❧ ❡♠ ❬●P✶✸❛✱ ▲✾✾✱ ▲✵✵❪✱ ❢♦✐ s✉✜❝✐❡♥t❡ ♣❛r❛ t❛❧ ❝❛r❛❝t❡r✐③❛çã♦✱ ♣♦rt❛♥t♦✱ ♦s r❡s✉❧t❛❞♦s ❛ss♦❝✐❛❞♦s ❛ ❝❛❞❛ ✉♠ ❞♦s ❝❛s♦s ♣❛r❛ ❛s ✐♥✈♦❧✉çõ❡s ♦r✐❡♥t❛❞❛s ♥ã♦ ❢♦r❛♠ ❡①♣❧✐❝✐t❛♠❡♥t❡ ✉t✐❧✐③❛❞♦s✱ ❡ ♣♦❞❡r✐❛♠ s❡r tr❛t❛❞♦s ❝♦♠♦ ❝♦r♦❧ár✐♦s ❞❡ss❛ ❡①t❡♥sã♦❀ s✉r♣r❡❡♥❞❡♥t❡♠❡♥t❡ t❛❧ ❢❛t♦ ♥ã♦ s❡ r❡♣❡t❡ ♥♦ t❡♠❛ ❛❜♦r❞❛❞♦ ♥❡st❡ ❝❛♣ít✉❧♦✱ ❞❡ss❛ ❢♦r♠❛✱ ❢❛③✲s❡ ♥❡❝❡ssár✐♦ ❛ ✐♥tr♦❞✉çã♦ ❞❡ ✉♠ ♥♦✈♦ s✉❜❣r✉♣♦C G ❞❛❞♦ ♣♦r

C ={xG:σ(x) =±1},

q✉❡ ❡ss❡♥❝✐❛❧♠❡♥t❡ é ❢♦r♠❛❞♦ ♣❡❧♦s ❡❧❡♠❡♥t♦s ❞♦ ❣r✉♣♦ s♦❜ ♦s q✉❛✐s ❛ ♦r✐❡♥t❛çã♦ ❣❡♥❡r❛❧✐✲ ③❛❞❛ s❡ ❝♦♠♣♦rt❛ ❝♦♠♦ ❛ ♦r✐❡♥t❛çã♦ ❝❧áss✐❝❛✱ s❡♥❞♦ ❛ss✐♠✱ ♦ ❡st✉❞♦ ❞❛ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❡❧❡♠❡♥t♦s s✐♠étr✐❝♦s s♦❜ ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ ❣❡♥❡r❛❧✐③❛❞❛ s❡r✐❛ ✐♠♣♦ss✐❜✐❧✐t❛❞♦ s❡♠ ♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞♦ ❝❛s♦ σ≡ {±1}✱ ❡♥❝♦♥tr❛❞♦ ❡♠ ❬●P✶✹❪✳

P♦r ✜♠✱ ♥♦ ❈❛♣ít✉❧♦ ✹✱ ✐♥✐❝✐❛r❡♠♦s ♦ ❡st✉❞♦ ❞❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❞♦s ❛♥t✐ss✐♠étr✐✲ ❝♦s ♣❛r❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ♦r✐❡♥t❛❞❛ ✉s✉❛❧✱ s❡♥❞♦ q✉❡ ❡st❡ é ♦ ú❧t✐♠♦ ❝❛s♦ ❛ s❡r tr❛t❛❞♦ ♣❛r❛ ♦ ❢❡❝❤❛♠❡♥t♦ ❞♦ q✉❛❞r♦ ❞❡ss❡ t✐♣♦ ❞❡ ✐♥✈♦❧✉çã♦ ✭♥❛ ✈❡r❞❛❞❡✱ ♣❛r❛ ❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛✱ ❥á ❡①✐st❡ ✉♠❛ ❝❛r❛❝t❡r✐③❛çã♦ q✉❡ ♣♦❞❡ s❡r ❡♥❝♦♥tr❛❞❛ ❡♠ ❬❇❏❘✵✼❪✮✳ ❊st✉❞❛r ❡ss❛ ♣❛rt✐❝✉❧❛r ✐♥✈♦❧✉çã♦ é ❥✉st✐✜❝❛❞❛ ❞❡✈✐❞♦ ❛♦ ❣r❛✉ ♠❛✐♦r ❞❡ ❝♦♠♣❧❡①✐❞❛❞❡ ❞❡ss❡ ❝❛s♦✱ ✈✐st♦ q✉❡ ❞❡s❞❡ ❛ ❛❜♦r❞❛❣❡♠ ♥ã♦ ♦r✐❡♥t❛❞❛✱ ❛ ❝❛r❛❝t❡r✐③❛çã♦ ❥á ❛♣r❡s❡♥t❛ ✉♠❛ ❝❧❛ss❡ ♠❛✐♦r ❞❡ ❣r✉♣♦s ❡ ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❛♥❡❧ R✱ ♦ q✉❡ t♦r♥❛ ❛✐♥❞❛ ♠❛✐s ❝♦♠♣❧❡①♦ ♦ ❡st✉❞♦ ❞♦s ❝❛s♦s ♦r✐❡♥t❛❞♦s❀

(16)

❛♥❡❧ R ♣♦ss✉✐ ❡❧❡♠❡♥t♦s ❞❡ t♦rsã♦ 2✱ ♦✉ s❡❥❛✱ ♦ ❝♦♥❥✉♥t♦

R2 :={r ∈R : 2r= 0},

(17)

❈❛♣ít✉❧♦ ✶

Pr❡❧✐♠✐♥❛r❡s

◆❡st❡ ❝❛♣ít✉❧♦✱ ✐r❡♠♦s ❛❜♦r❞❛r ❛❧❣✉♥s ❝♦♥❝❡✐t♦s ❡ r❡s✉❧t❛❞♦s ❢✉♥❞❛♠❡♥t❛✐s ♣❛r❛ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡ss❡ tr❛❜❛❧❤♦✳

✶✳✶ ❆♥é✐s ❞❡ ●r✉♣♦s

❏á ✈✐♠♦s ❝♦♠♦ ♦s ❛♥é✐s ❞❡ ❣r✉♣♦ ❢✉♥❝✐♦♥❛♠ ❝♦♠♦ ♣♦♥t♦ ❞❡ ❡♥❝♦♥tr♦ ❞❛s ❞✐✈❡rs❛s t❡♦r✐❛s ❛❧❣é❜r✐❝❛s✳ ■r❡♠♦s ❛❣♦r❛ ❛♣r❡s❡♥t❛r✱ ❝♦♠ ♠❛✐s ♣r❡❝✐sã♦✱ ❡ss❛ ❡str✉t✉r❛✳

❊♠❜♦r❛ ♥ã♦ s❡❥❛ ✉♠❛ ❡①✐❣ê♥❝✐❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ❛ ❞❡✜♥✐çã♦✱ ❛❞♠✐t✐r❡♠♦s s❡♠♣r❡ q✉❡ ♦ ❛♥❡❧R t♦♠❛❞♦ ♣♦r ❜❛s❡ ❞❡ ✉♠ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG✱ s❡❥❛ ❝♦♠✉t❛t✐✈♦ ❡ ❝♦♠ ✉♥✐❞❛❞❡

1R ∈R✳

❉❡✜♥✐çã♦ ✶✳✶✳ ❙❡❥❛♠G✉♠ ❣r✉♣♦ ❡R ✉♠ ❛♥❡❧ ❝♦♠ ✉♥✐❞❛❞❡✳ ❉❡♥♦t❡ ♣♦rRG ♦ ❝♦♥❥✉♥t♦

❞❡ t♦❞❛s ❛s ❝♦♠❜✐♥❛çõ❡s ❧✐♥❡❛r❡s ❢♦r♠❛✐s ❞❛ ❢♦r♠❛

α=X

g∈G

axx,

♦♥❞❡ ax ∈ R ❡ {ax}x∈G é q✉❛s❡ ♥✉❧❛✱ ♦✉ s❡❥❛✱ ax 6= 0 ❛♣❡♥❛s ♣❛r❛ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ í♥❞✐❝❡s✳ ❖ ❝♦♥❥✉♥t♦ (RG,+,·) ❞♦t❛❞♦ ❞❛s ♦♣❡r❛çõ❡s ❞❡ s♦♠❛ ❡ ♣r♦❞✉t♦ ❞❡✜♥✐❞♦ ❞❛

❢♦r♠❛ ❛ s❡❣✉✐r é ✉♠ ❛♥❡❧✱ ❝❤❛♠❛❞♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ ❞❡G s♦❜r❡ R✿

X

x∈G

axx !

+ X

x∈G

bxx !

=X

x∈G

(ax+bx)x;

X

x∈G

axx !

· X

x∈G

bxx !

= X

x,y∈G

(axby)(xy).

◆♦t❡ q✉❡ RG é ✉♠ ❛♥❡❧ ❝♦♠ ✐❞❡♥t✐❞❛❞❡✱ ♦♥❞❡ 1RG = 1R1G✱ q✉❡✱ ❞❡ ❛❣♦r❛ ❡♠ ❞✐❛♥t❡✱ ❞❡♥♦t❛r❡♠♦s s✐♠♣❧❡s♠❡♥t❡ ♣♦r 1✳

(18)

❉❛❞♦ α = X

x∈G

axx ∈ RG✱ ❝❤❛♠❛♠♦s ❞❡ s✉♣♦rt❡ ❞❡ α✱ supp(α)✱ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦sxG q✉❡ ❛♣❛r❡❝❡♠ ♥❛ ❝♦♠♣♦s✐çã♦ ❞❡α ❞❡ ❢♦r♠❛ ♥ã♦ tr✐✈✐❛❧✱ ♦✉ s❡❥❛✱

supp(α) ={xG:αx 6= 0}.

P♦❞❡♠♦s t❛♠❜é♠ ❞❡✜♥✐r ✉♠ ♣r♦❞✉t♦ ♣♦r ❡s❝❛❧❛r❡s ❞♦ ❛♥❡❧R❞❛ s❡❣✉✐♥t❡ ♠❛♥❡✐r❛

r· X

x∈G

axx !

=X

x∈G

raxx, ∀r∈R,

❡ ❢❛❝✐❧♠❡♥t❡ ✈❡r✐✜❝❛♠♦s q✉❡RG é ✉♠ R✲♠ó❞✉❧♦✳ ❆❞❡♠❛✐s✱ s❡R é ❝♦♠✉t❛t✐✈♦✱ ❡♥tã♦ RG

é ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ R✳

❉✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦✱ ❡ ♣❡❧♦ ❢❛t♦ ❞❡ R s❡r ❝♦♠✉t❛t✐✈♦✱ ♥ã♦ é ❞✐❢í❝✐❧ ✈❡r✐✜❝❛r

q✉❡RG s❡rá ❝♦♠✉t❛t✐✈♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ G ❢♦r ❛❜❡❧✐❛♥♦✳

❊①❡♠♣❧♦ ✶✳✷✳ ❙❡❥❛♠ G = C∞ ≃ {. . . , x−2, x−1, x0, x1, x2, . . .} ❡ R = R✱ ♦ ❝♦r♣♦ ❞♦s ♥ú♠❡r♦s r❡❛✐s✳ ❚❡♠♦s q✉❡ RG=RC é ✐s♦♠♦r❢♦ ❛♦ ❛♥❡❧ ❞♦s ♣♦❧✐♥ô♠✐♦s ❞❡ ▲❛✉r❡♥t✳

❯t✐❧✐③❛♥❞♦ ♦ ♠♦♥♦♠♦r✜s♠♦ ❞❡ ✐♥❝❧✉sã♦ i : R RG ❞❡✜♥✐❞♦ ♣♦r i(r) 7→ r1G✱

t❡♠♦s ♥❛t✉r❛❧♠❡♥t❡ q✉❡ RG ❝♦♥té♠ ✉♠ s✉❜❛♥❡❧ ✐s♦♠♦r❢♦ ❛ R✱ ♦ q✉❛❧ ❢r❡q✉❡♥t❡♠❡♥t❡

tr❛t❛r❡♠♦s ❝♦♠♦ ♦ ♣ró♣r✐♦ R❀ ❛♥❛❧♦❣❛♠❡♥t❡✱ ❝♦♠♦ R ♣♦ss✉✐ ✉♥✐❞❛❞❡✱ ❞❛❞♦ ❛ ✐♥❝❧✉sã♦ x 7→ 1Rx, ∀x ∈ G✱ t❡♠♦s q✉❡ ♦ ❣r✉♣♦ ❞❛s ✉♥✐❞❛❞❡s ❞♦ ❛♥❡❧ RG✱ U(RG)✱ ♣♦ss✉✐ ✉♠ s✉❜❣r✉♣♦ ✐s♦♠♦r❢♦ ❛♦ ❣r✉♣♦ G✱ q✉❡ t❛♠❜é♠ s❡rá ✐❞❡♥t✐✜❝❛❞♦ ♣♦rG✳

✶✳✷ ■♥✈♦❧✉çõ❡s

■r❡♠♦s ❛♣r❡s❡♥t❛r ♥❡st❛ s❡çã♦ ❛ ♣r✐♥❝✐♣❛❧ ❢❡rr❛♠❡♥t❛ ✉t✐❧✐③❛❞❛ ♥❡ss❡ tr❛❜❛❧❤♦ ♣❛r❛ ♦ ❡st✉❞♦ ❞♦s ❛♥é✐s ❞❡ ❣r✉♣♦s✳

✶✳✷✳✶ ■♥✈♦❧✉çõ❡s ❡♠ ●r✉♣♦s

❉❡✜♥✐çã♦ ✶✳✸✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❛♣❧✐❝❛çã♦∗:GG✱ ❡♠ ✉♠ ❣r✉♣♦ G✱ é ✉♠❛ ✐♥✈♦❧✉çã♦

❞❡ ❣r✉♣♦s s❡ ✈❡r✐✜❝❛ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿ ✭✐✮ (xy)∗ =yx

✭✐✐✮ (x∗)=x

♣❛r❛ t♦❞♦x, y G✳ ◆❡ss❡ ❝❛s♦ ❞✐③❡♠♦s q✉❡ G é ✉♠ ❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦

❊①❡♠♣❧♦ ✶✳✹✳ ❆ ❛♣❧✐❝❛çã♦x7→x−1 ❞❡✜♥❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ q✉❛❧q✉❡r ❣r✉♣♦

G✱ ❝❤❛♠❛❞❛

(19)

❊①❡♠♣❧♦ ✶✳✺✳ ❆ tr❛♥s♣♦s✐çã♦ ❞❡ ♠❛tr✐③❡s ❞❡✜♥❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ❞❡ ❣r✉♣♦s ♥♦ ❣r✉♣♦ ❣❡r❛❧ ❧✐♥❡❛r GL(n, K)✱ s❡♥❞♦ K ✉♠ ❝♦r♣♦✳

❊①❡♠♣❧♦ ✶✳✻✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ∗ ❡ H ≤G✳ ❙❡ h∗ H, hH✱ ❡♥tã♦

∗|H é ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ H✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ ❛ tr❛♥s♣♦s✐çã♦ é ✉♠❛ ✐♥✈♦❧✉çã♦ ♥♦ ❣r✉♣♦

H={M ∈GL(n, K) :det(M) =±1}✳

❊①❡♠♣❧♦ ✶✳✼✳ ❙❡❥❛ Dn=hx, y :xn=y2 = 1, yx =x−1yi ♦ ❣r✉♣♦ ❞✐❡❞r❛❧ ❞❡ 2n ❡❧❡♠❡♥✲ t♦s✳ ❆ ❛♣❧✐❝❛çã♦∗:Dn →Dn❞❡✜♥✐❞❛ ♣❡❧❛ ❡①t❡♥sã♦ ❛♦ ♣r♦❞✉t♦ ❛ ♣❛rt✐r ❞❡x∗ =x−1, y∗ =

xyx✱ é ✉♠❛ ✐♥✈♦❧✉çã♦ ♥❡ss❡ ❣r✉♣♦✳

❉✉r❛♥t❡ t♦❞♦ ♦ tr❛❜❛❧❤♦✱ ❞❡♥♦t❛r❡♠♦s ❛s ❝♦♥❥✉❣❛çõ❡s ❡ ❝♦♠✉t❛❞♦r❡s ❡♠ G ♣♦r xy =y−1

xy ❡(x, y) = x−1

y−1

xy, ∀x, y ∈G✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

❊①❡♠♣❧♦ ✶✳✽✳ ❙❡❥❛ ♦ Q8 ❣r✉♣♦ ❞♦s q✉❛tér♥✐♦s ❞❡ ♦r❞❡♠ 8✱ ❝✉❥❛ ♣r❡s❡♥t❛çã♦ é

Q8 =hx, y;x 4

= 1, x2

=y2

, xy =x−1

i.

❆ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛ ❡♠ Q8 é ❞❛❞❛ ❡①♣❧✐❝✐t❛♠❡♥t❡ ♣♦r

g∗ =

(

g , s❡ g ∈ {1, x2

}

x2

g , ❝❛s♦ ❝♦♥trár✐♦✳

◆♦t❡ q✉❡ {1, x2

}=Z(Q8) = Q′8 =hx 2

i✳

❖❜s❡r✈❡ ♥♦ ❡①❡♠♣❧♦ ❛❝✐♠❛ q✉❡ ❛ ✐♥✈♦❧✉çã♦ é ❞❛❞❛ ♣r❡❝✐s❛♠❡♥t❡ ♣❡❧❛ ♠✉❧t✐♣❧✐❝❛çã♦ ♣♦r ❡❧❡♠❡♥t♦s ❞♦ s✉❜❣r✉♣♦ ❞❡r✐✈❛❞♦ ❞❡ Q8✳ ❱❡r❡♠♦s ♠❛✐s ❛❞✐❛♥t❡ q✉❡ ✐♥✈♦❧✉çõ❡s ❞❡ss❡

t✐♣♦ sã♦ r❡❝♦rr❡♥t❡s ♥❛ ❧✐♥❤❛ ❞❡ ♣❡sq✉✐s❛ ❛❜♦r❞❛❞❛ ♥❡ss❡ t❡①t♦✳

❉❡✜♥✐çã♦ ✶✳✾✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❣r✉♣♦ ♥ã♦ ❛❜❡❧✐❛♥♦ G é ✉♠ ▲❈✲❣r✉♣♦✶✱ s❡ ♣❛r❛ t♦❞♦ x, y G t❛✐s q✉❡ xy=yx✱ ❡♥tã♦ x, y ♦✉ xy∈ Z(G)✳

❯♠❛ ❝❧❛ss❡ ❜❛st❛♥t❡ ✐♥t❡r❡ss❛♥t❡ ❞❡ ❣r✉♣♦s ❝♦♠ ✐♥✈♦❧✉çã♦ s✉r❣❡ q✉❛♥❞♦ ❛❞♠✐t✐✲ ♠♦s q✉❡ ✉♠ ▲❈✲❣r✉♣♦G♣♦ss✉✐ ✉♠ s✉❜❣r✉♣♦ ❞❡r✐✈❛❞♦ ❞❡ ♦r❞❡♠2✱ ♦✉ s❡❥❛G′ =hsi ≃C

2✳

◆❡st❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡G ♣♦ss✉✐ ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ s✳ ❖ s❡❣✉✐♥t❡ t❡♦r❡♠❛

❝❧❛ss✐✜❝❛ ❡ss❛ ❝❧❛ss❡ ❞❡ ❣r✉♣♦s✱ ❝❛r❛❝t❡r✐③❛♥❞♦✲♦s ♣♦r s❡r❡♠ ♠✉♥✐❞♦s ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ♠✉✐t♦ ♣❛rt✐❝✉❧❛r ❞♦ t✐♣♦ ❝✐t❛❞♦ ♥♦ ❊①❡♠♣❧♦ ✶✳✽✳

❚❡♦r❡♠❛ ✶✳✶✵ ✭❚❡♦r❡♠❛s ■■■✳✸✳✸ ❡ ■■■✳✸✳✻✱ ❬●❏P✾✻❪✮✳ ❙❡❥❛G ✉♠ ❣r✉♣♦ ♥ã♦ ❛❜❡❧✐❛♥♦✳ ❆s

s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s sã♦ ❡q✉✐✈❛❧❡♥t❡s✿

✭✐✮ G é ✉♠ ▲❈✲❣r✉♣♦ ❝♦♠ ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ s✳

(20)

✶✵

✭✐✐✮ G ♣♦ss✉✐ ✉♠❛ ✐♥✈♦❧✉çã♦ ❝♦♠ ❛ ♣r♦♣r✐❡❞❛❞❡x−1

yx∈ {y, y∗}, x, y G✱ q✉❡ ♥❡ss❡ ❝❛s♦ é ❞❛❞❛ ♣r❡❝✐s❛♠❡♥t❡ ♣♦r

x∗ =

(

x , s❡ x∈ Z(G)

sx , s❡ x /∈ Z(G).

✭✐✐✐✮ G/Z(G)C2 ×C2✳

❉❡✜♥✐çã♦ ✶✳✶✶✳ ❖s ▲❈✲❣r✉♣♦s ❝♦♠ ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ s ♠✉♥✐❞♦ ❞❛ ✐♥✈♦✲

❧✉çã♦ ∗ ❛♣r❡s❡♥t❛❞❛ ♥♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r sã♦ ❝❤❛♠❛❞♦s ❞❡ ❙▲❈✲❣r✉♣♦s✷✳ ❆ ✐♥✈♦❧✉çã♦ ∗ é

❝❤❛♠❛❞❛ ❞❡ ✐♥✈♦❧✉çã♦ ❝❛♥ô♥✐❝❛ ❞♦s ❙▲❈✲❣r✉♣♦s✳

❊①❡♠♣❧♦ ✶✳✶✷✳ Q8 ♠✉♥✐❞♦ ❝♦♠ ❛ ✐♥✈♦❧✉çã♦ ∗ ❞♦ ❊①❡♠♣❧♦ ✶✳✽ é ✉♠ ❙▲❈✲❣r✉♣♦✳ ◆♦t❡

q✉❡ ♥❡ss❡ ❝❛s♦ ❛ ✐♥✈♦❧✉çã♦ ❞♦s ❙▲❈✲❣r✉♣♦s ❝♦✐♥❝✐❞❡ ❝♦♠ ❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛ ❡♠ Q8✳

❊①❡♠♣❧♦ ✶✳✶✸✳ ❉❛❞♦ ❛ ♣r❡s❡♥t❛çã♦ ❞❡ D4 ♥♦ ❊①❡♠♣❧♦ ✶✳✼✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ ❡st❡

❣r✉♣♦ é ✉♠ ▲❈✲❣r✉♣♦ ❝♦♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ x2✱ ♣♦ré♠ ♥ã♦ ♣♦❞❡♠♦s ❞✐③❡r q✉❡

D4 ♠✉♥✐❞♦ ❞❛ ✐♥✈♦❧✉çã♦ ∗ ❞❡ss❡ ♠❡s♠♦ ❡①❡♠♣❧♦ s❡❥❛ ✉♠ ❙▲❈✲❣r✉♣♦✱ ❥á q✉❡ ❡ss❛ ♥ã♦ é

❞♦ t✐♣♦ ❞♦ t❡♦r❡♠❛ ❛♥t❡r✐♦r✳ P♦r ♦✉tr♦ ❧❛❞♦✱ ❡st❡ ♠❡s♠♦ t❡♦r❡♠❛ ❛✜r♠❛ q✉❡ ❛ ❛♣❧✐❝❛çã♦

w∗ =

(

w , s❡ w∈ {1, x2

}

x2

w , s❡ w /∈ {1, x2

},

é ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ D4❀ ❛♦ ♠✉♥✐r D4 ❝♦♠ ❡ss❛ ✐♥✈♦❧✉çã♦✱ ❞✐③❡♠♦s q✉❡ ❡st❡ é ✉♠ ❙▲❈✲

❣r✉♣♦✳

❊①❡♠♣❧♦ ✶✳✶✹✳ ❆ ♣r♦♣r✐❡❞❛❞❡ ❞❡ s❡r ✉♠ ▲❈✲❣r✉♣♦ é ✐♥❞❡♣❡♥❞❡♥t❡ ❞❡ ♣♦ss✉✐r ✉♠ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧✳ ❉❡ ❢❛t♦✱ ♥♦t❡ q✉❡ ♦ ❣r✉♣♦

G=x1, x2, x3 :x 4

i = (x

2

i, xj) = ((xi, xj), xk) = 1 ♣❛r❛ i, j, k ❞✐st✐♥t♦s

♣♦ss✉✐ três ❝♦♠✉t❛❞♦r❡s ♥ã♦ tr✐✈✐❛✐s(x1, x2),(x2, x3)❡(x1, x3)✱ ❡✱ ♣♦r ♦✉tr♦ ❧❛❞♦✱G/Z(G)

♣♦ss✉✐ ❡①♣♦❡♥t❡2✱ ♦ q✉❡ ❣❛r❛♥t❡ q✉❡ ♦ ♣r♦❞✉t♦ ❞❡ ❞♦✐s ❡❧❡♠❡♥t♦s ♥ã♦ ❝❡♥tr❛✐s é ❝❡♥tr❛❧✱ ❡

❝♦♥s❡q✉❡♥t❡♠❡♥t❡ ❛ ❝♦♥❞✐çã♦ ❞❡ ▲❈✲❣r✉♣♦ é ✈❡r✜❝❛❞❛✳ ◆♦t❡ q✉❡ ❞❡✈✐❞♦ à q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♠✉t❛❞♦r❡s✱ ❡st❡ ♥ã♦ s❡ ❡♥❝❛✐①❛ ♥❛ ❝❧❛ss❡ ❞♦s ❙▲❈✲❣r✉♣♦s✳

❖❜s❡r✈❡ q✉❡ ♥❡st❡ ❝❛s♦✱ H =hx1, x2i ❡ I =hx1, x2, x23isã♦ ❙▲❈✲❣r✉♣♦s ❞❡ ♦r❞❡♠

32❡ 64✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ s❡♥❞♦ s = (x1, x2) ♦ ú♥✐❝♦ ❝♦♠✉t❛❞♦r ♥ã♦ tr✐✈✐❛❧ ❞❡ H ❡ I✳

✶✳✷✳✷ ■♥✈♦❧✉çõ❡s ❡♠ ❆♥é✐s ❞❡ ●r✉♣♦

❉❡✜♥✐çã♦ ✶✳✶✺✳ ❉✐③❡♠♦s q✉❡ ✉♠❛ ❛♣❧✐❝❛çã♦ ∗ : R R✱ ❡♠ R é ✉♠ ❛♥❡❧✱ é ✉♠❛

✐♥✈♦❧✉çã♦ ❞❡ ❛♥é✐s s❡ ✈❡r✐✜❝❛ ❛s s❡❣✉✐♥t❡s ❝♦♥❞✐çõ❡s✿

(21)

✶✶

✭✐✮ (rs)∗ =sr ✭✐✐✮ (r+s)∗ =r+s ✭✐✐✐✮ (r∗)=r

♣❛r❛ t♦❞♦r, s∈R✳ ◆❡ss❡ ❝❛s♦ ❞✐③❡♠♦s q✉❡ R é ✉♠ ❛♥❡❧ ❝♦♠ ✐♥✈♦❧✉çã♦ ∗✳

❊①❡♠♣❧♦ ✶✳✶✻✳ ❆ tr❛♥s♣♦s✐çã♦ ❞❡ ♠❛tr✐③❡s ❞❡✜♥❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ♥♦ ❛♥❡❧ Mn(R) ❞❛s ♠❛tr✐③❡s n×n ❝♦♠ ❡♥tr❛❞❛s ❡♠ ✉♠ ❛♥❡❧ R✳

❊①❡♠♣❧♦ ✶✳✶✼✳ ❆ ❝♦♥❥✉❣❛çã♦ ❝♦♠♣❧❡①❛ é ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ C✳

❉❛s ♣♦ssí✈❡✐s ✐♥✈♦❧✉çõ❡s ❛ s❡r❡♠ ❞❡✜♥✐❞❛s ❡♠ ✉♠ ❛♥❡❧ ❞❡ ❣r✉♣♦RG✱ ❞❡st❛❝❛♠✲s❡

❛s ✐♥✈♦❧✉çõ❡s ♣r♦✈❡♥✐❡♥t❡s ❞❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗❞♦ ❣r✉♣♦ G✳

❊①❡♠♣❧♦ ✶✳✶✽ ✭■♥✈♦❧✉çã♦ ■♥❞✉③✐❞❛ ❡♠ ❆♥é✐s ❞❡ ●r✉♣♦s✮✳ ❙❡❥❛ RG ♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ ❞❡ Gs♦❜r❡R ❡ ∗✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ G✳ ❆ ❛♣❧✐❝❛çã♦✱ t❛♠❜é♠ ❞❡♥♦t❛❞❛ ♣♦r ∗✱ ∗:RG→RG✱

❞❡✜♥✐❞❛ ❛ ♣❛rt✐r ❞❛ ❡①t❡♥sã♦ ❧✐♥❡❛r

X

x∈G

αxx !∗

=X

x∈G

αxx∗,

é ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ RG✱ ❝❤❛♠❛❞❛ ❞❡ ■♥✈♦❧✉çã♦ ■♥❞✉③✐❞❛ ❞❡ G ❡♠ RG✳

❈♦♠♦ t♦❞♦ ❣r✉♣♦ é ♠✉♥✐❞♦ ❞❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♣♦❞❡♠♦s ❢❛③❡r ❛ ❡①t❡♥sã♦ ❧✐♥❡❛r ❞❡✜♥✐❞❛ ❛❝✐♠❛✱ ❢❛③❡♥❞♦ ❝♦♠ q✉❡ t♦❞♦ ❛♥❡❧ ❞❡ ❣r✉♣♦RGs❡❥❛ ♥❛t✉r❛❧♠❡♥t❡

✉♠ ♣♦rt❛❞♦r ❞❡ss❛ ✐♥✈♦❧✉çã♦✱ ❝❤❛♠❛❞❛ ❞❡ ■♥✈♦❧✉çã♦ ❈❛♥ô♥✐❝❛ ✭♦✉ ❈❧áss✐❝❛✮ ❡♠ ❆♥é✐s ❞❡ ●r✉♣♦s✳ ❊ss❛ ♣❛rt✐❝✉❧❛r ✐♥✈♦❧✉çã♦ é ♦❜❥❡t♦ ❞❡ ❣r❛♥❞❡ ✐♥t❡r❡ss❡ ❡♠ q✉❛❧q✉❡r ❡st✉❞♦ q✉❡ r❡❧❛❝✐♦♥❡ ❛♥é✐s ❞❡ ❣r✉♣♦s ❡ ✐♥✈♦❧✉çõ❡s✱ s❡♥❞♦ q✉❡ ❛ ♣❛rt✐r ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♥❡ss❡s ❝❛s♦s✱ ❜✉s❝❛♠✲s❡ ❡st❡♥❞ê✲❧♦s ♣❛r❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ✐♥❞✉③✐❞❛ q✉❛❧q✉❡r ♦✉ ❛té ♣❛r❛ ✉♠❛ ✐♥✈♦❧✉çã♦ ❣❡r❛❧ ❞❡ ❛♥é✐s✳ ❊①❡♠♣❧♦s ❞♦ ❡st✉❞♦ ❞❡ss❛ ✐♥✈♦❧✉çã♦ ❡ ❞❛ s✉❛ ❡①t❡♥sã♦✱ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬▲✾✾✱ ▲✵✵✱ ▲✶✵✱ ●P❙✵✾✱ ▲❙❙✵✾❪✳

❉❡✜♥✐çã♦ ✶✳✶✾✳ ❙❡❥❛♠ R ✉♠ ❛♥❡❧ ❡ ✉♠❛ ✐♥✈♦❧✉çã♦ ❡♠ R✳ ❯♠ ❡❧❡♠❡♥t♦ rR é ❝❤❛✲

♠❛❞♦ ❞❡ s✐♠étr✐❝♦ ♦✉ ❛♥t✐ss✐♠étr✐❝♦✱ s❡r∗ =r ♦✉ r=r✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❉❡♥♦t❛♠♦s ♣♦r R∗ ❡ R−∗ ♦ ❝♦♥❥✉♥t♦ ❞♦s ❡❧❡♠❡♥t♦s s✐♠étr✐❝♦s ❡ ❛♥t✐ss✐♠étr✐❝♦s ❞❡ R✱ ♦✉ s❡❥❛✱

R∗ ={r∈R;r∗ =r} ❡ R−∗ ={r ∈R;r∗ =−r}.

◆❛ ❧✐t❡r❛t✉r❛ é ❝♦♠✉♠ ❡♥❝♦♥tr❛r ❛ ♥♦t❛çã♦ R+✱ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ ❞♦s s✐♠étr✐✲

(22)

✶✷

♣r✐♠❡✐r❛ s❡ t♦r♥❛ ♠❛✐s ❛❞❡q✉❛❞❛✱ ❡♠ ❝♦♥tr❛♣❛rt✐❞❛✱ ❛ s❡❣✉♥❞❛ é ♠❛✐s ❡❝♦♥ô♠✐❝❛✳ ❯t✐❧✐✲ ③❛r❡♠♦s t❛♠❜é♠ ❛s ♥♦t❛çõ❡s (x−1

)∗ = x−∗, x G G

∗ ❛♦ ❝♦♥❥✉♥t♦ ❞♦s s✐♠étr✐❝♦s ❞♦ ❣r✉♣♦ G✱ ♦✉ s❡❥❛✱

G∗ ={x∈G:x=x∗}

❖❜s❡r✈❛çã♦ ✶✳✷✵✳ ◆♦t❡ q✉❡ 1 ∈ G∗ ❡ ♣♦rt❛♥t♦✱ s❡ x ∈ G∗✱ ❡♥tã♦ 1 = 1∗ = (xx−1)∗ =

x−∗x= x−∗x✱ ❛ss✐♠✱ x−∗ = x−1✱ ♦✉ s❡❥❛

x−1

∈ G∗✳ ❆♥❛❧♦❣❛♠❡♥t❡ ♠♦str❛♠♦s q✉❡

s❡ x / G∗✱ ❡♥tã♦ x−1

/

∈ G∗✱ ♦✉ s❡❥❛✱ x G∗ s❡✱ ❡ s♦♠❡♥t❡ s❡ x−1 ∈ G∗✳ ❆❞❡♠❛✐s

x−∗= (x)−1

❯t✐❧✐③❛r❡♠♦s ♦s ❢❛t♦ ❛❝✐♠❛ ❛♦ ❧♦♥❣♦ ❞♦ t❡①t♦ s❡♠ ❢❛③❡r r❡❢❡rê♥❝✐❛ ❛ ❡ss❛ ♦❜s❡r✈❛✲ çã♦✳

❙❡❥❛ RG ✉♠ ❛♥❡❧ ❞❡ ❣r✉♣♦ ❝♦♠ ✉♠❛ ✐♥✈♦❧✉çã♦ ∗ ✐♥❞✉③✐❞❛ ♣♦r G✱ ❛ss✐♠✱ ❞❛❞♦

α=X

x∈G

αxx∈RG✱ t❡♠♦s

α∗ =X

x∈G

αxx∗,

♣♦rt❛♥t♦✱ s❡αRG+✱ ❡♥tã♦

X

x∈G

αxx= X

x∈G

αxx∗.

❉❡✈✐❞♦ ❛♦ ❢❛t♦ ❞❡ RG s❡r ❧✐✈r❡♠❡♥t❡ ❣❡r❛❞♦ ❝♦♠♦ R✲♠ó❞✉❧♦ ♣♦r G✱ ♣❛rt✐❝✐♦♥❛♥❞♦ G=

G∗ ∪(G\G∗)✱ ♣♦❞❡♠♦s ✈❡r✐✜❝❛r q✉❡ s❡ α ∈ RG+✱ ❡♥tã♦ αx = αx∗ s❡ x /∈ G∗❀ ❛❞❡♠❛✐s✱

♥❡♥❤✉♠❛ ❝♦♥❞✐çã♦ é ❡①✐❣✐❞❛ ♣❛r❛ ♦s ❝♦❡✜❝✐❡♥t❡s ❛ss♦❝✐❛❞♦s ❛ ❡❧❡♠❡♥t♦s ❞❡ G∗✱ ❧♦❣♦✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ RG+ é ❣❡r❛❞♦ ❝♦♠♦ ✉♠

R✲♠ó❞✉❧♦ ♣♦r

G∗∪ {x+x∗ :x∈G\G∗}. ❉❡ ✉♠❛ ❢♦r♠❛ ❛♥á❧♦❣❛✱ é ❢á❝✐❧ ✈❡r✐✜❝❛r q✉❡RG− é ❣❡r❛❞♦ ♣♦r

{rx:xG∗, r∈R2} ∪ {x−x∗ :x∈G\G∗}, ♦♥❞❡R2 ={r∈R: 2r= 0}✳

❖ ✐♥t❡r❡ss❡ ❡♠ ❡st✉❞❛r ❡ss❡s ❝♦♥❥✉♥t♦s ❢♦✐ ✐♠♣✉❧s✐♦♥❛❞♦ ❛♣ós ❆♠✐ts✉r ♣r♦✈❛r✱ ❡♠ ❬❆✻✾❪✱ q✉❡ s❡R+ ♦✉

R− s❛t✐s❢❛③ ❛❧❣✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧✱ ❡♥tã♦ R t❛♠❜é♠ s❛t✐s❢❛③ ❛❧❣✉♠❛ ✐❞❡♥t✐❞❛❞❡✳ ◆❛t✉r❛❧♠❡♥t❡✱ ❡♠ ❣❡r❛❧✱ ❛s ✐❞❡♥t✐❞❛❞❡s ♣♦❞❡♠ ♥ã♦ s❡r ♣r❡s❡r✈❛❞❛s ❞♦s s✐♠étr✐❝♦s ♣❛r❛ t♦❞♦ ♦ ❛♥❡❧ R✱ s❡♥❞♦ ❛ss✐♠✱ ✉♠ ♣r♦❜❧❡♠❛ q✉❡ s✉r❣❡ ✐♠❡❞✐❛t❛♠❡♥t❡

❞❡ss❡ r❡s✉❧t❛❞♦ é ❝♦♥❤❡❝❡r q✉❛✐s ✐❞❡♥t✐❞❛❞❡s ✈❡r✐✜❝❛❞❛s ❡♠ R+ ♦✉

R− t❛♠❜é♠ ♦ s❡rã♦ ❡♠ R✱ ♥❡ss❡ ❝❛s♦✱ ❞✐③❡♠♦s q✉❡ ❛ ✐❞❡♥t✐❞❛❞❡ ❢♦✐ ❧❡✈❛♥t❛❞❛ ❞❡ R+ ♦✉

R− ♣❛r❛ R

❉❡♥♦t❛♥❞♦ ♦ ❝♦❧❝❤❡t❡ ❞❡ ▲✐❡ ❡♠ ✉♠ ❛♥❡❧R♣♦r[r, s] =rssr, r, sR✱ ❛ ♣❛rt✐r

❞❛ ✐t❡r❛❞❛ ❞❡ss❡s ❝♦❧❝❤❡t❡s✱

(23)

✶✸

∀ri ∈R✱ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛s s❡❣✉✐♥t❡s ✐❞❡♥t✐❞❛❞❡s

[r1, . . . , rn] = 0

[r, s, s, . . . , s | {z }

n−✈❡③❡s

] = 0,

❝❤❛♠❛❞❛s ❞❡ ▲✐❡ ♥✐❧♣♦tê♥❝✐❛ ❞❡ í♥❞✐❝❡n ❡ ▲✐❡ n✲❊♥❣❡❧✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳

◆♦ ❝❛s♦ ❞❡ K s❡r ✉♠ ❝♦r♣♦ ❞❡ ❝❛r❛❝t❡ríst✐❝❛ ❞✐❢❡r❡♥t❡ ❞❡ 2✱ ●✐❛♠❜r✉♥♦✱ ❙❡❤❣❛❧

❡ ▲❡❡✱ ♣r♦✈❛r❛♠✱ ❡♠ ❬●❙✾✸✱ ▲✵✵❪✱ q✉❡ s❡ Gé ✉♠ ❣r✉♣♦ q✉❡ ♥ã♦ ♣♦ss✉✐ 2✲❡❧❡♠❡♥t♦s ❡ é

❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛ ❡♠ G✱ ❡♥tã♦ ❛s ✐❞❡♥t✐❞❛❞❡s ❞❡ ▲✐❡ ♥✐❧♣♦tê♥❝✐❛ ❡ ▲✐❡ n✲❊♥❣❡❧ ♣♦❞❡♠

s❡r ❧❡✈❛♥t❛❞❛s ❞❡ KG+

KG− ♣❛r❛ t♦❞♦ ♦ KG✳ P♦st❡r✐♦r♠❡♥t❡ ●✐❛♠❜r✉♥♦✱ P♦❧❝✐♥♦ ❡ ❙❡❤❣❛❧✱ ♠♦str❛r❛♠✱ ❡♠ ❬●P❙✵✾❪✱ q✉❡ ♣❛r❛ ♦ ❝♦♥❥✉♥t♦ ❞♦s s✐♠étr✐❝♦s ❛ ❝♦♥❞✐çã♦ ❞❡

G ♥ã♦ ♣♦ss✉✐r 2✲❡❧❡♠❡♥t♦s ❝♦♥t✐♥✉❛ s❡♥❞♦ s✉✜❝✐❡♥t❡ ♣❛r❛ ❣❛r❛♥t✐r q✉❡ ❡ss❛s ✐❞❡♥t✐❞❛❞❡s

s❡❥❛♠ ❧❡✈❛♥t❛❞❛s ♥♦ ❝❛s♦ ❞❛ ✐♥✈♦❧✉çã♦ ∗ s❡r ✉♠❛ ✐♥✈♦❧✉çã♦ ✐♥❞✉③✐❞❛ q✉❛❧q✉❡r ❡♠ KG❀

❛❧é♠ ❞✐ss♦✱ ▲❡❡✱ ❡♠ ❬▲✾✾❪✱ ❡st❡♥❞❡✉ ✉♠ ❞♦s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ❡♠ ❬●❙✾✸❪✱ ❣❛r❛♥t✐♥❞♦ q✉❡ ♥♦ ❝❛s♦ ❞❛ ✐♥✈♦❧✉çã♦ ❝❧áss✐❝❛✱ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ❛✐♥❞❛ é ✈❡r✐✜❝❛❞♦ ❝❛s♦ ♦ ❣r✉♣♦ ❞♦s q✉❛tér♥✐♦sQ8 ♥ã♦ ❡st❡❥❛ ❝♦♥t✐❞♦ ❡♠ G✳

P♦r ♦✉tr♦ ❧❛❞♦✱ q✉❛♥❞♦ ♦s ❝♦♥❥✉♥t♦s RG+

RG− ✈❡r✐✜❝❛♠ ❛❧❣✉♠❛ ✐❞❡♥t✐❞❛❞❡ ♣♦❧✐♥♦♠✐❛❧ ❡ ♦ ❧❡✈❛♥t❛♠❡♥t♦ ♥ã♦ ♦❝♦rr❡✱ é ♣♦ssí✈❡❧ ❛✐♥❞❛ ❡①tr❛✐r ✐♥❢♦r♠❛çõ❡s ❛❝❡r❝❛ ❞❛ ❡str✉t✉r❛ ❞♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG✱ ❜✉s❝❛♥❞♦ ❝❛r❛❝t❡r✐③❛r ♦ ❣r✉♣♦ G q✉❡ ♦ ❝♦♠♣õ❡✱ ❛ss✐♠

❝♦♠♦ ❛❧❣✉♠❛s ♣r♦♣r✐❡❞❛❞❡s ❞♦ ❛♥❡❧R✳ ◆❡ss❡ s❡♥t✐❞♦ ♦s r❡s✉❧t❛❞♦s ❡♥❝♦♥tr❛❞♦s ❡♠ ❬▲✾✾✱

▲✵✵✱ ▲❙❙✵✾❪ ❝♦♠♣❧❡♠❡♥t❛♠ ♦ ❡st✉❞♦ ❝♦♠❡♥t❛❞♦ ♥♦ ♣❛rá❣r❛❢♦ ❛♥t❡r✐♦r✳ ❉❡ ✉♠❛ ❢♦r♠❛ r❡s✉♠✐❞❛✱ ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛ ❧♦❝❛❧✐③❛ ♦s r❡s✉❧t❛❞♦s q✉❡ ❛❝❛❜❛♠♦s ❞❡ ❛❜♦r❞❛r✿

▲✐❡ ♥✐❧♣♦tê♥❝✐❛ ▲✐❡ n✲❊♥❣❡❧ RG∗ ❬●❙✾✸✱ ▲✾✾✱ ●P❙✵✾✱ ▲❙❙✵✾❪ ❬▲✵✵✱ ●P❙✵✾❪

RG−

∗ ❬●❙✾✸❪ ❬▲✵✵❪

❚❛❜❡❧❛ ✶✳✶✿ Pr♦♣r✐❡❞❛❞❡s ❞❡ ▲✐❡ ♣❛r❛ ✐♥✈♦❧✉çõ❡s ✐♥❞✉③✐❞❛s

❊①✐st❡♠ ♦✉tr❛s ✐❞❡♥t✐❞❛❞❡s ❞❡ ✐♥t❡r❡ss❡ ♥❛ t❡♦r✐❛✱ ❞❡♥tr❡ ❡❧❛s✱ ❞❡st❛❝❛r❡♠♦s ❛q✉❡✲ ❧❛s q✉❡ ❡stã♦ ❞✐r❡t❛♠❡♥t❡ ❧✐❣❛❞❛s à ♣♦ss✐❜✐❧✐❞❛❞❡ ❞❡ ♠✉♥✐r ♦s ❝♦♥❥✉♥t♦sR+

R− ❞❡ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❛♥❡❧✳

❉❡✜♥✐çã♦ ✶✳✷✶✳ ❉✐③❡♠♦s q✉❡ ✉♠ s✉❜❝♦♥❥✉♥t♦ A R é ❛♥t✐❝♦♠✉t❛t✐✈♦ s❡ ab = ba✱

∀a, bA✱ ♦✉ s❡❥❛✱ ♦ ♦♣❡r❛❞♦r ❞❡ ❏♦r❞❛♥ ab =ab+ba é ✐❞❡♥t✐❝❛♠❡♥t❡ ♥✉❧♦ ❡♠ A✳

❖ ✐♠♣❡❞✐♠❡♥t♦ ♣❛r❛ q✉❡ ♦s ❝♦♥❥✉♥t♦sR+❡

(24)

✶✹

Pr♦♣♦s✐çã♦ ✶✳✷✷✳ ❙❡ R ✉♠ ❛♥❡❧ ❝♦♠ ✉♠❛ ✐♥✈♦❧✉çã♦✱ ❡♥tã♦ sã♦ ✈❡r❞❛❞❡✐r❛s✿

✭❛✮ R+ é ✉♠ s✉❜❛♥❡❧ ❞❡

R s❡✱ ❡ s♦♠❡♥t❡ s❡✱ R+ é ❝♦♠✉t❛t✐✈♦✳

✭❜✮ R− é ✉♠ s✉❜❛♥❡❧ ❞❡ R s❡✱ ❡ s♦♠❡♥t❡ s❡✱ Ré ❛♥t✐❝♦♠✉t❛t✐✈♦✳ ❉❡♠♦♥str❛çã♦✳ ✭❛✮ ❙❡❥❛♠a, bR+✳ ❊♥tã♦

(ab)∗ =ab=ab✳ ▲♦❣♦ ab R+✳

❆❣♦r❛✱ (ab)∗ =ba=ba✳ ❆ss✐♠✱ abR+ s❡✱ ❡ s♦♠❡♥t❡ s❡✱

ab=ba❀ ✐st♦ é✱ s❡✱ ❡ s♦♠❡♥t❡

s❡✱R+ é ❝♦♠✉t❛t✐✈♦✳ ❖ ✐t❡♠ ✭❜✮ s❡❣✉❡ ❞❡ ❢♦r♠❛ ❛♥á❧♦❣❛✳

❊ss❛ ♣r♦♣♦s✐çã♦ ♠♦t✐✈❛ ❛ ❜✉s❝❛ ♣♦r r❡s✉❧t❛❞♦s q✉❡ r❡❧❛❝✐♦♥❡♠ ❡st❛s ✐❞❡♥t✐❞❛❞❡s s♦❜r❡ ❡ss❡s s✉❜❝♦♥❥✉♥t♦s✱ q✉❡ é ❡①❛t❛♠❡♥t❡ ♦ ♣r♦♣ós✐t♦ ❞❡ss❛ t❡s❡ ♣❛r❛ ❝❡rt❛s ✐♥✈♦❧✉çõ❡s q✉❡ ✐r❡♠♦s ❡♠ ❜r❡✈❡ ❛❜♦r❞❛r✳

P♦❞❡♠♦s ❡♥❝♦♥tr❛r ♥❛ ❝r❡s❝❡♥t❡ ❧✐t❡r❛t✉r❛ ❛❝❡r❝❛ ❞❡ss❡ t❡♠❛✱ r❡s✉❧t❛❞♦s ❣❛r❛♥✲ t✐♥❞♦✱ ♣❛r❛ ❛ ✐♥✈♦❧✉çã♦∗ ✐♥❞✉③✐❞❛ ❡♠RG✱ ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ❛ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛♥t✐❝♦✲

♠✉t❛t✐✈✐❞❛❞❡ ❞❡ RG∗ ❡ RG−∗ s❡❥❛♠ ✈❡r✐✜❝❛❞❛s✳ ❚❛✐s r❡s✉❧t❛❞♦s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s✱ ♥♦ ❝❛s♦charR6= 2✱ ❝♦♥❢♦r♠❡ ❛ s❡❣✉✐♥t❡ t❛❜❡❧❛✿

❈♦♠✉t❛t✐✈✐❞❛❞❡ ❆♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡

RG∗ ❬❏❘✵✻❪

❬●P✶✸❛❪

RG−

∗ ❬❇❏P❘✵✾❪

❚❛❜❡❧❛ ✶✳✷✿ ❈♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ♣❛r❛ ✐♥✈♦❧✉çõ❡s ❧✐♥❡❛r❡s ❖❜s❡r✈❡ q✉❡ s❡charR= 2✱ ❡♥tã♦ ♦s ❝♦♥❥✉♥t♦sRG+

RG−❝♦✐♥❝✐❞❡♠✱ ❡ ♦ ♠❡s♠♦ ♦❝♦rr❡ ♣❛r❛ ❡ss❛s ✐❞❡♥t✐❞❛❞❡s✳ ❊ss❡ ❝❛s♦ ❢♦✐ ❡st✉❞❛❞♦ ❡♠[❏❘✵✻]✳

✶✳✸ ■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s

❖❜s❡r✈❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ❝✐t❛❞♦s ❛❝✐♠❛ ❛❝❡r❝❛ ❞❛s ✐❞❡♥t✐❞❛❞❡s ❞❡ ▲✐❡ ♥✐❧♣♦tê♥❝✐❛✱ ▲✐❡n✲❊♥❣❡❧✱ ❝♦♠✉t❛t✐✈✐❞❛❞❡ ❡ ❛♥t✐❝♦♠✉t❛t✐✈✐❞❛❞❡ ✈❡r✐✜❝❛❞❛s ♥♦s ❝♦♥❥✉♥t♦sRG+ ❡

RG− ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡✱ ♥✉♠ ❝❡rt♦ s❡♥t✐❞♦✱ ♣♦✉❝♦ t❡♠✲s❡ ❛ ❛❝r❡s❝❡♥t❛r ♥❡ss❡ ❡st✉❞♦✳ ❉❡ss❛ ❢♦r♠❛✱ ♣❛r❛ ♣r♦ss❡❣✉✐r♠♦s ❞❡✈❡♠♦s ❜✉s❝❛r ♥♦✈❛s ❢♦r♠❛s ❞❡ ❝♦♥str✉✐r ✐♥✈♦❧✉çõ❡s ♥♦ ❛♥❡❧ ❞❡ ❣r✉♣♦ RG ❛ ♣❛rt✐r ❞❡ ✐♥✈♦❧✉çõ❡s ❡♠ G✳ ◆❛ ❜✉s❝❛ ❞❡ss❛s ✐♥✈♦❧✉çõ❡s✱ s✉r❣❡ ♦ ❝♦♥❝❡✐t♦

❞❡ ■♥✈♦❧✉çõ❡s ❖r✐❡♥t❛❞❛s✱ ✐♥tr♦❞✉③✐❞♦s ♣♦r ◆♦✈✐❦♦✈ ♥♦ ❝♦♥t❡①t♦ ❞❛ K✲❚❡♦r✐❛✳

❉❡✜♥✐çã♦ ✶✳✷✸✳ ❙❡❥❛♠ R ✉♠ ❛♥❡❧ ❡ G ✉♠ ❣r✉♣♦ ❝♦♠ ✐♥✈♦❧✉çã♦ ✳ ❙❡❥❛ σ :G → {±1}

✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦s✱ ♦ q✉❛❧ ❝❤❛♠❛♠♦s ❞❡ ♦r✐❡♥t❛çã♦ ❞❡ G✱ ❝♦♠♣❛tí✈❡❧ ❝♦♠ ❛

✐♥✈♦❧✉çã♦ ∗✱ ♥♦ s❡♥t✐❞♦ ❞❡ xx∗ N := kerσ✳ ❊♥tã♦✱ ❛ ❛♣❧✐❝❛çã♦ σ : RG RG ❞❛❞❛ ♣♦r

X

x∈G

αxx !σ∗

=X

x∈G

σ(x)αxx∗,

Figure

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