❇é ❣✐➳♦ ❞ơ❝ ✈➭ ➤➭♦ t➵♦ ❱✐Ư♥ ❍➭♥ ❧➞♠ ❑❍ ✈➭ ❈◆ ✈✐Öt ♥❛♠ ✈✐Ö♥ ✈❐t ❧ý ❱â ❱➝♥ ❱✐➟♥ ◆❤ã♠ ➤è✐ ①ø♥❣ ❣✐➳♥ ➤♦➵♥ ❱➭ ❝➳❝ ♠➠ ❤×♥❤ ✸✲✸✲✶ ▲✉❐♥ ➳♥ t✐Õ♥ sÜ ✈❐t ❧ý ❍➭ ♥é✐✲✷✵✶✸ ❇é ❣✐➳♦ ❞ơ❝ ✈➭ ➤➭♦ t➵♦ ❱✐Ư♥ ❍➭♥ ❧➞♠ ❑❍ ✈➭ ❈◆ ✈✐Öt ♥❛♠ ✈✐Ö♥ ✈❐t ❧ý ❱â ❱➝♥ ✈✐➟♥ ◆❤ã♠ ➤è✐ ①ø♥❣ ❣✐➳♥ ➤♦➵♥ ✈➭ ❝➳❝ ♠➠ ❤×♥❤ ✸✲✸✲✶ ❈❤✉②➟♥ ♥❣➭♥❤✿ ❱❐t ❧ý ❧ý t❤✉②Õt ✈➭ ✈❐t ❧ý t♦➳♥ ▼➲ sè✿ ✻✷ ✹✹ ✵✶ ✵✶ ◆❣➢ê✐ ❤➢í♥❣ ❞➱♥✿ ●❙ ✲ ❚❙✳ ❍♦➭♥❣ ◆❣ä❝ ▲♦♥❣ ▲✉❐♥ ➳♥ t✐Õ♥ sÜ ❱❐t ❧ý ❍➭ ♥é✐ ✲ ✷✵✶✸ ▲ê✐ ❝➯♠ ➡♥ ▲✉❐♥ ➳♥ ♥➭② ➤➢ỵ❝ ❤♦➭♥ t❤➭♥❤ t➵✐ ❚r✉♥❣ t➞♠ ❱❐t ❧ý ❧ý t❤✉②Õt ✲ ❱✐Ư♥ ❱❐t ❧ý✱ ❞➢í✐ sù ❤➢í♥❣ ❞➱♥ ❝đ❛ ●❙ ✲ ❚❙✳ ❍♦➭♥❣ ◆❣ä❝ ▲♦♥❣✳ ❚➠✐ ①✐♥ ❜➭② tá ❧ß♥❣ ❜✐Õt ➡♥ ❝❤➞♥ t❤➭♥❤ ✈➭ s➞✉ s➽❝ ➤Õ♥ ●❙ ✲❚❙✳ ❍♦➭♥❣ ◆❣ä❝ ▲♦♥❣ ✲ ♥❣➢ê✐ ➤➲ ❤Õt ❧ß♥❣ tr✉②Ị♥ ❞➵②✱ ➤é♥❣ ✈✐➟♥✱ ❦❤Ý❝❤ ❧Ư ✈➭ ➤Þ♥❤ ❤➢í♥❣ ♥❣❤✐➟♥ ❝ø✉ ❝❤♦ t➠✐ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣ ✈➭ tõ♥❣ ❜➢í❝ ❤♦➭♥ ❝❤Ø♥❤ ❧✉❐♥ ➳♥✳ ❚➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❚❙✳ P❤ï♥❣ ỗ ị ì ú ➤ì t➠✐ r✃t ♥❤✐Ị✉ tr♦♥❣ ✈✐Ư❝ tÝ❝❤ ❧ị② ❦✐Õ♥ t❤ø❝ ✈➭ ❝➳❝ ❦ü t❤✉❐t tÝ♥❤ t♦➳♥✱ ❝ị♥❣ ♥❤➢ ♥❤÷♥❣ ➤ã♥❣ ❣ã♣ ❤Õt sø❝ ❜æ Ý❝❤ ❝❤♦ ❧✉❐♥ ➳♥✳ ❚➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ●❙✲❚❙✳ ➜➷♥❣ ❱➝♥ ❙♦❛✱ P●❙✲❚❙✳ ◆❣✉②Ô♥ ◗✉ú♥❤ ▲❛♥✱ ❚❤❙✳ ▲➟ ❚❤ä ❍✉Ö ✈➭ ❚❤❙✳ ❈❛♦ ❍♦➭♥ ◆❛♠ ✈× ➤➲ ❝ã ♥❤✐Ị✉ tr❛♦ ➤ỉ✐ ❜ỉ Ý❝❤ ✈Ị ❝❤✉②➟♥ ♠➠♥ ✈➭ sù đ♥❣ ❤é✱ ❣✐ó♣ ➤ì t➠✐ tr♦♥❣ q✉➳ tr×♥❤ ❤ä❝ t❐♣✳ ❚➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ❚❤❙✳ ◆❣✉②Ơ♥ ◆❣ä❝ ❚ù ✈➭ ❜➵♥ ❜❒✱ ➤å♥❣ ♥❣❤✐Ư♣ ✈× ➤➲ ❝❤✐❛ sÏ ❝➳❝ t➭✐ ❧✐Ư✉ t❤❛♠ ❦❤➯♦ ❜ỉ Ý❝❤ ❝❤♦ ❧✉❐♥ ➳♥✳ ❚➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ▲➲♥❤ ➤➵♦ ❱✐Ö♥ ❱❐t ❧ý ❍➭ ◆é✐✱ ❚r✉♥❣ t➞♠ ❱❐t ❧ý ý tết Pò ọ ì t ề ệ t ợ t tr q trì ọ t❐♣✳ ❚➠✐ ①✐♥ ❝❤➞♥ t❤➭♥❤ ❝➯♠ ➡♥ ▲➲♥❤ ➤➵♦ ❚r➢ê♥❣ ➜➵✐ ❤ä❝ ❚➞② ◆❣✉②➟♥✱ ❑❤♦❛ ❑❤♦❛ ❤ä❝ ❚ù ♥❤✐➟♥ ✈➭ ❈➠♥❣ ♥❣❤Ö ✈➭ ❇é ♠➠♥ ❱❐t ❧ý ✲ ♥➡✐ t➠✐ t ì t ọ ề ệ t ợ ❝❤♦ t➠✐ tr♦♥❣ s✉èt t❤ê✐ ❣✐❛♥ ❤ä❝ t❐♣ ✈➭ ❧➭♠ ✈✐Ư❝✳ ❚➠✐ ✈➠ ❝ï♥❣ ❜✐Õt ➡♥ ❣✐❛ ➤×♥❤ ✈➭ ♥❣➢ê✐ t❤➞♥ ➤➲ ❞➭♥❤ t×♥❤ ❝➯♠ ②➟✉ t❤➢➡♥❣✱ ❧✉➠♥ ➤é♥❣ ✈✐➟♥ ✈➭ t➵♦ ♠ä✐ ➤✐Ị✉ ❦✐Ư♥ tèt ♥❤✃t ➤Ĩ t➠✐ ❤♦➭♥ t❤➭♥❤ ❧✉❐♥ ➳♥ ♥➭②✳ ❍➭ ◆é✐✱ ♥❣➭② ✳✳✳t❤➳♥❣✳✳✳♥➝♠ ✷✵✶✸ ❱â ❱➝♥ ❱✐➟♥ ✐ ▲ê✐ ❝❛♠ ➤♦❛♥ ❚➠✐ ①✐♥ ❝❛♠ ➤♦❛♥ ➤➞② ❧➭ ❝➠♥❣ tr×♥❤ ♥❣❤✐➟♥ ❝ø✉ ❝đ❛ r✐➟♥❣ t➠✐✳ ❈➳❝ sè ❧✐Ư✉✱ ❦Õt q✉➯ ♠í✐ ♠➭ t➠✐ ❝➠♥❣ ❜è tr♦♥❣ ❧✉❐♥ ➳♥ ❧➭ tr✉♥❣ t❤ù❝ ✈➭ ❝❤➢❛ tõ♥❣ ➤➢ỵ❝ ❛✐ ❝➠♥❣ ❜è tr♦♥❣ ❜✃t ❦ú ❝➠♥❣ tr×♥❤ ♥➭♦ ❦❤➳❝✳ ❚➳❝ ❣✐➯ ❧✉❐♥ ➳♥ ❱â ❱➝♥ ❱✐➟♥ ✐✐ ❈➳❝ ❦ý ❤✐Ö✉ ❝❤✉♥❣ ❑Ý ❤✐Ư✉ ◆é✐ ❞✉♥❣ ▼❍❈ ▼➠ ❤×♥❤ ❝❤✉➮♥ ✸✸✶❘❍ ▼➠ ❤×♥❤ ✸✲✸✲✶ ✈í✐ ♥❡✉tr✐♥♦ ♣❤➞♥ ❝ù❝ ♣❤➯✐ ✸✸✶◆❋ ▼➠ ì r tr ò S3 ì ✈í✐ ♥❤ã♠ ➤è✐ ①ø♥❣ S3 S3 ▼➠ ❤×♥❤ ✸✸✶❘❍ ✈í✐ ♥❤ã♠ ➤è✐ ①ø♥❣ S3 S4 ▼➠ ❤×♥❤ ✸✸✶◆❋ ✈í✐ ♥❤ã♠ ➤è✐ ①ø♥❣ S4 ✸✸✶◆❋ ✸✸✶❘❍ ✸✸✶◆❋ ❍P❙ ❍❛rr✐s♦♥✲P❡r❦✐♥s✲❙❝♦tt ❱❊❱ ❱❛❝✉✉♠ ❊①♣❡❝t❛t✐♦♥ ❱❛❧✉❡ ✭❚r✉♥❣ ❜×♥❤ ❝❤➞♥ ❦❤➠♥❣✮ ❈❑▼ ❈❛❜✐❜❜♦✲❑♦❜❛②❛s❤✐✲▼❛s❦❛✇❛ ❉❖◆❯❚ ❉✐r❡❝t ❖❜s❡r✈❛t✐♦♥ ♦❢ t❤❡ ◆✉ ❚❛✉ ❈❊❘◆ P❉● ´e ❈♦♥s❡✐❧ ❊✉r♦♣ ´e ❡♥ ♣♦✉r ❧❛ ❘❡❝❤❡r❝❤❡ ◆✉❝❧ P❛rt✐❝❧❡ ❉❛t❛ ●r♦✉♣ ✐✐✐ ❛✐r❡ ❉❛♥❤ s➳❝❤ ❤×♥❤ ✈Ï ✶✳✶ ➜è✐ ①ø♥❣ S3 ❝đ❛ t❛♠ ❣✐➳❝ ➤Ò✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✹ ✶✳✷ ➜è✐ ①ø♥❣ S4 ❝đ❛ ❤×♥❤ ❧❐♣ ♣❤➢➡♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✶ ➜å t❤Þ ♠➠ t➯ sù ♣❤ơ t❤✉é❝ ❝đ❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✷ ➜å tị t ụ (8.713 ì 103 , 0.1) ✷✳✸ ➜å t❤Þ ♠➠ t➯ (0.1, 0.25) ✱ ✈➭ ✷✳✹ ➜å t❤Þ ♠➠ t➯ sù ✈➭ ♣❤ơ t❤Þ ♠➠ (0.085, 0.2) ✷✳✻ ➜å t❤Þ ♠➠ (0.2, 0.6) t➯ sù ✱ ✈➭ ✷✳✼ ➜å t❤Þ ♠➠ (0.085, 0.6) sù ✱ ✈➭ t➯ ❝đ❛ ♣❤ơ ♣❤ơ m1 , m2 , m3 t❤✉é❝ ❝ñ❛ t❤✉é❝ ❝ñ❛ m1 , m2 , m3 ✳ ✳ ✳ ✳ ✳ ✈➭♦ ✳ ✳ a ✳ ✈➭♦ ✳ a ✳ t❤✉é❝ ❝ñ❛ m1 , m2 , m3 t❤✉é❝ ❝ñ❛ t❤✉é❝ ✳ ❝ñ❛ ✳ ✳ ✳ m1 , m2 , m3 ✳ ✳ ✳ ✳ ✳ ✳ m1 , m2 , m3 t➯ sù ✈➭ a ∈ (−0.6, −0.085) ✳ ✳ ✐✈ ✳ ✳ ✳ ✳ ✳ ✈➭♦ ✳ ✳ ✳ ✈➭♦ ✳ ✳ ✳ ✈➭♦ ✳ ✳ ✳ a ✳ a ✳ a ✳ a ∈ ✈í✐ ✳ a ∈ (−0.6, −8.713 × 10−3 ) a ∈ (−0.6, −0.2) ♣❤ô ✳ m1 , m2 , m3 ✮✳ ♣❤ô a ✈➭♦ ✳ a ∈ (−0.2, −0.085 sù a ✈➭♦ a ∈ (−0.1, −8.713 × 10−3 ) ✳ ✳ ✱ ✈➭ ➜å a, b a ∈ (−0.25, 0.1) (8.713 × 10−3 , 0.6) ✷✳✺ t❤✉é❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✳ ✳ ✳ ✹✽ ✳ ✳ ✳ ✹✾ ✳ ✳ ✳ ✹✾ a ∈ ✈í✐ ✳ ✳ a ∈ ✈í✐ ✳ ✹✽ a ∈ ✈í✐ ✳ ✳ a ∈ ✈í✐ ✳ ✳ a ∈ ✈í✐ ✳ ✳ ✳ ✳ ✳ ✳ ✹✾ ❉❛♥❤ s➳❝❤ ❜➯♥❣ S3 ✶✳✶ ❈➳❝ ❧í♣ ❧✐➟♥ ❤ỵ♣ ❝đ❛ ♥❤ã♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✷ ❇➯♥❣ ➤➷❝ ❜✐Ĩ✉ ❝đ❛ ♥❤ã♠ S3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✶✳✸ ❇➯♥❣ ➤➷❝ ❜✐Ĩ✉ ❝đ❛ ♥❤ã♠ S4 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✶✳✹ ❈➳❝ ❧í♣ ❧✐➟♥ ❤ỵ♣ ❝đ❛ ♥❤ã♠ S4 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✸✳✶ ❈➳❝ ❦❤➯ ♥➝♥❣ ❦Õt ❝➷♣ ❝➬♥ t❤✐Õt s✐♥❤ ❦❤è✐ ❧➢ỵ♥❣ q✉❛r❦ ✳ ✳ ✳ ✳ ✳ ✺✺ ✈ ▼ô❝ ❧ô❝ ✶ ◆❤ã♠ ✶✳✶ ✸ ◆❤ã♠ ✈➭ ♠➠ ❤×♥❤ ✸✲✸✲✶ S3 , S4 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✶✳✶ ◆❤ã♠ ➤è✐ ①ø♥❣ S3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✶✳✶✳✷ ◆❤ã♠ ➤è✐ ①ø♥❣ S4 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✷ ✶✳✷ ▼➠ ì ì r tr ò ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✸✳✶ ❙ù s➽♣ ①Õ♣ ❤➵t ❝đ❛ ♠➠ ❤×♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✶✳✸✳✷ P❤➳ ✈ì ➤è✐ ①ø♥❣ tù ♣❤➳t ✈➭ ❦❤è✐ ❧➢ỵ♥❣ ❢❡r♠✐♦♥ ✳ ✳ ✳ ✳ ✳ ✷✸ ✳ ✳ ✳ ✳ ✳ ✷✺ ✶✳✹ ✷ S3 , S4 ✳ ✳ ✳ ❑Õt ❧✉❐♥ ❝❤➢➡♥❣ ✶ ➜è✐ ①ø♥❣ ✈Þ S4 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ tr ì r tr ò ự s ế t ủ ì ố ợ ❧❡♣t♦♥ ♠❛♥❣ ➤✐Ư♥ ✷✳✸ ❑❤è✐ ❧➢ỵ♥❣ ♥❡✉tr✐♥♦ ✷✳✹ ❑❤è✐ ❧➢ỵ♥❣ q✉❛r❦ ✳ ✷✳✺ ✷✳✻ ✷✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ❙ù ➤Þ♥❤ ❤➢í♥❣ ❝❤➞♥ ❦❤➠♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ❑Õt ❧✉❐♥ ❝❤➢➡♥❣ ✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ◆❤ã♠ ➤è✐ ①ø♥❣ ✈Þ S3 ✳ ✳ ✳ ✳ ✳ tr♦♥❣ ❝➳❝ ♠➠ ❤×♥❤ ✸✲✸✲✶ ✸✳✶ ❙ù s➽♣ ế t ủ ì ố ợ t ➤✐Ư♥ ✸✳✸ ❑❤è✐ ❧➢ỵ♥❣ q✉❛r❦ ✳ ✸✳✹ ❑❤è✐ ❧➢ỵ♥❣ ✈➭ tré♥ ❧➱♥ ♥❡✉tr✐♥♦ ✸✳✺ ●✐í✐ ❤➵♥ t❤ù❝ ♥❣❤✐Ư♠ ✈í✐ tr➢ê♥❣ ❤ỵ♣ ✶ ✸✳✻ ●✐í✐ ❤➵♥ t❤ù❝ ♥❣❤✐Ư♠ ✈í✐ sù ❦Õt ❤ỵ♣ ❝đ❛ tr➢ê♥❣ ❤ỵ♣ ✶ ✈➭ ✷ ✸✳✼ ◆❤❐♥ ①Ðt ✈Ị sù ♣❤➳ ✈ì✱ ❝➳❝ tr✉♥❣ ❜×♥❤ ❝❤➞♥ ❦❤➠♥❣ ✈➭ t❤❛♠ sè ✳ ✳ ✳ ✳ ✳ ✺✵ ✳ ✳ ✈✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✸ ✳ ✳ ✻✺ ρ ✼✵ S3 ✸✳✽ ➜è✐ ①ø♥❣ ✸✳✾ ❚❤Õ ✈➠ ❤➢í♥❣ νR tr♦♥❣ ♠➠ ❤×♥❤ ✸✲✸✲✶ ✈í✐ ♥❡✉tr✐♥♦ ♣❤➞♥ ❝ù❝ ♣❤➯✐ ✭ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ S3 ✸✳✾✳✶ ❚❤Õ ✈➠ ❤➢í♥❣ ❝đ❛ ♠➠ ❤×♥❤ ✸✸✶◆❋ ✸✳✾✳✷ ❚❤Õ ✈➠ ❤➢í♥❣ ❝đ❛ ♠➠ ❤×♥❤ ✸✸✶❘❍ ✸✳✶✵ ❑Õt ❧✉❐♥ ❝❤➢➡♥❣ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✼ S3 ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✮ ✼✶ ✳ S3 ❆ ❇✐Ĩ✉ ❞✐Ơ♥ ❝❤Ý♥❤ q✉② ❝đ❛ ✐ ❇ ❚×♠ ❤Ư ❙è ❈❧❡❜s❝❤✲●♦r❞❛♥ ❝đ❛ ♥❤ã♠ ❈ ❚×♠ ❤Ư sè ❈❧❡❜s❝❤ ✲ ●♦r❞❛♥ ❝đ❛ ♥❤ã♠ ❉ ❈➳❝ số ợ tử ủ ì S4 số ợ tử ủ ì S3 S3 S3 S4 ✐✐ ✈✐✐✐ ①✈ ①✈✐ ▼ë ➤➬✉ ▲ý ❞♦ ❝❤ä♥ ➤Ị t➭✐ ❚×♠ ❤✐Ĩ✉ t❤Õ ❣✐í✐ tù ♥❤✐➟♥ ❧➭ ♠ét tr♦♥❣ ♥❤÷♥❣ ♥❤✐Ư♠ ✈ơ ❧í♥ ♥❤✃t ❝đ❛ ❧♦➭✐ ♥❣➢ê✐ tr♦♥❣ q✉➳ tr×♥❤ ❝❤✐♥❤ ♣❤ơ❝ t❤✐➟♥ ♥❤✐➟♥✱ ♠➷❝ ❞ï✱ t❤❡♦ t❤ê✐ ❣✐❛♥ ❝➳❝❤ t❤ø❝ t✐Õ♣ ❝❐♥ ❝ã t❤Ĩ t❤❛② ➤ỉ✐✱ ✈➭ sù ❤✐Ó✉ ❜✐Õt ♣❤➳t tr✐Ó♥ tï② t❤❡♦ t❤ê✐ ➤➵✐ ✈➭ ❝➳❝ ♥Ị♥ ✈➝♥ ❤ã❛✳ ▼ơ❝ t✐➟✉ ❝đ❛ ✈❐t ❧ý ❧➭ ♠➠ t➯ ❝➳❝ ❤✐Ư♥ t➢ỵ♥❣ tù ♥❤✐➟♥ ❜➺♥❣ ❧ý t❤✉②Õt ✈➭ t❤ù❝ ♥❣❤✐Ư♠✳ ❱❐t ❧ý t❤ù❝ ♥❣❤✐Ư♠ ❝ã ✈❛✐ trß ❦✐Ĩ♠ ❝❤ø♥❣ ❝➳❝ t✐➟♥ ➤♦➳♥ ❝đ❛ ❝➳❝ ♠➠ ❤×♥❤ ✈❐t ❧ý ❧ý t❤✉②Õt ✈➭ ➤➢❛ r❛ ♥❤÷♥❣ t✐➟♥ ➤♦➳♥ ♠í✐✱ ✈❐t ❧ý ❧ý t❤✉②Õt ①➞② ❞ù♥❣ ❝➳❝ ♠➠ ❤×♥❤ ♠➠ t➯ ❝➳❝ ❦Õt q✉➯ t❤ù❝ ♥❣❤✐Ö♠✱ ➤å♥❣ t❤ê✐ ➤➢❛ r❛ ❝➳❝ t✐➟♥ ➤♦➳♥ ♠í✐✳ ❍❛✐ ❧Ü♥❤ ✈ù❝ ♥➭② tå♥ t➵✐ s s t ẽ ỗ trợ t❤ó❝ ➤➮② sù ♣❤➳t tr✐Ĩ♥ ❝đ❛ ♥❣➭♥❤ ✈❐t ❧ý✱ ❧➭ ➤é♥❣ ❧ù❝ ❝❤Ý♥❤ ❝❤♦ sù ❤✐Ĩ✉ ❜✐Õt ❝đ❛ ♥❤➞♥ ❧♦➵✐ ✈Ị t❤Õ ❣✐í✐ tù ♥❤✐➟♥ ❤✉②Ị♥ ❜Ý✳ ▼ét ❧ý t❤✉②Õt ✈❐t ❧ý tèt sÏ ♠➠ t➯ ➤ó♥❣ ❝➳❝ ❦Õt q✉➯ tí ệ ợ r ữ t✐➟♥ ➤♦➳♥ ➤➳♥❣ t✐♥ ❝❐② sÏ ➤➢ỵ❝ ❦✐Ĩ♠ tr❛ ❜➺♥❣ t❤ù❝ ♥❣❤✐Ư♠ tr♦♥❣ t➢➡♥❣ ❧❛✐✳ ❑❤✐ ❝➳❝ t✐➟♥ ➤♦➳♥ ➤➢ỵ❝ ①➳❝ ♥❤❐♥✱ ❧ý t❤✉②Õt trë ♥➟♥ ♥❣➭② ❝➭♥❣ ➤➢ỵ❝ ❝❤✃♣ ♥❤❐♥✳ ◆❣➢ỵ❝ ❧➵✐✱ ♥Õ✉ ❝ã ❝➳❝ q✉❛♥ s➳t t❤ù❝ ♥❣❤✐Ư♠ ♠➞✉ t❤✉➱♥✱ ❧ý t❤✉②Õt ❝➬♥ ♣❤➯✐ ➤➢ỵ❝ ①❡♠ ①Ðt ❧➵✐ ❤♦➷❝ ①➞② ❞ù♥❣ ♠ét ❧ý t❤✉②Õt ♠í✐ ♣❤ï ❤ỵ♣ ❤➡♥✳ ❚r♦♥❣ sè ❝➳❝ ❤➵t ❤×♥❤ t❤➭♥❤ ♥➟♥ ✈ị trơ✱ ❝ã ♠ét ❧♦➵✐ ❤➵t ➤ã♥❣ ✈❛✐ trß r✃t q✉❛♥ trä♥❣ tr♦♥❣ sù t✐Õ♥ ❤ã❛ ❝đ❛ ✈ị trơ ë t❤ê✐ ❦ú s➡ ❦❤❛✐✱ tr♦♥❣ q✉➳ tr×♥❤ s✐♥❤ ❧❡♣t♦♥✱ s✐♥❤ ❜❛r②♦♥✱ ✈➭ sù ❤×♥❤ t❤➭♥❤ ❜ø❝ ①➵ ♥Ị♥ ✈ị trơ✱ ❝ị♥❣ ♥❤➢ ✈❛✐ trò t t tố ó t tr ợ r✃t ❜Ð✱ ✈í✐ s♣✐♥ ❜➺♥❣ 2✱ ◆❡✉tr✐♥♦ ❧à ❤➵t ❦❤➠♥❣ ♠❛♥❣ ➤✐Ö♥✱ ❝ã ❦❤è✐ ❝❤Ø t➢➡♥❣ t➳❝ r✃t ②Õ✉ ✈➭ ❤✐Õ♠ ✈í✐ ❝➳❝ ✈❐t ❝❤✃t✳ ❙ù tå♥ t➵✐ ❝đ❛ ♥❡✉tr✐♥♦ ❧➬♥ ➤➬✉ t✐➟♥ ➤➢ỵ❝ ➤Ị ①✉✃t ❜ë✐ ❲♦❧❢❣❛♥❣ P❛✉❧✐✱ ✈➭♦ ♥➝♠ ✶✾✸✵✱ ➤Ĩ ❣✐➯✐ q✉②Õt ✈✃♥ ➤Ị ❜➯♦ t♦➭♥ ♥➝♥❣ ❧➢ỵ♥❣ ✈➭ ♠➠ ♠❡♥ ①✉♥❣ ❧➢ỵ♥❣ tr♦♥❣ ♣❤➞♥ r➲ ❜❡t❛✱ ✈í✐ t➟♥ ❣ä✐ ❧➭ ♥❡✉tr♦♥ ❬✹❪✱ s❛✉ ➤ã ➤➢ỵ❝ ❋❡r♠✐ ❣ä✐ ❧➭ ♥❡✉tr✐♥♦ ✈× ❤➵t ♥❡✉tr♦♥ t❤ù❝ sù ➤➲ ➤➢ỵ❝ ❦❤➳♠ ♣❤➳ ❜➺♥❣ t❤ù❝ ♥❣❤✐Ư♠ ❜ë✐ ❏❛♠❡s ❈❤❛❞✇✐❝❦ ✶ P❤ơ ❧ơ❝ ❇ ❚×♠ ❤Ư ❙è ❈❧❡❜s❝❤✲●♦r❞❛♥ ❝đ❛ ♥❤ã♠ ●ä✐ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ❝đ❛ ❧➢ì♥❣ t✉②Õ♥ ❧➭ x1 ∼ ≡ x1 |1 + x2 |2 , ∼ x2 ❑❤✐ ➤ã✱ tÝ❝❤ t❡♥s♦r x1 , x2 2⊗2 ✈➭ y1 y2 y1 , y2 S3 ✱ t❛ ❝ã✿ ≡ y1 |1 + y2 |2 ✭❇✳✶✮ ➤➢ỵ❝ ✈✐Õt ❞➢í✐ ❞➵♥❣✿ ⊗ ∼ (x1 |1 + x2 |2 ) ⊗ (y1 |1 + y2 |2 ) ∼ x1 y1 |1 |1 + x1 y2 |1 |2 + x2 y1 |2 |1 + x2 y2 |2 |2 ∼ (|1 |1 , |1 |2 , |2 |1 , |2 |2 ) (x1 y1 x1 y2 x2 y1 x2 y2 )T , tr♦♥❣ ➤ã✱ ❝➡ së✱ ✈➭ |e1 = |1 |1 , |e2 = |1 |2 , |e3 = |2 |1 , |e4 = |2 |2 x1 y1 , x1 y2 , x2 y1 , x2 y2 tÝ❝❤ trù❝ t✐Õ♣ 2⊗2 ✭❇✳✷✮ ❧➭ ❝➳❝ ✈❡❝t♦r ❧➭ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ❝ñ❛ ✈❡❝t♦r tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ✳ ➜➷t |e1 = (1 0 0)T , |e2 = (0 0)T , |e3 = (0 0)T , |e4 = (0 0 1)T ✭❇✳✸✮ ❑❤✐ ➤ã ✭❇✳✷✮ ➤➢ỵ❝ ✈✐Õt ❧➵✐ ❞➢í✐ ❞➵♥❣✿ ⊗ ∼ x1 y1 |e1 + x1 y2 |e2 + x2 y1 |e3 + x2 y2 |e4 , tr ể ễ ủ 22 ợ ị [D22 ]iµ,jν = [D2 (g)]ij [D2 (g)]µν , (i, j, µ, ν = 1, 2; g = e, a1 , , a5 ) ❳❡♠ ❝➷♣ ❝❤Ø sè iµ ❧➭ ❝❤Ø sè ❤➭♥❣✱ ✈➭ ❝➷♣ ❝❤Ø sè D2⊗2 (g) = ✭❇✳✹✮ jν ❧➭ ❝❤Ø sè ❝ét✱ t❛ ❝ã [D2 (g)]11 D2 (g) [D2 (g)]12 D2 (g) D2 (g)]21 D2 (g) [D2 (g)]22 D2 (g) ✐✐ ✭❇✳✺✮ , ✭❇✳✻✮ ❑Õt ❤ỵ♣ ♥❤ã♠ ✭✶✳✷✮ S3 ú t tì ợ tr❐♥ ❜✐Ó✉ √ √ 1 − , √ − −3 √ − √ − 3 √ −3 − 0 √ , 0 − 3 √ √ 3 √ − 1 √ , 4 −1 − √ − ❚♦➳♥ tư ❝❤✐Õ✉ ❜✐Ĩ✉ ❞✐Ơ♥ 2⊗2 √ ❝đ❛ −3 √ − −1 0 √ − √ − 3 √ √ , 0 , −1 √ √ − − −1 3 √ −1 √ ✈Ò ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ tè✐ ❣✐➯♥ ❝đ❛ P1 2⊗2 tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ t❤ù❝✿ 0 0 0 0 0 √ − √ 1 √ −3 √ 3 √ √ 1 −1 √ 4 3 √ − P1 ❞✐Ô♥ S3 √ √ ợ ị 0 a a ∗ 1 0 0 = χ1 (g)D2⊗2 (g) = D2⊗2 (g) = g=e g=e 2 0 0 0 0 0 −1 −1 1 1 = , P2 = −1 2 1 0 0 −1 0 ✐✐✐ 0 , 0 ✭❇✳✽✮ ❚♦➳♥ tö ❝❤✐Õ✉ t➳❝ ➤é♥❣ ❧➟♥ ❦❤➠♥❣ ❣✐❛♥ tÝ❝❤ t❡♥s♦r ➤➢ỵ❝ tÝ♥❤✿ xy 1 xy 2 P1 = (x1 y1 + x2 y2 ) x2 y x2 y xy 1 xy P1 x2 y x2 y 0 |e1 |e4 = √ (x1 y1 + x2 y2 ) √ + √ 2 0 1 |e ≡ √ (x1 y1 + x2 y2 ) √1 , ∼ (x1 y1 + x2 y2 ), 2 1 = (x1 y2 − x2 y1 ) , ∼ (x1 y2 − x2 y1 ), 1 xy 1 xy 2 P2 = (x1 y1 − x2 y2 ) x2 y1 x2 y2 ✭❇✳✾✮ ✭❇✳✶✵✮ 1 + (x1 y2 + x2 y1 ) 1 −1 ✭❇✳✶✶✮ ❚õ ❜✐Ĩ✉ t❤ø❝ ✭❇✳✶✶✮ ❝❤ó♥❣ t❛ t❤✃② ❤❛✐ tỉ ❤ỵ♣ ❝ã ❦❤➯ ♥➝♥❣ trë t❤➭♥❤ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ❝đ❛ ❧➢ì♥❣ t✉②Õ♥ ➤ã ❧➭ (x1 y1 − x2 y2 ) ✈➭ (x1 y2 + x2 y1 ) ✳ ❚✉② ✈❐② t❛ ❝➬♥ ➤✐Ị✉ ❦✐Ư♥ ♣❤ơ ➤Ĩ ①➳❝ ➤Þ♥❤ ❧➢ì♥❣ t✉②Õ♥✳ ❈❤ó ý r➺♥❣ ❝➳❝ ♣❤Ð♣ q✉❛② t ổ ợ ò é sÏ ❧➭♠ ❝❤ó♥❣ ➤ỉ✐ ❞✃✉✳ ❉♦ ✈❐② t❛ ❝➬♥ ①Ðt t➳❝ ➤é♥❣ ❝đ❛ ❜✃t ❦ú t❤➭♥❤ ♣❤➬♥ ♥➭♦ ❝đ❛ ❧í♣ ♥❤➢ a4 ✳ C3 ✲ ❧í♣ ❝➳❝ ②Õ✉ tè ♣❤➯♥ ①➵✱ ✈Ý ❞ơ ❇✐Ĩ✉ ❞✐Ơ♥ ❤❛✐ ❝❤✐Ị✉ ø♥❣ ✈í✐ ②Õ✉ tè ♥➭② ❧➭ ❤❛✐ ❧➢ì♥❣ t✉②Õ♥ ❞➢í✐ ♣❤Ð♣ ❜✐Õ♥ ➤ỉ✐ x1 x2 y1 y2 → → x1 x2 y1 y2 ✳ ❳Ðt ❜✐Õ♥ ➤ỉ✐ ❝đ❛ D2 (a4 ) ✿ = D2 (a4 ) = D2 (a4 ) x1 √ = x2 √ (y + 3y2 ) √ ( 3y1 − y2 ) ✐✈ (x1 + √ ( 3x1 3x2 ) − x2 ) , ✭❇✳✶✷✮ ◆❤➢ ✈❐② ❞➢í✐ ♣❤Ð♣ ❜✐Õ♥ ổ D2 (a4 ) ỗ xi yj (i, j = 1, 2) ❜✐Õ♥ ➤æ✐ t❤❡♦ q✉② ❧✉❐t ♥❤➢ s❛✉✿ √ √ x1 y1 → x1 y1 = (x1 + 3x2 )(y1 + 3y2 ), √ √ x1 y2 → x1 y2 = (x1 + 3x2 )( 3y1 − y2 ), √ √ x2 y1 → x2 y1 = ( 3x1 − x2 )(y1 + 3y2 ), √ √ x2 y2 → x2 y2 = ( 3x1 − x2 )( 3y1 − y2 ) ❚õ ❜✐Ĩ✉ t❤ø❝ ✭❇✳✶✸✮✱ ❞Ơ ❞➭♥❣ t❤✃② r➺♥❣ ❞➢í✐ t➳❝ ❞ơ♥❣ ❝đ❛ ✭❇✳✶✸✮ D2 (a4 ) ✿ D2 (a4 )(x1 y1 + x2 y2 ) = x1 y1 + x2 y2 = x1 y1 + x2 y2 D2 (a4 )1 = ✭❇✳✶✹✮ D2 (a4 )(x1 y2 − x2 y1 ) = −(x1 y2 − x2 y1 ) ✭❇✳✶✺✮ D2 (a4 )1 = −1 ✭❇✳✶✻✮ ❑❤✐ ➤ã✱ D2 (a4 ) 2∼ x2 y2 − x1 y1 = − x y2 + x y x2 y2 − x1 y1 x2 y2 − x1 y1 x1 y2 + x2 y1 ≡ (22 − 11, 12 + 21) x y2 + x y ✭❇✳✶✼✮ ❚❤ù❝ ❤✐Ư♥ ❝➳❝ ❜➢í❝ tÝ♥❤ t♦➳♥ t➢➡♥❣ tù ❝❤♦ ❝➳❝ tÝ❝❤ t❡♥s♦r ❦❤➳❝✱ ❝❤ó♥❣ t➠✐ t❤✉ ➤➢ỵ❝ ❝➳❝ ❤Ư sè ❈❧❡❜s❝❤ ✲ ●♦r❞❛♥ ❝đ❛ ♥❤ã♠ S3 tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ t❤ù❝ ♥❤➢ s❛✉✿ ⊗ = 1(11), ⊗ = 1(11), ⊗ = (11), ⊗ = 2(11, 12), ⊗ = 2(11, −12), ⊗ = 1(11 + 22) ⊕ (12 − 21) ⊕ 2(22 − 11, 12 + 21) ✭❇✳✶✽✮ tr♦♥❣ ➤ã✱ ❝❤ó♥❣ t➠✐ ➤➲ sư ❞ơ♥❣ ❦ý ❤✐Ư✉ ❝➳❝ t❤➭♥❤ ♣❤➬♥ ❝đ❛ ➤➡♥ t✉②Õ♥ ✈➭ ❧➢ì♥❣ t✉②Õ♥ ❞➢í✐ ❞➵♥❣ ♥❣➽♥ ❣ä♥ xi ≡ i, xi yj ≡ ij, xi yj ± xk yl ≡ ij ± kl ✳ ❚r♦♥❣ ♥❤✐Ị✉ tr➢ê♥❣ ❤ỵ♣✱ ➤Ĩ t❤✉❐♥ t✐Ư♥ ♥❣➢ê✐ t❛ ❧➭♠ ✈✐Ư❝ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♣❤ø❝✳ ❱✐Ư❝ ❝❤✉②Ĩ♥ tõ ❦❤➠♥❣ ❣✐❛♥ t❤ù❝ s❛♥❣ ❦❤➠♥❣ ❣✐❛♥ ♣❤ø❝ ➤➢ỵ❝ t❤ù❝ ❤✐Ư♥ ✈ ♥❤ê ♠❛ tr❐♥ ✉♥✐t❛✿ √i − √i2 √1 √ U = √1 − √i2 + ,U = √1 i √ ✭❇✳✶✾✮ ❑❤✐ ➤ã✱ ❝➳❝ ♠❛ tr❐♥ ❜✐Õ♥ ➤æ✐ t➢➡♥❣ ø♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♣❤ø❝✱ U D2 (g)U + ✱ ❝ã ❞➵♥❣ ♥❤➢ ✭✶✳✸✮✳ ❚➢➡♥❣ tù ♥❤➢ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ t❤ù❝✱ tõ ❜✐Ó✉ t❤ø❝ ✭✶✳✸✮ ✈➭ ✭❇✳✻✮✱ t❛ t❤✉ ➤➢ỵ❝ ❝➳❝ ♠❛ tr❐♥ ❜✐Ĩ✉ ❞✐Ô♥ tù e, a1 , , a5 2⊗2 tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♣❤ø❝ t❤❡♦ t❤ø ✿ 0 0 0 0 0 0 0 0 , 0 0 , 0 0 ω 0 0 0 0 ω 0 0 , ω ω , 0 0 ω 0 0 0 0 ω2 0 0 , ω ω 0 0 ✭❇✳✷✵✮ tr♦♥❣ ➤ã✱ √ √ i4π 3 1 ω = e = − + i , ω2 = e = − − i , 2 2 ω = 1, ω = ω, ω + ω + = i2π ❚♦➳♥ tư ❝❤✐Õ✉ ❜✐Ĩ✉ ❞✐Ơ♥ 2⊗2 ✭❇✳✷✶✮ ë ✭❇✳✷✵✮ ✈Ị ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ tè✐ ❣✐➯♥ ❝đ❛ ♥❤ã♠ S3 ❧➬♥ ❧➢ỵt ➤➢ỵ❝ ①➳❝ ➤Þ♥❤✿ P1 = χ∗1 (g)D2 g P1 0 1 −1 = −1 0 0 1 0 (g) = 2 0 0 1 , P2 = 2 0 ✈✐ 0 0 , 0 0 0 ✭❇✳✷✷✮ 0 0 0 0 0 ✭❇✳✷✸✮ ❑❤✐ ➤ã✱ xy 1 xy 2 P1 = (x1 y2 + x2 y1 ) x2 y1 x2 y2 1 1 = (x1 y2 + x2 y1 ) (|e2 + |e3 ) , |ei = |ei , (i = 2, 3), ∼ (x1 y2 + x2 y1 ) ≡ 12 + 21 ✭❇✳✷✹✮ xy 1 xy 2 P1 = (x1 y2 − x2 y1 ) (|e2 − |e3 ) , ∼ 12 − 21 x2 y1 x2 y2 xy 1 xy x2 y x2 y2 2 ≡ (22, 11) P2 ,2∼ = (|e1 , |e4 ) x2 y1 x y x y 1 1 x2 y2 ❍♦➭♥ t♦➭♥ t➢➡♥❣ tù ❝❤♦ ❝➳❝ tÝ❝❤ t❡♥s♦r ❦❤➳❝✱ ❈❧❡❜s❝❤ ✲ ●♦r❞❛♥ ❝đ❛ ♥❤ã♠ S3 ✭❇✳✷✺✮ ✭❇✳✷✻✮ ❝❤ó♥❣ t➠✐ t❤✉ ➤➢ỵ❝ ❝➳❝ ❤Ư sè tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ♣❤ø❝ ♥❤➢ s❛✉✿ ⊗ = 1(11), ⊗ = 1(11), ⊗ = (11), ⊗ = 2(11, 12), ⊗ = 2(11, −12), ⊗ = 1(12 + 21) ⊕ (12 − 21) ⊕ 2(22, 11) tr♦♥❣ ➤ã✱ tr♦♥❣ ❝➳❝ ❞✃✉ ♥❣♦➷❝ ➤➡♥✱ ❝❤Ø sè t❤ø ♥❤✃t ➤➢ỵ❝ ❤✐Ĩ✉ ❧➭ 1, 2) ✱ ❝❤Ø sè t❤ø ❤❛✐ ➤➢ỵ❝ ❤✐Ĩ✉ ❧➭ ij ± kl ✱ ❝❤➻♥❣ ❤➵♥✱ ✭❇✳✷✼✮ i ≡ xi (i = j ≡ yj (j = 1, 2) xi yj ≡ ij, xi yj ± xk yl ≡ ✱ 1(11) ≡ 1(x1 y1 ); 2(11, 12) ≡ 2(x1 y1 , x1 y2 ), ✈✐✐ P❤ơ ❧ơ❝ ❈ ❚×♠ ❤Ư sè ❈❧❡❜s❝❤ ✲ ●♦r❞❛♥ ❝ñ❛ ♥❤ã♠ Da (g) ●ä✐ ❧➭ 1, 2, , m) Db (g) ✱ 1, 2, , n) ✳ ❑❤✐ ❜✐Ĩ✉ m ❞✐Ơ♥ ❧➭ ❜✐Ĩ✉ ❞✐Ơ♥ ➤ã✱ ❜✐Ĩ✉ ❞✐Ơ♥ n ❝❤✐Ị✉ t➳❝ ❞ơ♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ ❝❤✐Ị✉ t➳❝ ❞ơ♥❣ tr♦♥❣ ❦❤➠♥❣ ❣✐❛♥ tÝ❝❤ t❡♥s♦r m×n ❝❤✐Ị✉✱ S4 |i , (i = |µ (µ = ✱ Da (g) ⊗ Db (g) ≡ Da⊗b (g) ✱ ❝ã ❝➳❝ ②Õ✉ tố ợ ị [Dab (g)]ià,j = [Da (g)]ij [Db (g)]µν , (i, j = 1, , m; µ, ν = 1, , n) ❚r♦♥❣ tr➢ê♥❣ ❤ỵ♣ m=n=3 ✱ t❛ ❝ã ❜✐Ĩ✉ ❞✐Ơ♥ tÝ❝❤ trù❝ t✐Õ♣ 3⊗3 ✭❈✳✶✮ ✿ [D3 (ai )]11 D3 (ai ) [D3 (ai )]12 D3 (ai ) [D3 (ai )]13 D3 (ai ) D3⊗3 (ai ) = [D (a )] D (a ) [D (a )] D (a ) [D (a )] D (a ) i 22 i i 23 i i 21 i [D3 (ai )]31 D3 (ai ) [D3 (ai )]32 D3 (ai ) [D3 (ai )]33 D3 (ai ) ✭❈✳✷✮ ❑Õt ❤ỵ♣ ✭✶✳✶✶✮ ✈➭ ✭❈✳✷✮✱ t❛ t❤✉ ➤➢ỵ❝ ❝➳❝ ❜✐Ĩ✉ ❞✐Ơ♥ tÝ❝❤ trù❝ t✐Õ♣ ø♥❣ ✈í✐ ❝➳❝ ♣❤➬♥ tư ❝đ❛ S4 t❤❡♦ t❤ø tù e, a1 , a2 , , a23 ✈✐✐✐ ♥❤➢ s❛✉✿ 3⊗3 t➢➡♥❣ C1 : 0 0 0 0 0 0 0 , C2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0 0 C3 : 0 0 0 0 0 −1 0 0 0 0 0 0 0 −1 0 0 −1 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 −1 0 0 −1 0 0 0 0 0 0 0 0 0 , 0 0 −1 0 0 −1 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 1 0 : ✐① 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 , 0 0 −1 0 0 0 0 −1 0 1 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 , 0 −1 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 −1 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 , 0 0 0 ① , 0 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 −1 0 1 , −1 0 0 0 0 0 0 0 C4 : 0 0 0 , 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 −1 0 0 0 −1 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 −1 ①✐ 0 0 0 0 0 0 −1 0 0 0 −1 0 −1 0 0 0 , 0 0 0 0 0 0 0 , 0 0 0 0 0 −1 0 0 0 0 −1 0 0 0 0 0 0 0 , 0 0 0 0 −1 0 0 0 0 0 0 0 −1 0 −1 0 −1 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 C5 : 0 0 0 0 , 0 0 0 −1 0 0 0 0 0 0 −1 0 0 0 0 0 0 −1 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 , −1 0 0 0 −1 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 0 , 0 0 0 ①✐✐ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 , 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 , ✭❈✳✸✮ ❚♦➳♥ tư ❝❤✐Õ✉ ❜✐Ĩ✉ ❞✐Ơ♥ 0 3⊗3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 6 0 0 6 0 0 0 0 0 0 0 0 6 0 6 0 0 0 0 0 0 0 0 0 ✈Ị ❝➳❝ ❜✐Ĩ✉ ❜✐Ơ♥ tè✐ ❣✐➯♥ 0 0 0 , 0, −6 0 0 0 − 16 0 0 0 0 0 − 61 0 , 0 0 0 0 0 0 0 0 0 1, , 2, 3, ❧➬♥ ❧➢ỵt ❧➭✿ 0 − 16 0 − 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 − 16 , 0 − 16 0 0 0 0 0 − 16 0 0 0 0 0 0 0 0 0 0 − 16 0 0 0 0 0 0 0 − 16 − 61 0 0 0 − 16 0 0 0 ✭❈✳✹✮ ➜➷t X = (a1 b1 a1 b2 a1 b3 a2 b1 a2 b2 a2 b3 a3 b1 a3 b2 a3 b3 )T ✳ ❑❤✐ ➤ã✱ t❛ ❝ã✿ (a1 b1 + a2 b2 + a3 b3 )(|11 + |22 + |33 ), ∼ a1 b1 + a2 b2 + a3 b3 ≡ 11 + 22 + 33 P1 X = ✭❈✳✺✮ 1 P2 X = = (2a1 b1 − a2 b2 − a3 b3 )|11 + (−a1 b1 + 2a2 b2 − a3 b3 )|22 6 + (−a1 b1 − a2 b2 + 2a3 b3 )|33 ✭❈✳✻✮ ❈➳❝ ➤➵✐ ❧➢ỵ♥❣ a2 b2 + 2a3 b3 ) (2a1 b1 − a2 b2 − a3 b3 ) (−a1 b1 + 2a2 b2 − a3 b3 ) ✱ ❜✐Õ♥ ➤ỉ✐ ♥❤➢ ❧➢ì♥❣ t✉②Õ♥ ✷ ❝đ❛ ①✐✐✐ ✱ ✈➭ S4 (−a1 b1 − ✳ ❉➢í✐ ♣❤Ð♣ ❜✐Õ♥ ➤æ✐ D2 (a4 ) ✱ t❛ ❝ã✿ x1 y1 + x2 y2 + x3 y3 D2 (a4 ) = x1 y1 + x2 y2 + x3 y3 x1 y1 + ω x2 y2 + ωx3 y3 x1 y1 + ωx2 y2 + ω x3 y3 11 + ω 22 + ω33) 2∼ ✭❈✳✼✮ 11 + ω22 + ω 33 a2 b3 + a3 b2 a3 b + a2 b 6P3 X = (|23 |31 |12 ) + (|32 |13 |21 ) a b + a b a b + a b a1 b2 + a2 b1 a2 b + a1 b a2 b3 + a3 b2 23 + 32 a3 b1 + a1 b3 ≡ 31 + 13 ≡ (23 + 32, 31 + 13, 12 + 21) a1 b2 + a2 b1 12 + 21 ❚❤➭♥❤ ♣❤➬♥ ❜✐Õ♥ ➤æ✐ ♥❤➢ ✱ t❛ ✈✐Õt✿ ∼ (23 + 32, 31 + 13, 12 + 21) a2 b − a3 b 6P3 X = (|23 |31 |12 ) a b − a b + (|32 a1 b − a2 b a2 b3 − a3 b2 a3 b − a2 b a3 b1 − a1 b3 a1 b3 − a3 b1 a1 b2 − a2 b1 a2 b − a1 b ❈➳❝ t❤➭♥❤ ♣❤➬♥ ✈➭ ✭❈✳✽✮ a3 b2 − a2 b3 |13 |21 ) a b − a b a2 b1 − a1 b2 ❜✐Õ♥ ➤æ✐ ♥❤➢ 3 ∼ (23 − 32, 31 − 13, 12 − 21) ✱ ✭❈✳✾✮ ❑Õt ❤ỵ♣ ✭❈✳✺✮✱ ✭❈✳✻✮✱ ✭❈✳✼✮✱ ✭❈✳✽✮ ✈➭ ✭❈✳✾✮ t❛ t❤✉ ➤➢ỵ❝✿ ⊗ = 1(11 + 22 + 33) ⊕ 2(11 + ω 22 + ω33, 11 + ω22 + ω 33) ✭❈✳✶✵✮ ⊕ 3s (23 + 32, 31 + 13, 12 + 21) ⊕ 3a (23 − 32, 31 − 13, 12 − 21), ❍♦➭♥ t♦➭♥ t➢➡♥❣ tù✱ ❝❤ó♥❣ t➠✐ tÝ♥❤ ➤➢ỵ❝ ❝➳❝ ❤Ư sè ❈❧❡❜s❝❤✲ ●♦r❞❛♥ ❝❤♦ ❝➳❝ tÝ❝❤ t❡♥s♦r ❝ñ❛ ♥❤ã♠ S4 ♥❤➢ tr♦♥❣ ✭✶✳✶✼✮✳ ①✐✈ P❤ơ ❧ơ❝ ❉ ❈➳❝ sè ❧➢ỵ♥❣ tư ❝đ❛ ♠➠ ❤×♥❤ ✸✸✶◆❋ S4 ❈❤ó♥❣ t➠✐ ❧✐Ưt ❦➟ sè ❧❡♣t♦♥ L ✱ ✈➭ ➤è✐ ①ø♥❣ ❧❡♣t♦♥ ❤×♥❤ ✸✸✶◆❋ ❞ù❛ tr➟♥ ♥❤ã♠ ➤è✐ ①ø♥❣ S4 Pl ✱ ❝ñ❛ ❝➳❝ ❤➵t tr♦♥❣ ♠➠ ✭❝➳❝ ❝❤Ø sè t❤Õ ❤Ư ➤➢ỵ❝ ❜á q✉❛✮✿ L ❈➳❝ ❤➵t − − + 0 0 0 NR u d φ+ φ1 φ2 φ2 η1 η1 η2 η2 χ3 σ33 s33 ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✵ Pl ✶ + + + + 0 0∗ 0 νL l U D∗ φ+ φ3 η3 η3 χ1 χ2 σ13 σ23 s13 s23 −1 −1 ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ++ + ++ 0 σ12 σ22 s11 s+ σ11 12 s22 ✱ ✱ ✱ ✱ ①✈ ✱ ✱ ✱ ✱ −2 ✶ Pụ ụ số ợ tử ủ ì ✸✸✶◆❋ S3 ✈➭ ✸✸✶❘❍ S3 ▼➠ ❤×♥❤ ✸✸✶◆❋S3 L Pl ✵ ✶ + + + + 0 χ0∗ χ2 s13 s23 s13 s23 ρ1 ρ2 −1 ✲✶ + ++ ++ ρ+ s011 s+ 12 s22 s11 s12 s22 −2 ✶ ❈➳❝ ❤➵t L Pl νL νR l ✲✶ ✵ ✶ −2 ✶ ❈➳❝ ❤➵t + 0 u d NR W Z φ+ φ2 φ1 φ2 ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ η10 η2− η10 η2− χ03 s033 s33 ✱ ✱ ✱ ✱ ✱ ✱ + 0 νL∗ l∗ U D∗ X 0∗ Y + φ+ φ3 η3 η3 ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ▼➠ ❤×♥❤ ✸✸✶❘❍S3 ✱ ✱ + 0 u d W Z φ+ φ2 φ1 φ2 ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ + η10 η2− η10 η2− χ03 s013 s+ 23 s13 s23 ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ + + 0 0∗ U D∗ X 0∗ Y + φ+ φ3 η3 η3 χ1 χ2 ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ ✱ + ++ + ++ 0∗ 0 s0∗ 33 s33 s11 s12 s22 s11 s12 s22 ✱ ✱ ✱ ✱ ✱ ①✈✐ ✱ ✱ ✱ ... 21) ⊕ 3a ( 23 − 32 , 31 − 13 , 12 − 21) , ⊗ = 1( 11 + 22 + 33 ) ⊕ 2 (11 + ω 22 + ? ?33 , 11 + ω22 + ω 33 ) ⊕ 3s ( 23 + 32 , 31 + 13 , 12 + 21) ⊕ 3a ( 23 − 32 , 31 − 13 , 12 − 21) , ⊗ = (11 + 22 + 33 ) ⊕ 2 (11 + ω... ⊗ = 3( 11 , 12 , 13 ) , ⊗ = (11 , 12 , 13 ) , ✭✶✳✶✹✮ ⊗ = (11 , 12 , 13 ) , ⊗ = 3( 11 , 12 , 13 ) , ⊗ = 1( 12 + 21) ⊕ (12 − 21) ⊕ 2(22, 11 ), ⊗ = (1 + 2 )1, ω (1 + ω2)2, ω (1 + ω 2 )3 ⊕ (1 − 2 )1, ω (1 − ω2)2, ω (1 −... ψL ≡ ? ?1, 2,3L = ? ?1, 2,3L , l1,2,3L , N1,2,3R T ∼ [3, ? ?1 /3, 2 /3, 3] , l1R ∼ [1, ? ?1, 1, 1] , lR ≡ l2,3R ∼ [1, ? ?1, 1, 2], Q3L = (u3L d3L UL )T ∼ [3, 1 /3, ? ?1 /3, 1] , QL ≡ Q1,2L = (d1,2L , −u1,2L , D1,2L