✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ✣➄◆● ✣Ù❈ ❚❘➚◆❍ P❍❹◆ ❚➑❈❍ ❇❘❯❍❆❚ ❱⑨ Ù◆● ❉Ö◆● ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆ ❍➴❈ ❍⑨ ◆❐■ ✲ ✷✵✶✶ ✣❸■ ❍➴❈ ◗❯➮❈ ●■❆ ❍⑨ ◆❐■ ❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈ ❚Ü ◆❍■➊◆ ✣➄◆● ✣Ù❈ ❚❘➚◆❍ P❍❹◆ ❚➑❈❍ ệ số ỵ t❤✉②➳t sè ▼➣ sè✿ ✻✵✹✻✵✺ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❑❍❖❆ ❍➴❈ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈ P●❙✳ ❚❙✳ ▲➊ ▼■◆❍ ❍⑨ ▼ư❝ ❧ư❝ ▲í✐ ❝↔♠ ì♥ ▲í✐ ♥â✐ ✤➛✉ ❇↔♥❣ ❦➼ ❤✐➺✉ ✵ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✐ ✐✐ ✐✈ ✶ ✵✳✶ P❤➨♣ t❤➳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✵✳✷ ❚ø ✈➔ ♥❤â♠ ❝→❝ ♣❤➨♣ t❤➳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻ ✵✳✸ ◆❤â♠ t✉②➳♥ t➼♥❤ tê♥❣ q✉→t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✵✳✹ P❤↕♠ trò✱ ❤➔♠ tû ✶ P❤➙♥ t➼❝❤ ❇r✉❤❛t ✶✳✶ P❤➨♣ t❤➳ ❏♦r❞❛♥ ✶✳✷ ◆❣➠♥ ❙❝❤✉❜❡rt ✶✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✶✳✸ P❤➙♥ t➼❝❤ ❇r✉❤❛t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻ ✶✳✹ ◆❤â♠ ❝♦♥ P❛r❛❜♦❧✐❝ ✸✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ Ù♥❣ ❞ö♥❣ ✸✼ ✷✳✶ ✣↕✐ sè ■✇❛❤♦r✐✲❍❡❝❦❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ✷✳✷ ▼æ✤✉♥ ❙t❡✐♥❜❡r❣ ✹✽ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❑➳t ❧✉➟♥ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ✺✹ ✺✺ ✈✐ ❈❤÷ì♥❣ ✵ ❑✐➳♥ t❤ù❝ ❝❤✉➞♥ ❜à ✵✳✶ P❤➨♣ t❤➳ ❈❤÷ì♥❣ ♥➔② ❞➔♥❤ ✤➸ tr➻♥❤ ❜➔② ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝➛♥ t❤✐➳t s➩ ✤÷đ❝ ❞ị♥❣ ✤➳♥ tr♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✳ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ tỉ✐ ❝❤õ ②➳✉ tr➻♥❤ ❜➔② t❤❡♦ ◆✳ P✳ ❙tr✐❝❦❧❛♥❞ ❬✶✶❪ ✈➔ ◆❣✉②➵♥ ❍✳❱✳ ❍÷♥❣ ❬✺❪✳ ●✐↔ sû tø T T ❧➔ ♠ët t➟♣ ❤ñ♣ ❤ú✉ ❤↕♥ ♥➔♦ ✤â✳ ❚➟♣ ❤ñ♣ S(T ) t➜t ❝↔ ❝→❝ s♦♥❣ →♥❤ ✈➔♦ ❝❤➼♥❤ ♥â ❝ị♥❣ ✈ỵ✐ ♣❤➨♣ ❤đ♣ t❤➔♥❤ →♥❤ ①↕ ❧➟♣ t❤➔♥❤ ♠ët ♥❤â♠✳ P❤➛♥ tû ✤ì♥ ✈à S(T ) ỗ t idT ✣à♥❤ ♥❣❤➽❛ ✵✳✶✳ ◆❤â♠ S(T ) ✤÷đ❝ ❣å✐ ❧➔ ♥❤â♠ ✤è✐ ①ù♥❣ tr➯♥ t➟♣ ❤ñ♣ T ✳ ▼é✐ ♥❤â♠ ❝♦♥ ❝õ❛ S(T ) ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♥❤â♠ ❝→❝ ♣❤➨♣ t❤➳ tr➯♥ T ◆➳✉ T = {1, , n} t❤➻ ♥❤â♠ S(T ) ✤÷đ❝ ❣å✐ ❧➔ ♥❤â♠ ✤è✐ ①ù♥❣ tr➯♥ n ♣❤➛♥ tû ✈➔ ✤÷đ❝ ❦➼ ❤✐➺✉ ❧➔ Σn ▼ët ♣❤➛♥ tû ❝õ❛ Σn ✤÷đ❝ ❣å✐ ❧➔ ♠ët ♣❤➨♣ t❤➳ tr➯♥ n ♣❤➛♥ tû✳ ✣à♥❤ ♥❣❤➽❛ ✵✳✷✳ ❈❤♦ ♣❤➨♣ t❤➳ σ ∈ Σ , ❦➼ ❤✐➺✉ L(σ) ❧➔ t➟♣ ❝→❝ ♥❣❤à❝❤ t❤➳ ❝õ❛ n ♥â ✈➔ σ(i) > σ(j)}, ❱➔ (σ) = |L(σ)|, ❣å✐ ❧➔ ✤ë ❞➔✐ ❝õ❛ ♣❤➨♣ t❤➳ σ✳ ❑➼ ❤✐➺✉ L+(σ) ✤÷đ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉ L(σ) = {(i, j) | < i < j ≤ n L+ (σ) = L(σ) {(i, i) | < i ≤ n} = {(i, j) | < i ≤ j ≤ n ✈➔ σ(i) ≥ σ(j)} ✶ ❱➼ ❞ö ✵✳✸✳ ❳➨t ♣❤➨♣ t❤➳ σ = (135)(24)✳ ❚❛ ❝â L(σ) = {(1, 4), (1, 5), (2, 4), (2, 5), (3, 4), (3, 5), (4, 5)} ❙✉② r❛ (σ) = 7✳ ✣æ✐ ❦❤✐ ❝❤♦ t✐➺♥ tr♦♥❣ ✈✐➺❝ sû ❞ö♥❣ t❛ ✤➦t L(σ) = {{i, j} | (i, j) L()} L() t ủ ỗ tt ❝→❝ t➟♣ ❝♦♥ ❝â ♣❤➛♥ tû T ❝õ❛ t➟♣ {1, , n} σ : T → σT ❧➔ ♥❣❤à❝❤ t❤➳✳ ❱➻ i < j ✈ỵ✐ ♠å✐ (i, j) ∈ L(σ)✱ t❛ ❝â ♠ët s♦♥❣ →♥❤ ◆❤÷ ✈➟②✱ ✈➔ h : L(σ) −→ L(σ) {i, j} −→ (i, j) ❞♦ õ |L()| = () ú ỵ t→❝ ✤ë♥❣ ❧➯♥ Z[x1 , , xn ] ❜➡♥❣ ❝→❝❤ ❤♦→♥ ✈à ❧↕✐ x1 , , xn ❚❛ ①➨t ♣❤➛♥ tû σ ∈ Σn , ❦❤✐ ✤â σ (xi − xj ) ∈ Z[x1 , , xn ] Σn = i l k>l ❤❛② {i, j} ∈ L(τ ) τ∗−1 L(σ)✳ t❤➻ {i, j} ∈ L(τ )✳ ✈➔ σ(k) > σ(l) ❚❛ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤ ♥➯♥ t❛ ❝â {k, l} ∈ L(σ)✳ {i, j} ∈ ▼➦t ❦❤→❝ t❛ ❝â τ∗−1 {k, l} = {τ −1 (k), τ −1 (l)} = {i, j}, ♥➯♥ {i, j} ∈ τ∗−1 L(σ) ❤❛② {i, j} ∈ L(τ ) τ∗−1 L(σ)✳ ❱➟② tr♦♥❣ ❝↔ ❤❛✐ tr÷í♥❣ ❤đ♣ t❛ ✤➲✉ ❝â L(στ ) ⊆ L(τ ) τ∗−1 L(σ) ❚✐➳♣ t❤❡♦ t❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ L(τ ) τ∗−1 L(σ) ⊆ L(στ ) ❚❤➟t ✈➟② ♥➳✉ {i, j} ∈ L(τ ) τ∗−1 L(σ), ❦❤✐ ✤â t❛ ❝ơ♥❣ ❝â ❤❛✐ tr÷í♥❣ ❤đ♣✳ ❚r÷í♥❣ ❤đ♣ ✿ ◆➳✉ ✈➔ τ (i) > τ (j)✳ ♥➯♥ {i, j} ∈ L(τ ) t❤➻ {i, j} ∈ τ∗−1 L(σ)✳ ❱➻ {i, j} ∈ L(τ ) ♥➯♥ i < j ▼➦t ❦❤→❝ t❛ ❝â {τ (i), τ (j)} ∈ L(σ) ❤❛② i = τ −1 τ (i) στ (i) > στ (j)✳ ✸ ✈➔ j = τ −1 τ (j) ❚ø ✤â s✉② r❛ ✱ ♠➔ {i, j} ∈ τ∗−1 L(σ) {i, j} ∈ L(στ )✳ ❚r÷í♥❣ ❤đ♣ {i, j} ∈ L(τ ) t❤➻ {i, j} ∈ τ∗−1 L(σ)✳ ❱➻ {i, j} ∈ L(τ ) ♥➯♥ i < j ✿ ◆➳✉ ✈➔ τ (i) < τ (j)✳ ♥➯♥ t❛ ❝â ▼➦t ❦❤→❝ t❛ ❝â {τ (i), τ (j)} ∈ L(σ) i = τ −1 τ (i) tù❝ ❧➔ ✈➔ j = τ −1 τ (j)✱ στ (i) > στ (j)✳ ♠➔ {i, j} ∈ τ∗−1 L(σ) {i, j} ∈ L(στ )✳ ❚ø ✤â s✉② r❛ ❚r♦♥❣ ❤❛✐ tr÷í♥❣ ❤ñ♣✱ t❛ ✤➲✉ ❝â L(τ ) τ∗−1 L(σ) ⊆ L(στ ) ❇ê ✤➲ ✵✳✽✳ ◆➳✉ σ(k) < σ(k + 1) t❤➻ σ(k + 1) t❤➻ (σsk ) = (σ) − 1✳ ❈❤ù♥❣ ♠✐♥❤✳ (σsk ) = (σ) + 1✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ σ(k) > L(sk ) = {{k, k + 1}}✱ ❦➨♦ t❤❡♦ (σ) − ♥➳✉ {k, k + 1} ∈ s−1 L(σ) k∗ (σsk ) = (σ) + tr♦♥❣ ❝→❝ tr÷í♥❣ ❤đ♣ ❦❤→❝ ❱➻ sk∗ {k, k + 1} = {k, k + 1}✱ t❛ ❝â {k, k + 1} ∈ s−1 k∗ L(σ) ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ {k, k + 1} L() õ tữỡ ữỡ ợ (k) > σ(k + 1)✳ ❱➻ ❙û ❞ö♥❣ ❝→❝ ❜ê ✤➲ tr➯♥✱ t❛ ❝â ♠➺♥❤ ✤➲ s❛✉✳ ▼➺♥❤ ✤➲ ✵✳✾✳ ♣❤➨♣ t❤➳ ❝➜♣✳ ❈❤ù♥❣ ♠✐♥❤✳ (σ) ◆➳✉ ❧➔ sè ♥❤ä ♥❤➜t s❛♦ ❝❤♦ σ ❝â t❤➸ ✈✐➳t ✤÷đ❝ t❤➔♥❤ ❝õ❛ σ = si1 sir ◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû t❤➻ tø ❇ê ✤➲ ✵✳✼✱ t❛ ❝â (σ) ≤ r✳ (σ) = r✱ t❛ sû ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣ t❤❡♦ r ♠✐♥❤ ♠➺♥❤ ✤➲✳ ❚❤➟t ✈➟②✱ ♥➳✉ r=0 σ t ự t tự tỹ ỗ ♥❤➜t ♥➯♥ ♠➺♥❤ ✤➲ ✤ó♥❣✳ ●✐↔ sû ♠➺♥❤ ✤➲ ✤ó♥❣ ✈ỵ✐ ♠å✐ ♣❤➨♣ t❤➳ ❝â ✤ë ❞➔✐ ♥❤ä ❤ì♥ ♣❤↔✐ ✤✐ ❝❤ù♥❣ ♠✐♥❤ ♠➺♥❤ ✤➲ ✤ó♥❣ ✈ỵ✐ ♣❤➨♣ t❤➳ ❝â ✤ë ❞➔✐ ❜➡♥❣ r>0 t❤➻ σ (σ) ♣❤↔✐ ❝â ➼t t ởt t tỗ t k r r✱ t❛ ❚❤➟t ✈➟② ✈ỵ✐ σ(k) > σ(k + 1)✳ ❇ê (σsk ) = r − ♥➯♥ t❤❡♦ ❣✐↔ t❤✐➳t q✉② ♥↕♣ t❤➻ σsk = si1 sir −1 ✈ỵ✐ ❝→❝ ♣❤➨♣ t❤➳ ❝➜♣ si1 , , sir −1 ♥➔♦ ✤â✳ ❚ø ✤â s✉② r❛ σ = si1 sir −1 sk ✳ ✤➲ ✵✳✽ ♥â✐ r➡♥❣ ✣à♥❤ ♥❣❤➽❛ ✵✳✶✵✳ ❳➨t ♠ët tø w = s (π(w)) = r ❑➼ ❤✐➺✉ t❤➻ t❛ ♥â✐ w ❧➔ tø rót ❣å♥✳ ρ ❧➔ ♣❤➨♣ t❤➳ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ i (j) ♥❣❤➽❛ ✵✳✶✶✳ ❈❤♦ ≤ m < k ≤ n − 1✱ ❦➼ ❤✐➺✉ t = sm sm+1 sk−1 ✳ k m ❉➵ t❤➜② tkm = (m, m + 1, , k − 1, k)✱ tm m = id ✤➦t ▼➺♥❤ ✤➲ ✵✳✶✷✳ ❱ỵ✐ ộ tỗ t t ởt ❞➣② m , m , , m ❝→❝ n sè tü ♥❤✐➯♥ t❤ä❛ ♠➣♥ ≤ mk ≤ n ✈➔ n σ = tnmn tn−1 mn−1 tm2 tm1 ❍ì♥ ♥ú❛✱ n (k − mk ) (σ) = k=1 ❈❤ù♥❣ ♠✐♥❤✳ τ ♥❤÷ ♠ët ♣❤➛♥ t❤❡♦ n✳ mn = σ(n) ✈➔ τ = (tnmn )−1 σ ✳ ❑❤✐ ✤â τ (n) = n✱ ♥➯♥ ❝â t❤➸ ①❡♠ tû ❝õ❛ Σn−1 ✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❜ê ✤➲ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣ ✣➦t ❚❤➟t ✈➟② ✈ỵ✐ n=1 t❤➻ ❤✐➸♥ ♥❤✐➯♥ ✤ó♥❣✳ ●✐↔ sû ✤ó♥❣ ✈ỵ✐ n − 1✱ t❛ ❝â n−1 τ = tm t2m2 t1m1 n−1 n−1 ✈ỵ✐ m1 , m2 , , mn−1 ♥➔♦ ✤â ✈➔ (k − mk )✳ (τ ) = ❚ø ✤â s✉② r❛ k=1 σ = tnmn τ = tnmn tn−1 mn−1 tm2 tm1 ❚❛ ✈➝♥ ❝â L(tnmn ) = {{i, n} | mn ≤ i < n} ♥➯♥ τ∗−1 L(tnmn ) = {{τ −1 (i), n} | mn ≤ i < n} Ð ✤➙② L(τ ) ❦❤æ♥❣ ❝❤ù❛ ❝➦♣ ❞↕♥❣ {τ −1 (i), n}✳ ◆❣❤➽❛ ❧➔ L(τ ) ✈➔ τ∗−1 L(tnmn ) ❧➔ rí✐ ♥❤❛✉ ✈➔ (σ) = |L(τ )| + |τ∗−1 L(tnmn )| = (τ ) + (n − mn ) n (k − mk ) = k=1 ❉♦ σ(n) ❧➔ t tỗ t t m1 , m2 , , mn ♥➯✉✳ ❱➼ ❞ö ✵✳✶✸✳ ◆➳✉ σ = (135)(24) t❤➻ σ = t t t t t ✱ t❤➟t ✈➟②✱ t❛ ❝â 1 t51 t41 t33 t22 t11 = (12345)(1234) = (135)(24) = σ ✺ ♥❤÷ ❜ê ✤➲ ✤➣ ✵✳✷ ❚ø ✈➔ ♥❤â♠ ❝→❝ ♣❤➨♣ t❤➳ ✣à♥❤ ♥❣❤➽❛ ✵✳✶✹✳ ✣➦t Σ˜ ❧➔ ♥❤â♠ tü ❞♦ s✐♥❤ ❜ð✐ s , , s ✈ỵ✐ ❝→❝ q✉❛♥ ❤➺ n n si = 1, si si+1 si = si+1 si si+1 ✈➔ si sj = sj si ♥➳✉ |i − j| > ❑✐➸♠ tr❛ trü❝ t✐➳♣✱ t❛ t❤➜② ❝→❝ t÷ì♥❣ ù♥❣ q✉❛♥ ❤➺ tr ụ ữủ tọ tr n tỗ t ỗ n n : si ✈➔♦ si ❚❛ ❝ơ♥❣ s➩ sû ❞ư♥❣ ❦➼ ❤✐➺✉ tkm = sm sm+1 sk tr♦♥❣ ˜n Σ ♥❤÷ tr♦♥❣ Σn ✳ ▼➺♥❤ ✤➲ ✵✳✶✺✳ ε : Σ˜ ❈❤ù♥❣ ♠✐♥❤✳ n −→ Σn ❧➔ ♠ët ✤➥♥❣ ❝➜✉✳ n−1 ˜ n ❧➔ t➟♣ ❝→❝ ♣❤➛♥ tû ❝â ❞↕♥❣ tnm tm t2m2 t1m1 ✈ỵ✐ Xn ⊆ Σ n n−1 ˜ n −→ Σn ❧➔ t♦➔♥ ≤ mk ≤ k ✳ ❚❛ ❝â |Xn | ≤ n!✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✵✳✶✷ t❤➻ ε|Xn : Σ ˜ n ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ q✉② ♥↕♣✳ →♥❤✱ ❞♦ ✤â ❧➔ s♦♥❣ →♥❤✳ ự Xn = t n ú ỵ r➡♥❣ tnm Xn−1 ✳ Xn = ●✐↔ sû ˜ n−1 ✱ Xn−1 = Σ t❛ ❝â Xn ✤â♥❣ ✈ỵ✐ ♣❤➨♣ Xn ✤â♥❣ ✈ỵ✐ ♣❤➨♣ m=1 ♥❤➙♥ ❜➯♥ ♣❤↔✐ ❜ð✐ ♥❤➙♥ ❜➯♥ ♣❤↔✐ ❜ð✐ ✈➔ ❞♦ ˜ n−1 Σ ˜ n−1 ✳ Σ sn−1 ✳ ◆➳✉ ˜n Σ s✐♥❤ ❜ð✐ ˜ n−1 Σ ✈➔ sn−1 t❤➻ ✣✐➲✉ ♥➔② ✤÷đ❝ s✉② r❛ trỹ t tứ ữợ ✈ỵ✐ tkm ✈ỵ✐ ♠å✐ k < n − 1✳ ❇ê ✤➲ ✵✳✶✻✳ ◆➳✉ k ≤ n ✈➔ l < n t❤➻ tr♦♥❣ Σ˜ t❛ ❝â n tnk tn−1 sn−1 = tnk tnl = l tn tn−1 l+1 l tn tn−1 l k−1 ✻ ✈ỵ✐ k ≤ l ✈ỵ✐ k > l ❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû ❚rü❝ t✐➳♣ tø ✤à♥❤ ♥❣❤➽❛ t❛ ❝â l < n✱ tnk tln−1 sn−1 = tnk tnl ✳ t❛ ❝â (tnl+1 )−1 tnl = tln1 (tnl )1 rữợ t t t ❦✐➸♠ tr❛ ✈ỵ✐ l=5 ✈➔ n = 10✳ ❱✐➳t k t❤❛② ❝❤♦ sk ✈➔ ❦➼ ❤✐➺✉ e ❧➔ →♥❤ ①↕ ỗ t õ 10 (t10 ) t5 = 987656789 (✈➻ 656 = 565) = 987565789 (✈➻ [5, 7] = [5, 8] = [5, 9] = e) = 598767895 (✈➻ 767 = 676) = 598676895 (✈➻ [6, 8] = [6, 9] = e) = 5698787965 (✈➻ 878 = 787) = 567989765 (✈➻ [7, 9] = e) = 567898765 (✈➻ 989 = 898) −1 = t95 (t10 ) ❚ê♥❣ q✉→t ❦❤✐ q✉② ♥↕♣ t❤❡♦ l = n−1 l✳ t❤➻ ✤➥♥❣ t❤ù❝ ✤ó♥❣✳ ◆➯♥ t❛ ❝â t❤➸ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ❳➨t q✉❛♥ ❤➺ sl+1 sl sl+1 = sl sl+1 sl ❚❛ ♥❤➙♥ ❜➯♥ ♣❤↔✐ ✈ỵ✐ tnl+2 ✈➔ ❜➯♥ tr→✐ ✈ỵ✐ (tnl+1 )−1 ✳ ❱➻ sl+1 tnl+2 = tnl+1 (tnl+2 )−1 sl+1 = (tnl+1 )−1 ✈➔ sl sl+1 tnl+2 = tnl ▲↕✐ ❝â sl ❣✐❛♦ ❤♦→♥ ✈ỵ✐ tnl+2 ♥➯♥ (tnl+2 )−1 sl+1 sl sl+1 tnl+2 = (tnl+1 )−1 tnl ❚ø ✤â s✉② r❛ (tnl+1 )−1 tnl = sl (tnl+2 )−1 sl+1 tnl+2 sl ❙û ❞ö♥❣ q✉❛♥ ❤➺ sl+1 tnl+2 = tnl+1 ✈➔ ❣✐↔ t❤✐➳t q✉② ♥↕♣ t❛ ❝â n−1 n sl (tnl+2 )−1 sl+1 tnl+2 sl = sl tl+1 (tl+1 )−1 sl ✼ ♥➯♥ t❛ ❝â Ui+1 /Ui−1 ∼ = F2p ✳ ❣✐❛♥ ❝♦♥ ♠ët ❝❤✐➲✉ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❞♦ ✤â Ti (a) = pa ❱➻ ✈➟② t❛ ❝â |F | = p + ✈➔ ❱➟② t❛ ❝â (Ti − p)(Ti + 1)[W ] = (Ti − p)(a) = ✣à♥❤ ♥❣❤➽❛ ✷✳✼✳ ✣➦t eˆ : Z −→ Z(p) [F lag(V)], (p) [F lag(V)] t❤ù❝ eˆ[U ] = |F lag(V)|−1 ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ [W ] W F lag(V) ú ỵ r e[U ] ✤ë❝ ❧➟♣ ✈ỵ✐ U✳ ❚ø ✤à♥❤ ♥❣❤➽❛ t❛ t❤➜② eˆ ∈ H ❚❛ ✤à♥❤ ♥❣❤➽❛ →♥❤ ①↕ ξˆ : H −→ Z(p) , ①→❝ ✤à♥❤ ❜ð✐ ˆ ξ( nσ p (σ) nσ Tσ ) = σ∈Σn σ∈Σn ▼➺♥❤ ✤➲ ✷✳✽✳ eˆ ❧➔ ♠ët ❧ơ② ✤➥♥❣ t➙♠ tr♦♥❣ H, ✈ỵ✐ eˆ t = eˆ✳ ˆ t ) = ξ(a) ˆ ˆ e ✈ỵ✐ ♠å✐ a ∈ H ❝➜✉ ✈➔♥❤ ✈ỵ✐ ξ(a ✈➔ aˆe = eˆa = ξ(a)ˆ ❈❤ù♥❣ ♠✐♥❤✳ ❳➨t V ∈ V ✱ ❦➼ ❤✐➺✉ x = eˆ[U ] = |F lag(V)|1 ỡ ỳ ỗ [W ] ∈ Z(p) [F lag(V)] W ∈F lag(V) ✈➔ eˆ[W ] = x ✈ỵ✐ ♠å✐ W ∈ F lag(V) ❚❤❡♦ ✣à♥❤ ♥❣❤➽❛ ✷✳✼ t❛ ❝â eˆ(x) = eˆ(|F lag(V)|−1 [W ]) W ∈F lag(V) = (|F lag(V)|−1 eˆ[W ]) W ∈F lag(V) (|F lag(V)|−1 = (|F lag(V)|−1 W ∈F lag(V) = (|F lag(V)|−1 U ∈F lag(V) (|F lag(V)|−1 (|F lag(V)|[W ]) W ∈F lag(V) = |F lag(V)|−1 [U ]) [W ] W ∈F lag(V) = x ✹✷ ❙✉② r❛ eˆ(x) = x ✈➔ ❞♦ ✤â eˆ2 = eˆ✳ ▼➦t ❦❤→❝ t❛ ❝â < eˆ[U ], [W ] > =< |F lag(V)|−1 [W ], [W ] > W ∈F lag(V) = |F lag(V)|−1 < [W ], [W ] > W ∈F lag(V) = |F lag(V)|−1 ✈➔ < [U ], eˆ[W ] > =< [U ], |F lag(V)|−1 [U ] > U ∈F lag(V) = |F lag(V)|−1 < [U ], [U ] > U ∈F lag(V) = |F lag(V)|−1 ❉♦ ✤â < eˆ[U ], W >=< [U ], eˆ[W ] >, ∀ U , W ∈ F lag(V) ◆❤÷ ✈➟② eˆ ❧➔ tü ❧✐➯♥ ❤ñ♣✱ eˆt = eˆ ❚✐➳♣ t❤❡♦ t❛ t❤➜② eˆTσ [U ] = eˆ[W ] δ(U ,W )=σ = |{W | δ(W , U ) = σ −1 }|x = p (σ −1 ) x = p (σ) eˆ[W ] ❙✉② r❛ eˆTσ = p (σ) eˆ ❍ì♥ ♥ú❛ ✈ỵ✐ a∈H ❜➜t ❦ý✱ a= nσ Tσ , t❤➻ σ∈Σn eˆa = σ∈Σn ❉♦ ✤â ˆ e eˆa = ξ(a)ˆ ▼➦t ❦❤→❝ ◆➳✉ a, b σ∈Σn ˆ eb = ξ(a) ˆ ξ(b)ˆ ˆ e eˆab = (ˆ ea)b = ξ(a)ˆ = Tσ−1 ♥➯♥ ˆ H t❤➻ eˆab = ξ(ab)ˆ e ˆ ˆ ˆ ξ(ab) = ξ(a)ξ(b), ♥➯♥ ξˆ ❧➔ ❧➔ ❤❛✐ ♣❤➛♥ tû ❜➜t ❦ý tr♦♥❣ ❉♦ ✤â ˆ t )ˆ a ❜ð✐ at t❤➻ t❛ ❝ô♥❣ ❝â at eˆ = ξ(a e✳ ❍ì♥ ♥ú❛ t❤❡♦ t ˆ ˆ ξ(a ) = ξ(a)✳ ❱➟② aˆ e = eˆa ❤❛② eˆ ụ t ỗ t Tσ nσ p (σ) eˆ eˆ(nσ Tσ ) = ✹✸ ỗ t H = End ✱ t❛ ❝â (Z(p)[G] (Z(p) [G/B]) Tσ−1 [gB] = Tt [gB] = [gxB] xX() ự rữợ t t ❇ê ✤➲ ✷✳✸✱ Tσ−1 [U ] = [W ], δ(U ,W )=σ −1 ♥➯♥ ✈ỵ✐ ♠å✐ g∈G t❛ ❝â gTσ−1 [U ] = [gW ] δ(U ,W )=σ −1 = [gW ] δ(gU,gW )=σ −1 = Tσ−1 [gU ] ❚ø ✤â t❛ ❝â I=∅ Tσ−1 ❧➔ ♠ët G✲→♥❤ ①↕✳ ▼➦t ❦❤→❝✱ t❤❡♦ ❇ê ✤➲ ✶✳✸✹✱ tr♦♥❣ tr÷í♥❣ ❤đ♣ t❛ ❝â G/B = F lag(V) ✭q✉❛ t÷ì♥❣ ù♥❣ gB −→ gE ✮ ▼➦t ❦❤→❝✱ Tσ−1 [gB] = Tσ−1 [gE] = [W ] δ(gE,W )=σ −1 [W ] = δ(W ,gE)=σ [W ] = δ(g −1 W ,E)=σ ❇➯♥ ❝↕♥❤ ✤â t❤❡♦ ▼➺♥❤ ✤➲ ✶✳✶✼ t❛ ❝â x ∈ X(σ) s❛♦ ❝❤♦ g −1 W = xσE ✳ ❙✉② r❛ δ(g −1 W , E) = σ W = gxσE Tσ−1 [gB] = Tt [gB] = tỗ t t õ [gxB] xX() ú ỵ ợ ộ ❝ì sð ❝õ❛ V ✤÷đ❝ ❝♦✐ ♥❤÷ ♠ët ✤➥♥❣ ❝➜✉ tø F ✈➔♦ V, ♥➯♥ ♠é✐ ♣❤➛♥ tû g ∈ G = Aut(Fnp) ❝❤♦ t❛ ♠ët ✤➥♥❣ ❝➜✉ ∗ ∗ ∗ (gh) = h g ✹✹ n p g ∗ : Base −→ Base, ✈ỵ✐ ❍➺ q✉↔ ✷✳✶✶✳ ❚❛ õ ỗ Z(p) [Base] Z(p) [F lag] T t (xσ)∗ σ x∈X(σ) π Z(p) [Base] −−−→ Z(p) [F lag] tr♦♥❣ ✤â π ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ π : Base −→ F lag π(v1 , , ) −→ (0 < Fp {v1 } < · · · < Fp {v1 , , } = V) ❈❤ù♥❣ ♠✐♥❤✳ [V, A] = {Z(p) [G]✲♠æ✤✉♥} ❙✉② r❛ ✤✐➲✉ ❚ø ❇ê ✤➲ ✷✳✾✱ ✈➔ ✤➥♥❣ t❤ù❝ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ❍➺ q✉↔ ✷✳✶✷✳ ❈❤♦ ❤❛✐ ♣❤➨♣ t❤➳ σ, τ ∈ Σ ✱ ♥➳✉ n (στ ) = (σ) + (τ ) t❤➻ Tστ = Tσ Tτ ❈❤ù♥❣ ♠✐♥❤✳ ❱➻ t Tστ = T(στ )−1 = Tτ −1 σ−1 , ♥➯♥ t❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ t Tτt Tσt [gB] = Tστ [gB] ❚❤❡♦ ❇ê ✤➲ ✷✳✾ t❛ ❝â t Tστ [gB] = [gzστ B] z∈X(στ ) ▼➦t ❦❤→❝ tø ▼➺♥❤ ✤➲ ✶✳✷✶ ✈ỵ✐ ♠é✐ X(σ), y ∈ X(τ ) z ∈ X(στ ) t❤➻ ♥➯♥ [gxσyσ −1 στ B] [gzστ B] = z∈X(στ ) x∈X(σ) y∈X(τ ) = [gxσyτ B] x∈X(σ) y∈X(τ ) Tτt [gxσB] = x∈X(σ) = Tτt Tσt [gB] ❱➟② t❛ ♥❤➟♥ ✤÷đ❝ t [gB], Tτt Tσt [gB] = Tστ ❤❛② z = xσyσ −1 Tστ = Tσ Tτ ✹✺ tr♦♥❣ ✤â x ∈ ❍➺ q✉↔ ✷✳✶✸✳ ◆➳✉ s ❈❤ù♥❣ ♠✐♥❤✳ Σn ✳ ❱➻ i1 sir si1 sir ❧➔ ♠ët tø rót ❣å♥ ❝õ❛ σ t❤➻ Ti ❧➔ tø rót ❣å♥ ❝õ❛ σ ♥➯♥ Tir = Tσ σ = si1 sir ✈ỵ✐ si1 , , s ir ∈ ❉♦ ✤â t❤❡♦ ❍➺ q✉↔ ✷✳✶✷ t❛ ❝â Ti1 Tir = Tσ ▼➺♥❤ ✤➲ ✷✳✶✹✳ H ❧➔ ✈➔♥❤ s✐♥❤ ❜ð✐ ❝→❝ ♣❤➛♥ tû T tr➯♥ Z ✱ ✈➔ ❝→❝ q✉❛♥ ❤➺ i (i) Ti2 (p) = p + (p − 1)Ti , (ii) Ti Ti+1 Ti = Ti+1 Ti Ti+1 , (iii) Ti Tj = Ti Tj i > j ú ỵ r q t✐➯♥ ❝â t❤➸ ✈✐➳t ❧➔ ❈❤ù♥❣ ♠✐♥❤✳ σ (Ti + 1)(Ti − p) = σ ∈ Σn ✱ ❣✐↔ sû σ ❝â ✤ë ❞➔✐ ❧➔ r✱ t❛ ❝❤å♥ ♠ët ❜✐➸✉ ❞✐➵♥ ❝õ❛ si1 sir ❝õ❛ σ ✳ ❚❤❡♦ ❍➺ q✉↔ ✷✳✶✸ t❛ ❝â ❱ỵ✐ ♠å✐ t❤➔♥❤ tø rót ❣å♥ Ti1 Tir = Tσ Tσ ▼➦t ❦❤→❝ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✺ ❝→❝ →♥❤ ①↕ Ti s✐♥❤ r❛ ❝❤♦ t❛ ❝ì sð ❝õ❛ H tr➯♥ Z(p) ✳ ❉♦ ✤â H✳ ❚✐➳♣ t❤❡♦ t❛ ❝❤ù♥❣ ♠✐♥❤ ❝→❝ q✉❛♥ ❤➺ ✤÷đ❝ t❤ä❛ ♠➣♥✳ ❚❤❡♦ ❍➺ q✉↔ ✷✳✶✷ t❛ ❝â Ti Ti+1 Ti = Ti+1 Ti Ti+1 = Tsi si+1 si ✈➔ ✈ỵ✐ |i − j| > Ti Tj = Tsi sj Tj Ti = Tsj si ❳➨t Ti2 ✳ ❈è ✤à♥❤ ❝í U ∈ F lag(V) ✈➔ ✤➦t A = {W | Ui−1 < W < Ui+1 }, t❛ ❝â |A| = p + 1✳ ❑➼ ❤✐➺✉ ϕ(W ) = (0 = U0 < · · · < Ui−1 < W < Ui+1 · · · < Un = V) ❑❤✐ ✤â ϕ(Ui ) = U ✈➔ δ(W , U ) = si ✈ỵ✐ W ∈ F lag(V) ✤â s✉② r❛ Ti [U ] = [ϕ(W )], W ∈A\Ui ✹✻ t❤ä❛ ♠➣♥ W ∈ A\Ui ✳ ❚ø ♥❣❤➽❛ ❧➔ Ti2 [U ] = Ti [ϕ(W )] W =Ui = [ϕ(W )] W =Ui W =W = ( [ϕ(W )] − [ϕ(W )]) W =Ui W = ( [ϕ(W )]) − W =Ui W [ϕ(W )] W =Ui = p[U ] + (p − 1) [ϕ(W )] W =Ui = (p + (p − 1)Ti )[U ] ❱➟② t❛ ❝â ✣➦t Ti2 = p + (p − 1)Ti ✳ H s✐♥❤ ❜ð✐ ❝→❝ ❦➼ ❤✐➺✉ ❑❤✐ ✤â t❛ ❝â θ : H −→ H ❚✐➳♣ t❤❡♦ t❛ ①➨t T1 , , Tn−1 ữủ n ợ q ❤➺ ♥❤÷ tr♦♥❣ ♠➺♥❤ ✤➲✳ θ(Ti ) = Ti ❈❤å♥ tø rót ❣å♥ ❧➔ t♦➔♥ →♥❤✳ u = si1 sir s❛♦ ❝❤♦ π(u) = σ Tσ = Ti1 Tir ∈ H ✳ ❚❤❡♦ ✣à♥❤ ỵ t t ❍➺ q✉↔ ✷✳✶✸ t❛ ❝â θ(Ti ) = Ti ✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ →♥❤ ①↕ ϕ : H −→ H ✤÷đ❝ ❝❤♦ ❜ð✐ ✈➔ ϕ(Tσ ) = Tσ ✱ t❤➻ t❛ ❝â ❚❛ ❦➼ ❤✐➺✉ ❜ð✐ Ti θϕ = idH A = ϕ(H) ≤ H ✳ ✈ỵ✐ ♠å✐ i✳ ♥➯♥ ϕ ❧➔ ✤ì♥ →♥❤✳ ❚❛ ❦❤➥♥❣ ✤à♥❤ A ✤â♥❣ ✈ỵ✐ ♣❤➨♣ ♥❤➙♥ ❜➯♥ ♣❤↔✐ Tσ ∈ A✳ ◆➳✉ σ(i) > σ(i + 1) t❤➻ t❛ ❝â (σsi ) = (σ) + 1✳ w ❝õ❛ σ ✳ ❑❤✐ ✤â wsi ❧➔ tø rót ❣å♥ ❝õ❛ σsi ♥➯♥ t❤❡♦ ❍➺ q✉↔ ❳➨t ❚❛ ❝❤å♥ tø rót ❣å♥ ✷✳✶✷ t❛ ❝â Tσ Ti = Tσsi ∈ A ◆➳✉ σ(i) < σ(i + 1) ❧➔ tø rót ❣å♥ ❝õ❛ σ✳ t❛ ❝❤å♥ tø rót ❣å♥ w ❝❤♦ ♣❤➨♣ t❤➳ τ = σsi ✳ ❈❤ó þ r➡♥❣ wsi A=H, s✉② ❙✉② r❛ Tσ Ti = Tτ (Ti )2 = Tτ (p + (p − 1)Ti ) = pTτ + (p − 1)Tτ si ∈ A ữ r A õ ợ ♣❤↔✐ ❜ð✐ ❝→❝ ❧➔ ✤➥♥❣ ❝➜✉✳ ▼➦t ❦❤→❝ θϕ = idH Ti ✳ ❱➻ ❧➔ ✤➥♥❣ ❝➜✉ ♥➯♥ ✹✼ θ 1∈A ♥➯♥ ❧➔ ✤➥♥❣ ❝➜✉✳ ✷✳✷ ▼æ✤✉♥ ❙t❡✐♥❜❡r❣ ▼æ✤✉♥ ❙t❡✐♥❜❡r❣ ỏ tr tữớ ữủ ỵ ❤✐➺✉ ❧➔ ❙t✱ ❧➔ ♠ët ❜✐➸✉ ❞✐➵♥ t✉②➳♥ t➼♥❤ ❝õ❛ ♥❤â♠ ✤↕✐ sè r❡❞✉❝t✐✈❡ tr➯♥ tr÷í♥❣ ❤ú✉ ❤↕♥ ❤♦➦❝ tr÷í♥❣ ữỡ ố ợ õ tr trữớ ỳ ❜✐➸✉ ❞✐➵♥ ♥➔② ✤÷đ❝ ❘♦❜❡rt ❙t❡✐♥❜❡r❣ ①➙② ❞ü♥❣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ tr♦♥❣ ❧♦↕t ❜➔✐ ❜→♦ ❬✽❪✱ ❬✾❪✱ ❬✶✵❪❀ ✤➛✉ t✐➯♥ ❧➔ ❝❤♦ ♥❤â♠ t✉②➳♥ t➼♥❤ tê♥❣ q✉→t✱ s❛✉ ✤â ❧➔ ❝→❝ ♥❤â♠ ❝ê ✤✐➸♥✱ ✈➔ ❝✉è✐ ❝ò♥❣ ❧➔ ❝→❝ ♥❤â♠ ❈❤❡✈❛❧❧❡②✳ ▼æ✤✉♥ ❙t❡✐♥❜❡r❣ ①✉➜t ❤✐➺♥ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ❝õ❛ t♦→♥ ❤å❝✳ ❚r♦♥❣ ❬✹❪✱ ❍✉♠♣❤r❡②s ✤➣ tr➻♥❤ ❜➔② ♠ët ❜➔✐ tê♥❣ q✉❛♥ r➜t ❝æ♥❣ ♣❤✉ ✈➔ ❝❤✐ t✐➳t ✈➲ ♠ỉ✤✉♥ ❙t❡✐♥❜❡r❣ ✈➔ ♥❤ú♥❣ ù♥❣ ❞ư♥❣ r➜t ✤❛ ❞↕♥❣ ❝õ❛ ♥â✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔②✱ ❝❤ó♥❣ tỉ✐ ①➨t tr÷í♥❣ ❤đ♣ ♠ỉ✤✉♥ ❙t❡✐♥❜❡r❣ ❝❤♦ ♥❤â♠ t✉②➳♥ t➼♥❤ tê♥❣ q✉→t✳ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✺✳ ❈❤♦ V ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì n ❝❤✐➲✉✳ ✣➦t π : Base −→ F lag, π(v1 , , ) −→ (0 < Fp {v1 } < · · · < Fp {v1 , , } = V) ð ✤â (v1, , vn) ❧➔ ♠ët ❝ì sð ♥➔♦ ✤â ❝õ❛ V ❑❤✐ ✤â π ❝↔♠ s✐♥❤ ❤❛✐ ♣❤➨♣ ❜✐➳♥ ✤ê✐ tü ♥❤✐➯♥ π : Z(p) [Base] −→ Z(p) [F lag] ✈➔ ❧✐➯♥ ❤ñ♣ π t : Z(p) [F lag] −→ Z(p) [Base] ❚❛ ✤à♥❤ ♥❣❤➽❛ ♣❤➨♣ ❜✐➳♥ ✤ê✐ tü ♥❤✐➯♥ ω = |G/B|−1 sgn(σ)σ ∗ : Z(p) [Base] Z(p) [Base] n ú ỵ r ( )t = (σ −1 )∗ ✈➔ ❞♦ ✤â ωt = ω✳ ❚❛ ✤➦t µ = πω : Z(p) [Base] −→ Z(p) [F lag] St = Im(à) ú ỵ r➡♥❣ St(Fnp ) ❧➔ ♠ët Z(p) [G]✲♠æ✤✉♥ ♠æ✤✉♥ ❙t❡✐♥❜❡r❣✳ ❚❛ ❦➼ ❤✐➺✉ e = µπ t = πωπ t = et : Z(p) [F lag] −→ Z(p) [F lag] ✹✽ ữủ eSt = t = π t πω : Z(p) [Base] −→ Z(p) [Base] e∈H ❚❛ ❝â ✈➔ eSt ∈ End(Z(p) [Base]) = Z(p) [G]op eSt ❚❛ ❣å✐ ❧➔ ❧ô② ✤➥♥❣ ❙t❡✐♥❜❡r❣✳ ▼➺♥❤ ✤➲ ✷✳✶✻✳ ⑩♥❤ ①↕ ξ : H −→ Z ❝❤♦ ❜ð✐ (T ) = sgn() ởt ỗ (p) ✈➔♥❤ ✈➔ ξ(a ) = ξ(a) ✈ỵ✐ ♠å✐ a ∈ H✳ ❍ì♥ ♥ú❛ ✈ỵ✐ ♠å✐ a ∈ H t❛ ❝â aµ = ξ(a)µ ✈➔ ae = ea = ξ(a)e, ♥➯♥ e ❧➔ ♣❤➛♥ tû t❤✉ë❝ t➙♠ ❝õ❛ H✳ t ❈❤ù♥❣ ♠✐♥❤✳ (1 + Ti )[U ] ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ Uj ✈ỵ✐ j = i✳ ∗ ▼➦t ❦❤→❝✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ t❤➻ πsi (v1 , , ) ❝❤➾ ❦❤→❝ π(v1 , , ) ð ❦❤æ♥❣ ❣✐❛♥ t❤ù i ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✻ t❤➻ ♥➯♥ t❛ ❝â (1 + Ti )π = (1 + Ti )πs∗i ❙✉② r❛ (1 + Ti )µ = (1 + Ti )πs∗i ω ▼➦t ❦❤→❝ ✈➻ s∗i ω = −ω ♥➯♥ t❛ ❝â (1 + Ti )πs∗i ω = −(1 + Ti )µ ❚ø ✤â s✉② r❛ ❈❤♦ (1 + Ti )µ = 0✱ σ ∈ Σn Ti = si1 sir t t ữủ tø rót ❣å♥ tr♦♥❣ Σn s❛♦ ❝❤♦ σ = si1 sir ✳ ❑❤✐ ✤â✱ Tσ µ = Ti1 Tir µ = (−1)r µ = ξ(Tσ )µ ❱➻ Tσ ❧➔ ♠ët tỷ s H aà = (a)à ợ ♠å✐ a✳ ❙✉② r❛ ξ(ab)µ = ξ(a)ξ(b)µ, ∀µ = t õ ỗ t ❦❤→❝ ❞♦ ✈➔ sgn(σ −1 ) = sgn(σ) ♥➯♥ t ξ(a ) = ξ(a) ❚✐➳♣ t❤❡♦✱ ✈➻ e = µπ t ✈➔ aµ = ξ(a)µ ♥➯♥ ae = ξ(a)e✳ t t ❚❤❡♦ ✣à♥❤ ♥❣❤➽❛ ✷✳✶✺ e = e ♥➯♥ t❛ ♥❤➟♥ ✤÷đ❝ ✤➥♥❣ t❤ù❝ t÷ì♥❣ ✤÷ì♥❣ ea = ξ(a)e✳ ❚❤❛② a ❜ð✐ at t t r❛ ea = ξ(a )e✳ ❱➻ ξ(a ) = ξ(a) ♥➯♥ ea = ξ(a)e✳ ❱➟② e t❤✉ë❝ t➙♠ ❝õ❛ H t❛ ❝â s✉② Tσt = Tσ−1 ▼➺♥❤ ✤➲ ✷✳✶✼✳ ❚❛ ❝â e = |G/B|−1 sgn(σ)p (σ σ∈Σn ✹✾ −1 ρ) Tσ ❈❤ù♥❣ ♠✐♥❤✳ ❚ø ❍➺ q✉↔ ✶✳✶✻ t❛ ❝â < πσ ∗ π t [U ], [W ] >=< σ ∗ π t [U ], [W ] >= |G/B|p (σ−1 ρ) ♥➳✉ δ(U , W ) = σ, 0 ♥➳✉ δ(U , W ) = σ ❚ø ✤â s✉② r❛ πσ ∗ π t = |G/B|−1 sgn(σ)p (σ −1 ρ) Tσ σ∈Σn ❉♦ ✤â sgn(σ)πσ ∗ π t = |G/B|−1 e = |G/B|−1 sgn(σ)p (σ −1 ρ) Tσ σ∈Σn σ∈Σn ❍➺ q✉↔ ✷✳✶✽✳ e = |G/B|−1 sgn(σ)p (σ −1 ρ) [gxσ −1 B] x∈X(σ −1 ) σ∈Σn ▼➺♥❤ ✤➲ õ à = ỗ e ✈➔ e ✤➲✉ ❧➔ ❝→❝ ❧ô② ✤➥♥❣ ✈➔ t St Im(e) = St✳ ⑩♥❤ ①↕ µ : Z(p)[Base] −→ St ❤↕♥ ❝❤➳ ①✉è♥❣ Im(eSt) ❝↔♠ s✐♥❤ ♠ët ✤➥♥❣ ❝➜✉ Im(eSt) −→ St✱ ✈ỵ✐ ♥❣❤à❝❤ ✤↔♦ ❧➔ πt ❈❤ù♥❣ ♠✐♥❤✳ rữợ t t õ (e) = |G/B|1 p ( ρ) Tσ σ∈Σn X(σ −1 ρ)| = |G/B|−1 | σ∈Σn = |G/B|−1 | X(τ )| τ ∈Σn = ❚ø ✤â s✉② r❛ ˆ µπ t µ = eµ = ξ(e)µ = µ, e2 = µπ t µπ t = µπ t = e ✈➔ e2St = π t µπ t µ = π t µ = eSt ✺✵ ❉♦ ✤â e µ = eµ s✉② r❛ ✈➔ eSt ❧➔ ❧ơ② ✤➥♥❣✳ ❱➻ Im(µ) ≤ Im(e)✳ e = µπ t ✱ t❛ ❝â Im(e) ≤ Im(µ)✳ ▼➦t ❦❤→❝ ❞♦ ❱➻ ✈➟② t❛ ❝â Im(µ) = Im(e) = St ✣➦t St = Im(eSt ) ✈➔ ①➨t →♥❤ ①↕ α : St −→ St ❧➔ ❤↕♥ ❝❤➳ ❝õ❛ µ✳ ❱➻ eSt = π t µ ♥➯♥ π t (St) = Im(π t µ) = St ◆➳✉ β ❧➔ ❤↕♥ ❝❤➳ ❝õ❛ ①↕ ❤↕♥ ❝❤➳ ❝õ❛ ❝➜✉ ♥➯♥ α ✈➔ β µπ t πt ❧➯♥ ❧➯♥ St St β t❤➻ ❝❤♦ t ởt t ỗ t t❛ ❝â ❧➔ ♥❣❤à❝❤ ✤↔♦ ❝õ❛ ♥❤❛✉✳ ❉♦ ✤â α β : St −→ St ✳ ❱➻ →♥❤ αβ = id✳ ❧➔ t♦➔♥ ❱➻ β ❧➔ ✤➥♥❣ ❝➜✉✳ ▼➺♥❤ ✤➲ ✷✳✷✵✳ St ❧➔ ♠ët G✲♠æ✤✉♥ ①↕ ↔♥❤✱ tü ✤è✐ ♥❣➝✉ tr♦♥❣ [V, A] ✈➔ rank(St) = pn(n−1)/2 ❈❤ù♥❣ ♠✐♥❤✳ ❈â ✤➥♥❣ ❝➜✉ ❨♦♥❡❞❛ Hom[V,A] (Z(p) [Base], A) = A(Fnp ) ♥➯♥ Z(p) [Base] ❧➔ ①↕ ↔♥❤✳ tû trü❝ t✐➳♣ tr♦♥❣ St Z(p) [Base] ✤➥♥❣ ❝➜✉ ✈ỵ✐ ↔♥❤ ❝õ❛ ♥➯♥ St eSt ✈➔ ✈➻ ↔♥❤ ❝õ❛ eSt ❧➔ ❤↕♥❣ ①↕ ↔♥❤✳ ▼➦t ❦❤→❝✱ t❛ ❝â →♥❤ ①↕ s♦♥❣ t✉②➳♥ t➼♥❤ ϕ : Z(p) [F lag] × Z(p) [F lag] −→ Z(p) , (x, x ) −→< x, x > ❑❤✐ ✤â ♥â ❝❤♦ t❛ ❣❤➨♣ ❝➦♣ ❤♦➔♥ ❤↔♦ φ1 : Z(p) [F lag] −→ Z(p) [F lag]∗ = HomZ(p) (Z(p) [F lag], Z(p) ) x −→< x, f > tr♦♥❣ ✤â f ∈ HomZ(p) (Z(p) [F lag], Z(p) ), ✈➔ φ1 : Z(p) [F lag] −→ Z(p) [F lag]∗ = HomZ(p) (Z(p) [F lag], Z(p) ) x −→< f, x > ❧➔ ❝→❝ ✤➥♥❣ ❝➜✉✳ ❍ì♥ ♥ú❛✱ ❞♦ e tr♦♥❣ ✤â f ∈ HomZ(p) (Z(p) [F lag], Z(p) ), ❧➔ ❧ơ② ✤➥♥❣ tü ❧✐➯♥ ❤đ♣✱ tù❝ ❧➔ e2 = e ✈➔ < ex, x >=< x, ex >✱ ♠➔ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✾ t❛ ❝â St ❧➔ ↔♥❤ ❝õ❛ e tr➯♥ Z(p) [F lag]✳ ❉♦ ✤â t❛ ❝â φ1 |St : St −→ St∗ ✺✶ ✈➔ φ2 |St : St −→ St∗ , ❧➔ ❝→❝ ✤➥♥❣ ❝➜✉✳ ❉♦ ✤â e✳ ❝❤➼♥❤ ❧➔ ✈➳t ❝õ❛ ◆➳✉ St ❧➔ tü ✤è✐ ♥❣➝✉✳ ❱➻ St = Im(e), σ = t❤➻ δ(U , U ) = σ ✳ ▼➦t ❦❤→❝ ❞♦ Tσ [U ] = ❞♦ ✤â ❤↕♥❣ ❝õ❛ St ✤à♥❤ ♥❣❤➽❛ [W ] δ(U ,W )=σ ♥➯♥ t❛ ❝â ✈➳t ❝õ❛ Tσ ❧➔ 0✳ ◆➳✉ σ = t❤➻ ✈➳t ❝õ❛ Tσ ◆➯♥ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✼ t❛ ❝â ✈➳t ❝õ❛ ▼➺♥❤ ✤➲ ✷✳✷✶✳ ⑩♥❤ ①↕ tü ♥❤✐➯♥ Z N = Im(1 e) t e ỗ t ợ p (ρ) = pn(n−1)/2 −→ End[V,A] (St) (p) |F lag| = |G/B|✳ ❧➔ ♠ët ✤➥♥❣ ❝➜✉✳ ◆➳✉ Hom[V,A] (St, N ) = 0, ✈➔ Hom[V,A] (N, St) = ❈❤ù♥❣ ♠✐♥❤✳ St ✭❤♦➦❝ tø Z(p) [F lag] = St ⊕ N, ỗ f : St tứ N St õ t rở tỹ ỗ rữợ t t õ St N f : Z(p) [F lag] −→ Z(p) [F lag]✳ H✳ ❝õ❛ ❍❛② ♥â✐ ❝→❝❤ ❦❤→❝ ▼➦t ❦❤→❝ ♠å✐ ♣❤➛♥ tû a ∈ H f ❧➔ ♠ët ♣❤➛♥ tû ♥➔♦ ✤â ❧✉æ♥ t❤ä❛ ♠➣♥ ae = ξ(a)e = ea✱ α ∈ Im(e) t tỗ t [U ] Z(p) [F lag] s ❝❤♦ α = e[U ] aα = ae[U ] = ξ(a)e[U ] ◆➯♥ t❤❡♦ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ ξ t❤➻ ξ(a)e[U ∈ Im(e) ♥ú❛ ✈ỵ✐ ♠å✐ s✉② r❛ a ❜↔♦ t♦➔♥ St = Im(e) t ữ ổ ữợ tr St ❤ì♥ ❉♦ ✤â ❚ø ✤â ❱➻ ✈➟② t❛ ❝â End[V,A] (St) = Z(p) ❚✐➳♣ t❤❡♦ t❛ ✤✐ ❝❤ù♥❣ ♠✐♥❤ ❝â a ❜↔♦ t♦➔♥ N = ker(e) ❚❤❡♦ ✤à♥❤ ♥❣❤➽❛ t❛ ❝â ♥➯♥ t❛ ❝â ❚❛ Hom[V,A] (St, N ) = ❚÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ ❝ơ♥❣ ①➨t ♣❤➛♥ tû ❜➜t ❦ý a t❤✉ë❝ Hom[V,A] (St, N ) = N = Im(1 − e) aα = (1 − e)[U ] tr♦♥❣ ✤â α ✈➔ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✾ t❛ ❝â t❤✉ë❝ Im(e)✱ [U ] t❤✉ë❝ St = Im(e) Z(p) [F lag] ❙✉② r❛ eaα = e(1 − e)[U ], ✈➻ e = e ♥➯♥ t❛ ❝â eaα = 0, ♠➦t ❦❤→❝ t❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✾ ❝â eaα = aeα = ξ(a)eα ❉♦ ✤â ξ(a)eα = ❤❛② ξ(a) = s✉② r❛ a = ❱➟② t❛ t❛ ❝â Hom[V,A] (St, N ) = ▲➟♣ ❧✉➟♥ ❤♦➔♥ t♦➔♥ t÷ì♥❣ tü ♥❤÷ tr➯♥ t❛ ❝ơ♥❣ ❝â Hom[V,A] (N, St) = ❱➼ ❞ư ✷✳✷✷✳ ▼ỉ t↔ ♠ỉ✤✉♥ ❙t❡✐♥❜❡r❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ n = 2✱ V ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ✈➨❝tì ❝❤✐➲✉ tr➯♥ tr÷í♥❣ Fp ❑❤✐ n = t❤➻ Σ2 = {e, s1} ▼➦t ❦❤→❝ t❤❡♦ ❇ê ✤➲ ✺✷ ✵✳✷✶✱ |G| = p(p + 1)(p − 1)2 ✈➔ |B| = p(p − 1)2, s✉② r❛ |G/B|−1 = p+1 ❚❤❡♦ ▼➺♥❤ ✤➲ ✷✳✶✾ t❤➻ St = Im(e), ♥➯♥ ✤➸ t➼♥❤ St t❛ ❝➛♥ ✤✐ ①→❝ ✤à♥❤ Im(e) ❚ø ▼➺♥❤ ✤➲ ✷✳✶✼ ✈ỵ✐ [U ] ∈ Z(p)[F lag(V)] e[U ] = |G/B|−1 sgn(σ)p (σ −1 ρ) Tσ [U ] σ∈Σ2 (pTe [U ] − T1 [U ]) p+1 = (p[U ] − [W ]) p+1 = [W ]=[U ] = [(p + 1)[U ] − p+1 = [U ] − p+1 ❱➟② St = {[U ] − [W ]] [W ]∈F lag(V) [W ] [W ]∈F lag(V) p+1 tr♦♥❣ ✤â [U ], [W ] ∈ F lag(V) ✺✸ [W ]} [W ]∈F lag(V) ❑➳t ❧✉➟♥ ❚r➯♥ ✤➙② ❧➔ ♠ët sè ❦✐➳♥ t❤ù❝ ♠➔ ♥❣÷í✐ ✈✐➳t t❤✉ ✤÷đ❝ tr♦♥❣ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❧➔♠ ❧✉➟♥ ✈➠♥✳ ▲✉➟♥ ✈➠♥ ♥➔② t➟♣ tr✉♥❣ ự ỡ s ỵ tt õ t t➼♥❤ tê♥❣ q✉→t tr➯♥ tr÷í♥❣ Fp ✳ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤õ ②➳✉ ❧➔ ◆✳ P✳ ❙tr✐❝❦❧❛♥❞ ❬✶✶❪✳ ◆❣÷í✐ ✈✐➳t ✤➣ ✤å❝ ❤✐➸✉ ✈➔ ❝è ❣➢♥❣ tr➻♥❤ ❜➔② ❧↕✐ rã r➔♥❣ ♥ë✐ ❞✉♥❣ ❝õ❛ t➔✐ ❧✐➺✉ ♥➔②✳ ❚r♦♥❣ ❦❤✉æ♥ ❦❤ê ữớ t ợ tr ❝❤✐ t✐➳t ❝→❝❤ ①➙② ❞ü♥❣ ✤↕✐ sè ■✇❛❤♦r✐✲ ❍❡❝❦❡✱ ♠æ✤✉♥ ❙t❡✐♥❜❡r❣ tø ♣❤➙♥ t➼❝❤ ❇r✉❤❛t✳ ❚✉② ♥❤✐➯♥✱ ✈✐➺❝ ①➙② ❞ü♥❣ ✤↕✐ sè ■✇❛❤♦r✐✲ ❍❡❝❦❡✱ ♠ỉ✤✉♥ ❙t❡✐♥❜❡r❣ ❝❤÷❛ ♥â✐ ❧➯♥ ✤÷đ❝ ù♥❣ ❞ư♥❣ ❝õ❛ ✈✐➺❝ t➼♥❤ t♦→♥ tr♦♥❣ tỉ♣ỉ ✤↕✐ sè✳ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② ❝á♥ ❝â ♠ët sè ✈➜♥ ✤➲ ❝➛♥ ♥❣❤✐➯♥ ❝ù✉ t✐➳♣ ❧➔✿ ❚➼♥❤ t♦→♥ tr♦♥❣ tæ♣æ ✤↕✐ sè✳ ✣➸ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✈➜♥ ✤➲ tr➯♥ ❝➛♥ ❝â t❤➯♠ ❝→❝ ❦✐➳♥ t❤ù❝ ♠➔ tr➻♥❤ ✤ë ❝õ❛ ♥❣÷í✐ ✈✐➳t ❝❤÷❛ t❤➸ ✤→♣ ù♥❣ ✤÷đ❝✳ ◆❣÷í✐ ✈✐➳t ❤② ✈å♥❣ ❝â t❤➸ t✐➳♣ tö❝ ♥❣❤✐➯♥ ❝ù✉ t✐➳♣ ✤➲ t➔✐ ♥➔② tr♦♥❣ t❤í✐ ❣✐❛♥ tỵ✐✳ ✺✹ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ❏✳ ▲✳ ❆❧♣❡r✐♥✱ ❘✳ ❇✳ ❇❡❧❧✱ ●r♦✉♣s ❛♥❞ ❘❡♣r❡s❡♥t❛t✐♦♥s ✱ ❙♣r✐♥❣❡r ✲ ❱❡r❧❛❣✱ ◆❡✇ ❨♦r❦✱ ✶✾✾✺✳ ❬✷❪ ❋✳❇r✉❤❛t✱ ❙✉r ❧❡s r❡♣r❡s❡♥t❛t✐♦♥s ✐♥❞✉✐t❡s ❞❡s ❣r♦✉♣❡s ❞❡ ▲✐❡✱ ▼❛t❤✳ ❋r❛♥❝❡ ✽✹ ✭✶✾✺✻✮✱ ✾✼✲✷✵✺✳ ❬✸❪ ❈✳ ❈❤❡✈❛❧❧❡②✱ ❇✉❧❧✳ ❙♦❝✳ ❈❧❛ss✐❢✐❝❛t✐♦♥ ❞❡s ●r♦✉♣❡s ❆❧❣➨❜r✐q✉❡s ❙❡♠✐✲s✐♠♣❧❡s✱ ❙♣r✐♥❣❡r✲ ❱❡r❧❛❣✱ ❇❡r❧✐♥✱ ✭✷✵✵✺✮✳ ❬✹❪ ❍✉♠♣❤r❡②s✱ ❏✳❊✳ ❚❤❡ ❙t❡✐♥❜❡r❣ r❡♣r❡s❡♥t❛t✐♦♥ ✱ ❇✉❧❧✳ ❆♠❡r✱ ▼❛t❤✳ ❙♦❝✳ ✭◆✳❙✳✮ ✶✻ ✭✷✮✭✶✾✽✼✮✱ ✷✸✼✲✷✻✸✳ ❬✺❪ ◆❣✉②➵♥ ❍ú✉ ❱✐➺t ❍÷♥❣✱ ✣↕✐ sè ✤↕✐ ❝÷ì♥❣✱ ◆❤➔ ①✉➜t ❜↔♥ ●✐→♦ ❞ö❝✱ ❍➔ ◆ë✐✱ ✶✾✾✾✳ ❬✻❪ ◆✳ ■✇❛❤♦r✐✱ ❖♥ t❤❡ str✉❝t✉r❡ ♦❢ ❛ ❍❡❝❦❡ r✐♥❣ ♦❢ ❛ ❈❤❡✈❛❧❧❡② ❣r♦✉♣ ♦✈❡r ❛ ❢✐♥✐t❡ ❢✐❡❧❞✱ ❏✳ ❋❛❝✳ ❙❝✐✳ ❯♥✐✈✳ ❚♦❦②♦ ✶✵ ✭✶✾✻✹✮✱ ♣♣✳ ✷✶✺✲✷✸✻✳ ❬✼❪ ●✳ ❙❤✐♠✉r❛✱ ❙✉r ❧❡s ✐♥t➨❣r❛❧❡s ❛tt❛❝❤➨❡s ❛✉① ❢♦r♠❡s ❛✉t♦♠♦r♣❤❡s✱ ❙♦❝✳ ❏❛♣❛♥ ✶✶ ✭✶✾✺✾✮✱ ✷✾✶✲✸✶✶✳ ❏✳ ▼❛t❤✳ ❬✽❪ ❙t❡✐♥❜❡r❣✱ ❘♦❜❡rt✱ ❆ ❣❡♦♠❡tr✐❝ ❛♣♣r♦❛❝❤ t♦ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ ❢✉❧❧ ❧✐♥❡❛r ❣r♦✉♣ ♦✈❡r ❛ ●❛❧♦✐s ❢✐❡❧❞✱ ❙♦❝✐❡t② ✼✶ ✭✷✮ ✭✶✾✺✶✮✱ ✷✼✹✕✷✽✷✱ ❚r❛♥s❛❝t✐♦♥s ♦❢ t❤❡ ❆♠❡r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❬✾❪ ❙t❡✐♥❜❡r❣✱ ❘♦❜❡rt✱ Pr✐♠❡ ♣♦✇❡r r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❢✐♥✐t❡ ❧✐♥❡❛r ❣r♦✉♣s ❞✐❛♥ ❏♦✉r♥❛❧ ♦❢ ▼❛t❤❡♠❛t✐❝s ✽✭✶✾✺✻✮✱ ✺✽✵✕✺✾✶✳ ❬✶✵❪ ❙t❡✐♥❜❡r❣✱ ❘✳ Pr✐♠❡ ♣♦✇❡r r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ ❢✐♥✐t❡ ❧✐♥❡❛r ❣r♦✉♣s ■■✱ ❏✳ ▼❛t❤✳ ✾✭✶✾✺✼✮✱ ✸✹✼✕✸✺✶✳ ✺✺ ❈❛♥❛✲ ❈❛♥❛❞✳ ❬✶✶❪ ◆✳ P✳ ❙tr✐❝❦❧❛♥❞✱ ❚❤❡ ❙t❡✐♥❜❡r❣ ▼♦❞✉❧❡ ❛♥❞ t❤❡ ❍❡❝❦❡ ❛❧❣❡❜r❛✱ str✐❝❦❧❛♥❞✳st❛❢❢✳s❤❡❢✳❛❝✳✉❦✴r❡s❡❛r❝❤✴♣r❡♣r✐♥ts✳❤t♠❧ ✳ ✺✻ ❤tt♣✿✴✴♥❡✐❧✲