講演集 pdf file

第６１回トポロジーシンポジウム講演集２０１４年７月於東北大学
ϧʔτ‫͔ܥ‬Βఆ·ΔτʔϦοΫଟ༷ମͷ
ίϗϞϩδʔ‫ͱ؀‬Ϡϯάਤ
Ѩ෦ ୓
େࡕࢢཱେֶ ਺ֶ‫ॴڀݚ‬
1
ಋೖ
·ͣ͸ϧʔτ‫ ͕ܥ‬An−1 ‫ܕ‬ͷ৔߹ͷྫΛ༻͍ͯɼͲͷΑ͏ͳ໰୊Λѻ͏͔Λઆ໌͢Δɽෳ
ૉϕΫτϧۭؒ Cn ͷ‫ض‬ଟ༷ମ F l(Cn ) ͱ͸ Cn ͷઢ‫ܗ‬෦෼ۭؒͷྻͷ੒ۭؒ͢Ͱ͋Δɽ
F l(Cn ) = {V0 ⊂ V1 ⊂ · · · ⊂ Vn | Vi ͸ Cn ͷઢ‫ܕ‬෦෼ۭؒͰ dimC Vi = i}
SLn (C) Λ n ࣍ͷಛघઢ‫͠ͱ܈ܗ‬ɼT ⊂ SLn (C) Λର֯੒෼ͷΈ͔ΒͳΔ෦෼‫͢ͱ܈‬Δɽ͜
ͷͱ͖ɼSLn (C) ͸‫ض‬ଟ༷ମ F l(Cn ) ʹࣗવʹ࡞༻͠ɼτʔϥε T ͸ͦͷ੍‫͢༻࡞ͯ͠ͱݶ‬
Δɽ͜ͷτʔϥε࡞༻ͷҰൠ‫ي‬ಓͷดแ͸τʔϦοΫଟ༷ମͰ͋Δ͜ͱ͕஌ΒΕ͍ͯΔɽ
ҎԼɼ͜ͷτʔϦοΫଟ༷ମΛ X ͱॻ͘ɽ
‫ض‬ଟ༷ମ F l(Cn ) ͸ n ࣍ஔ‫ ܈׵‬Sn ͷ‫ ݩ‬w ͝ͱʹγϡʔϕϧτ๔ମ Cw ͱ‫ݺ‬͹ΕΔ๔ମΛ
΋ͪɼX ͱͷ‫ڞ‬௨෦෼ X ∩ Cw Λߟ͑Δͱ͜ΕΒ͸ X ͷ๔ମ෼ׂΛ༩͑ΔɽࠓɼX ∩ Cw ͷ
X ʹ͓͚ΔดแΛ Xw ͱॻ͘ͱ͖ɼXw ͷϙΞϯΧϨ૒ର [Xw ] ͸ X ͷಛҟίϗϞϩδʔ
ͷ‫ج‬ఈΛ੒͢ɿH ∗ (X; Z) = w∈Sn Z[Xw ]ɽͦ͜ͰɼΧοϓੵͷల։
[Xu ][Xv ] =
cw
uv [Xw ]
w∈Sn
w
ʹ͓͍ͯల։܎਺ cw
uv ∈ Z ͕‫ݱ‬ΕΔ͕ɼ͜ͷ਺ cuv ͸ͲͷΑ͏ʹ‫ه‬ड़͞ΕΔ਺ͩΖ͏͔ɽ
ͦΕʹ͍ͭͯߟ͍͑ͨɽ͜ͷ໰୊Λ೦಄ʹஔ͍ͨ͏͑Ͱɼ͜ͷ༧ߘͰ͸ίϗϞϩδʔ‫؀‬
H ∗ (X; Z) ͱϠϯάਤͷؔ܎ʹ͍ͭͯઆ໌͢Δɽಛʹɼ֤ [Xw ] ΛϠϯάਤͰද͢ํ๏΍ɼ
ߏ଄ఆ਺ cw
uv ͱಉ஋ͳྔͰ͋Δަࠥ਺Λ͜ΕΒͷϠϯάਤͰ‫͢ࢉܭ‬Δํ๏ʹ͍ͭͯड़΂Δɽ
͜͜Ͱ͸ɼཧ༝͸͓͖ͯ͞ͲͷΑ͏ͳϠϯάਤΛߟ͑Δ͔͚ͩड़΂͓ͯ͘ɽงғ‫͚ͩؾ‬
Ͱ΋఻ΘΕ͹ͱࢥ͏ɽஔ‫ ׵‬w ∈ Sn ʹର͠ɼw(i) > w(i + 1) Λຬͨ͢ i ∈ [n] ʹ஫໨͠ɼͦ
ͷ৘ใΛ΋ͱʹϠϯάਤΛ࡞Δɽͦͯͦ͠ͷΑ͏ͳ w(i) ͷ৘ใ͕ࣦΘΕͳ͍Α͏ʹϠϯά
ਤͷ্ʹ w(i) ୡΛฒ΂Δɽྫ͑͹
w=
4153
1 2 3 4 5
ͳΒ͹ɺ[Xw ] ͸
Ͱද͞ΕΔ
4 1 5 3 2
ͱ͍͏۩߹Ͱ͋Δʢৄ͍͠ߏ੒๏͸ 5 અͰઆ໌͢Δʣɽ
第６１回トポロジーシンポジウム講演集２０１４年７月於東北大学
2
τʔϦοΫଟ༷ମ
τʔϦοΫଟ༷ମͱ͸ɼਖ਼‫ن‬1 ୅਺ଟ༷ମͰ͋Γ୅਺తτʔϥε (C× )n ΛͦͷβϦεΩʔ
։ू߹ʹ΋ͪɼ(C× )n ͷ‫ߏ܈‬଄Λ֦ு͢ΔΑ͏ͳ (C× )n -࡞༻Λۭ࣋ͭؒͷ͜ͱͰ͋Δɽτʔ
ϦοΫଟ༷ମ͕‫ڵ‬ຯΛ࣋ͨΕ͍ͯΔओͳཧ༝͸ɼͦͷ‫ز‬Կ΍τϙϩδʔΛઔͱ‫ݺ‬͹ΕΔ૊
Έ߹Θͤ࿦తͳର৅Λ༻͍ͯ‫ه‬ड़͢Δ͜ͱ͕Ͱ͖Δͱ͍͏఺Ͱ͋Δɽ
τʔϦοΫଟ༷ମ͸ઔΛ༻͍ͯߏ੒Ͱ͖Δ͕ɼ͜͜Ͱ͸·ͣઔͱ͸ͲͷΑ͏ͳର৅Ͱ͋
Δ͔ɼ·ͨɼઔͷੑ࣭ͱτʔϦοΫଟ༷ମͷ‫ز‬Կతੑ࣭͕ͲͷΑ͏ʹ݁ͼ͍͍ͭͯΔ͔Λ
֓‫͢؍‬Δɽ
Rn ʹ͓͚Δਲ਼ σ ͕༗ཧ‫ڧ‬ತଟ໘ਲ਼Ͱ͋Δͱ͸ɼਲ਼ͱͯ͠ͷੜ੒‫ݩ‬Λ Zn ͷ༗‫ݸݶ‬ͷ‫͔ݩ‬
ΒબͿ͜ͱ͕Ͱ͖ʢ༗ཧੑʣɼ͔ͭ σ ∩ (−σ) = {0}ʢ‫ڧ‬ತੑʣ͕ຬͨ͞ΕΔͱ͖Λ͍͏ɽ
Rn ʹ͓͚Δ༗ཧ‫ڧ‬ತଟ໘ਲ਼ͷͳ͢༗‫ ߹ूݶ‬Δ Ͱ࣍ͷ৚݅Λຬͨ͢΋ͷΛઔͱ͍͏ɽ
(1) σ ∈ Δ ͷͱ͖ɼσ ͷ೚ҙͷ໘͸ Δ ͷ‫ݩ‬
(2) σ, τ ∈ Δ ͳΒ͹ͦͷ‫ڞ‬௨෦෼ σ ∩ τ ͸ σ, τ ͦΕͧΕͷ໘Ͱ͋Δɽ
͜͜Ͱ͸ৄࡉ͸ল͕͘ɼઔ͔ΒτʔϦοΫଟ༷ମΛߏ੒͢Δ͜ͱ͕ग़དྷΔɽಛʹɼઔͱ
τʔϦοΫଟ༷ମͷಉ‫ྨܕ‬͸ 1 ର 1 ʹରԠ͢Δ͜ͱ͕஌ΒΕ͍ͯΔɽ
ઔ
1:1
τʔϦοΫଟ༷ମͷಉ‫ྨܕ‬
R2 ʹ͓͚Δઔͷྫͱͯ͠ҎԼͷΑ͏ͳ΋ͷ͕͋Δɽࠨଆͷઔ͸ෳૉࣹӨฏ໘ CP 2 ʹର
Ԡ͠ɼӈଆͷઔ͸͋Δ Hirzebruch ‫ۂ‬໘ʹରԠ͢Δɽ
ਤ 1: R2 ʹ͓͚Δઔͷྫ
͜͜Ͱ‫͡ײ‬औͬͯ௖͖͍ͨ͜ͱ͸ɼ͜ͷରԠͷԼͰɼτʔϦοΫଟ༷ମ X(Δ) ͷ‫ز‬Կ΍
τϙϩδʔ͕ઔ Δ ͷ‫ݴ‬༿Ͱ‫͑׵͍ݴ‬ΒΕΔͰ͋Ζ͏ͱ͍͏͜ͱͰ͋Δɽ࣮ࡍɼྫ͑͹࣍ͷ
Α͏ͳରԠ͕͋Δɽ
• Δ ͷਲ਼ͷू·Γ͕ Rn શମΛ෴͏ ⇐⇒ X(Δ) ͕ίϯύΫτ
• Δ ͷ֤ਲ਼ͷੜ੒‫ ͕ݩ‬Zn ͷ Z ্ͷ‫ج‬ఈͷҰ෦ʹͱΕΔ ⇐⇒ X(Δ) ͸ඇಛҟ
1
ਖ਼‫ن‬ੑΛԾఆ͠ͳ͍ఆ͕ٛ࠷΋ҰൠతͰ͋Δ͕ɼ͜ͷ༧ߘͰ͸ਖ਼‫ن‬ੑΛఆٛʹೖΕΔɽ
第６１回トポロジーシンポジウム講演集２０１４年７月於東北大学
• ઔ Δ ͷ n ࣍‫ݩ‬ਲ਼ͷ਺ = τʔϦοΫଟ༷ମ X(Δ) ͷΦΠϥʔ਺
͜͜ʹ‫ز‬Կ΍τϙϩδʔͱ૊Έ߹Θͤ࿦ͷௐ࿨Λ‫ͱͯݟ‬Δ͜ͱ͕Ͱ͖Δɽ
3
ϧʔτ‫͔ܥ‬Βఆ·ΔτʔϦοΫଟ༷ମ
E n Λ࣮ n ࣍‫ݩ‬ϢʔΫϦουۭؒͱ͠ɼΦ ⊂ E n Λϧʔτ‫ܥ‬ɼW ΛͦͷϫΠϧ‫͢ͱ܈‬Δɽ
Φ ͔ΒτʔϦοΫଟ༷ମΛߏ੒͠Α͏ɽΠ = {α1 , · · · , αn } Λ Φ ͷఈʢ·ͨ͸୯७ϧʔτͷ
ू߹ʣͱ͢Δͱɼͦͷ૒ର‫ج‬ఈ Π∗ = {ω1 , · · · , ωn } ͕૒ରۭؒ (E n )∗ ͷதʹܾ·Δɽಛʹ
ωi (αj ) = δij Ͱ͋Γɼωi ͸‫ج‬ຊί΢ΣΠτͱ‫ݺ‬͹ΕΔɽ͜ͷͱ͖
σ0 : = {f ∈ (E n )∗ | i = 1, · · · , n ʹର͠ f (αi ) ≥ 0}
= {c1 ω1 + · · · + cn ωn ∈ (E n )∗ | i = 1, · · · , n ʹର͠ ci ∈ R≥0 }
ͱ͓͖ɼ֤ u ∈ W ʹର͠ σu := uσ0 ͱఆΊΔɽͨͩ͠ɼWeyl ‫ ܈‬W ͷ (E n )∗ ΁ͷ࡞༻͸ E n
΁ͷ࡞༻ͷ૒ର࡞༻Ͱ༩͑Δɽ֤ σu ͸ Weyl ͷ෦԰ͱ‫ݺ‬͹ΕΔɽ
֤ σu ͸ (E n )∗ ʹ͓͚Δਲ਼Ͱ͋Γɼશͯͷ u ∈ W ʹ͍ͭͯਲ਼ σu ͱͦͷ໘ʢ͜Ε΋·ͨਲ਼
Ͱ͋Δʣͷू·ΓΛߟΔͱͦΕ͸ (E n )∗ ʹ͓͚ΔઔΛͳ͢ɽͨͩ͠ɼ֨ࢠ఺͸ί΢ΣΠτ
֨ࢠ N := ⊕n Zωi (∼
= Zn ) ʹΑͬͯ༩͑Δɽ͜ͷಘΒΕͨઔΛ Δ(Φ) ͱॻ͘͜ͱʹ͢Δɽ
i=1
ྫ 3.1. ϧʔτ‫ ܥ‬Φ ͕ A2 ‫ܕ‬ͷ৔߹ɼզʑͷઔ Δ(A2 ) ͸࣍ͷΑ͏ʹͳΔɽ
(E 2 )∗
E2
α2
ω2
α1
ω1
ͯ͞ɼզʑ͸ϧʔτ‫ ܥ‬Φ ⊂ E n ͔Βͦͷ૒ରۭؒ (E n )∗ ʹ͓͚Δઔ Δ(Φ) Λߏ੒ͨ͠ɽ͜
ͷઔʹରԠ͢ΔτʔϦοΫଟ༷ମΛ X(Φ) ͱॻ͘͜ͱʹ͢Δɽඪ‫ޠ‬తʹ‫ͱ͏ݴ‬ɼ
ʮX(Φ) ͸ Weyl ͷ෦԰ͱί΢ΣΠτ֨ࢠͷ੒͢ઔʹରԠ͢ΔτʔϦοΫଟ༷ମʯ
Ͱ͋Δɽ͜ͷτʔϦοΫଟ༷ମ X(Φ) ͕࣮͸ 1 અʹग़͖ͯͨτʔϦοΫଟ༷ମ X ͦͷ΋ͷ
ͳͷͰ͋Δɽ͢ͳΘͪɼX(Φ) ͸ҰൠԽ͞Εͨ‫ض‬ଟ༷ମ΁ͷඪ४తͳτʔϥε࡞༻ͷҰൠ
‫ي‬ಓͷดแͱͯ͠‫ه‬ड़͢Δ͜ͱ΋Ͱ͖ΔɽX(Φ) ͷҰൠతͳੑ࣭ʹ͍ͭͯ͸ Klyachko [6]
΍ Batyrev-Blume [2] Λࢀর͞Ε͍ͨɽ
ઔ Δ(Φ) ͷߏ੒͔ΒɼϫΠϧ‫ ܈‬W ͕ Δ(Φ) ʹ࡞༻͠ɼΏ͑ʹ X(Φ) ʹ W ͕ࣗવʹ࡞༻͢
Δɽ͜Ε͸ W ͕ H ∗ (X(Φ); C) ্ʹද‫ݱ‬Λ࣋ͭ͜ͱΛҙຯ͢Δɽ Procesi [8]ɼDolgachev-
Lunts [4]ɼStembridge [9] Β͕͜ͷද‫ݱ‬Λৄ͘͠ௐ΂͍ͯΔɽ
第６１回トポロジーシンポジウム講演集２０１４年７月於東北大学
·ͨɼྫ͑͹ϧʔτ‫ ܥ‬Φ ͕ An ‫ܕ‬ͷ৔߹ɼX(An ) ͸ෳૉࣹӨۭؒ CP n ͷ͢΂ͯͷ࠲ඪ
෦෼ۭؒʹԊͬͯ࣍‫ݩ‬ͷ௿͍΋ͷ͔ΒॱʹϒϩʔΠϯάΞοϓΛ‫܁‬Γฦͯ͠ಘΒΕΔۭؒ
Ͱ͋Δ ([8])ɽLosev-Manin [7] ΍ Batyrev-Blume [2] ͳͲ΋ X(An ) ͷ‫ز‬Կʹ͍ͭͯ‫͠ڀݚ‬
͍ͯΔɽ
ઔʹ͓͍ͯಛʹ 1 ࣍‫ݩ‬ਲ਼͸τʔϥε࡞༻Ͱෆมͳෳૉ༨࣍‫ ݩ‬1 ͷ෦෼ଟ༷ମͱ 1 ର 1 ʹର
Ԡ͠ɼτʔϦοΫଟ༷ମͷ‫ز‬Կ΍τϙϩδʔΛ‫ه‬ड़͢Δ͏͑ͰॏཁͰ͋Δɽզʑͷઔ Δ(Φ)
ͷ৔߹ɼ1 ࣍‫ݩ‬ਲ਼ͷੜ੒‫ݩ‬શମ͸ί΢ΣΠτͷ੒͢ू߹
Φ∗ := ∪u∈W {uω1 , · · · , uωn } ⊂ (E n )∗
Ͱ༩͑ΒΕΔɽઔ Δ(Φ) ͷੑ࣭͔ΒɼτʔϦοΫଟ༷ମ X(Φ) ͸ίϯύΫτͰඇಛҟͳτʔ
ϦοΫଟ༷ମͰ͋Δ͜ͱ͕ै͏ɽΔ(Φ) ͷߏ੒͔Β͸໌Β͔Ͱ͸ͳ͍͕ɼ࣮͸ X(Φ) ͸ࣹ
Өଟ༷ମͰ͋Δ2 ɽ
ຊདྷɼτʔϦοΫଟ༷ମ͸ઔͷσʔλΛ༻͍ͯΞϑΟϯτʔϦοΫଟ༷ମͷுΓ߹Θͤ
ͱͯ͠ఆٛ͢Δͷ͕ҰൠͷఆٛͰ͋Δ͕ɼզʑͷτʔϦοΫଟ༷ମ X(Φ) ͸ίϯύΫτͳ
ͷͰɼͦͷΑ͏ͳτʔϦοΫଟ༷ମʹରͯ͠࢖͏͜ͱͷͰ͖Δผͷ‫ه‬ड़ʹΑΓ X(Φ) Λ
∗
ఆ͓ٛͯ͘͠ɽCΦ Λ Φ∗ ͷ‫ʹͱ͝ݩ‬੒෼Λ࣋ͭෳૉϕΫτϧͷશମͱ͢Δɽ͢ͳΘͪɼ
∗
∗
CΦ = {(zx )x∈Φ∗ | zx ∈ C} Ͱ͋ΔɽCΦ ͷ‫ ݩ‬z ʹର͠ɼ੒෼͕ 0 ʹͳ͍ͬͯΔ࠲ඪͷू·
ΓΛ Φ∗ (z) = {x ∈ Φ∗ | zx = 0} ͱ͓͖ɼ͜ΕΛ༻͍ͯ
∗
U (Φ) = {z ∈ CΦ | Φ∗ (z) ͸ઔ Δ(Φ) ʹ͓͍ͯਲ਼ΛுΔ }
∗
= {z ∈ CΦ | Φ∗ (z) ⊂ {uω1 , · · · , uωn } ͱͳΔ u ∈ W ͸ଘࡏ͠ͳ͍ }
ͱఆΊΔɽ͜ͷͱ͖ɼ
X(Δ(Φ)) = U (Φ)/ ker ν
(1)
ͨͩ͠ɼ
∗
ν : (C× )Φ → (C× )n
;
t = (tx )x∈Φ∗ →
n ,x
1 ,x
(tα
, · · · , tα
)
x
x
x∈Φ∗
∗
ߏ੒͔Βɼn ࣍‫ݩ‬ͷ঎τʔϥε (C× )Φ / ker ν (͜Ε͸४ಉ‫ ܕ‬ν ʹΑΓ (C× )n ͱಉ‫͕ )ܕ‬
X(Δ(Φ)) ʹࣗવʹ࡞༻͠ɼ͜ͷ࡞༻ʹΑΓ X(Φ) ͸τʔϦοΫଟ༷ମͱͳΔɽ
4
τʔϦοΫଟ༷ମ X(Φ) ͷίϗϞϩδʔ‫؀‬
Φ Λϧʔτ‫͠ͱܥ‬ɼX(Φ) ΛલઅͰߏ੒ͨ͠τʔϦοΫଟ༷ମͱ͢ΔɽX(Φ) Λ‫ه‬ड़͢Δ
ઔ Δ(Φ) ͷ 1 ࣍‫ݩ‬ਲ਼ͷੜ੒‫ݩ‬ͷू߹͕ Φ∗ = ∪u∈W {uω1 , · · · , uωn } Ͱ͋Δ͜ͱ͸ 1 અͰ͢Ͱ
ʹ‫ͨݟ‬ɽࣜ (1) ʹ͓͍֤ͯ x ∈ Φ∗ ͝ͱʹɼୈ x ൪໨ͷ࠲ඪ͕θϩͱ͍͏৚݅Ͱఆ·Δෳૉ
༨࣍‫ ݩ‬1 ͷ෦෼ଟ༷ମ Dx ⊂ X(Φ) ͕ఆ·Δʢಛੑ෦෼ଟ༷ମͱ‫ݺ‬͹ΕΔʣɽDx ͷϙΞϯ
2
‫ض‬ଟ༷ମͷӡಈྔଟ໘ମʹରԠ͢ΔඇಛҟࣹӨతτʔϦοΫଟ༷ମͰ͋Δ͜ͱ͕஌ΒΕ͍ͯΔɽ
第６１回トポロジーシンポジウム講演集２０１４年７月於東北大学
ΧϨ૒ରΛ τx := [Dx ] ͱॻ͘ͱɼτx ∈ H 2 (X(Φ); Z) Ͱ͋ΔɽίϯύΫτ͔ͭඇಛҟͳτʔ
ϦοΫଟ༷ମͷಛҟίϗϞϩδʔ‫؀‬Λଟ߲ࣜ‫؀‬ͷ঎ͱͯ͠‫ه‬ड़͢Δํ๏͸Α͘஌ΒΕ͓ͯ
Γ ([5])ɼզʑͷτʔϦοΫଟ༷ମ X(Φ) ͷ৔߹ʹ͸࣍ͷΑ͏ʹදࣔ͞ΕΔɽ
H ∗ (X(Φ); Z) = Z[τx | x ∈ Φ∗ ]/I
͜͜ͰɼI ͸࣍ͷ‫ܗ‬ͷ‫Ͱݩ‬ੜ੒͞ΕΔ Z[τx | x ∈ Φ∗ ] ͷΠσΞϧͰ͋Δɽ
(i) τx1 · · · τxk ͨͩ͠ xi = uωi (i = 1, · · · , k) ͱͳΔΑ͏ͳ u ∈ W ͸ଘࡏ͠ͳ͍
(ii)
n
i=1 α, xτx
(α ∈ Φ)
ͯ͞ɼϫΠϧ‫܈‬ͷ‫ ݩ‬u ∈ W ʹର͠ɼX(Φ) ͷ෦෼ଟ༷ମ Xu ͕࣍ͷΑ͏ʹͯ͠ఆ·Δɽ
Xu :=
Duωi
i∈[n]
uαi ∈Φ−
ͨͩ͠ɼ[n] = {1, 2, · · · , n} Ͱ͋ΓɼΦ− ͸ෛϧʔτͷू߹Ͱ͋ΔɽXu ͷϙΞϯΧϨ૒ର
[Xu ] ͸ίϗϞϩδʔ‫ ؀‬H ∗ (X(Φ); Z) ͷ‫ݩ‬ΛఆΊΔɽަΘΓ͕ۭͰͳ͍૬ҟͳΔಛੑ෦෼ଟ
༷ମ Dx1 , · · · , Dxk ͸ԣஅతʹަΘΔͷͰɼ[Xu ] ͸࣍ͷΑ͏ʹͱॻ͘͜ͱ͕Ͱ͖Δɽ
[Xu ] =
τuωi
∈ H ∗ (X(Φ); Z)
(2)
i∈[n]
uαi ∈Φ−
໋୊ 4.1. (De Mari-Procesi-Shayman [3]) [Xu ] ୡ͸ X(Φ) ͷίϗϞϩδʔ‫܈‬ͷ‫ج‬ఈΛ੒͢ɿ
H ∗ (X(Φ); Z) = u∈W Z[Xu ]ɽ
ͦ͜ͰɼΧοϓੵͷల։
[Xu ][Xv ] =
w
cw
uv [Xw ] (cuv ∈ Z)
w∈W
w
∗
ʹ͓͍ͯల։܎਺ cw
uv ∈ Z ͕‫ݱ‬ΕΔɽ͜ͷ܎਺ cuv ͸ίϗϞϩδʔ‫ ؀‬H (X(Φ); Z) ͷʢ‫ج‬
ఈ {[Xw ]}w∈W ʹؔ͢Δʣߏ଄ఆ਺ͱ‫ݺ‬͹ΕΔɽংઆͰ΋ॻ͍͕ͨɼߏ଄ఆ਺ cw
uv ͕ͲͷΑ
͏ʹ‫ه‬ड़͞ΕΔ͔Λௐ΂͍ͨ3 ɽ
গؒ͠઀తʹ͸ͳΔ͕ɼҰͭͷํ๏ͱͯ͠ަࠥ਺Λ༻͍Δํ๏͕͋ΔɽX(Φ) ͷϗϞϩ
δʔ‫ج‬ຊྨΛ μX(Φ) ͱॻ͘ͱ͖ɼϗϞϩδʔͱίϗϞϩδʔͷϖΞϦϯά ( , ) Λ༻͍ͯ
X(Φ) ʹ͓͚Δ Y w , Xu , Xv ͷަࠥ਺Λ
w
Iuv
:= (μX(Φ) , [Y w ][Xu ][Xv ]) ∈ Z
3
͜ͷ໰୊͸‫ض‬ଟ༷ମ΍άϥεϚϯଟ༷ମͷγϡʔϕϧτΧϧΩϡϥεͷτʔϦοΫଟ༷ମʹ͓͚Δྨࣅ
໰୊ͱ‫ͯͬݴ‬Α͍ͱࢥ͏͕ɼߏ଄ఆ਺͸ෛʹͳΔ͜ͱ΋͋Δɽ
第６１回トポロジーシンポジウム講演集２０１４年７月於東北大学
ʹΑΓఆΊΔɽͨͩ͠ɼY w := w0 Xw0 w Ͱ͋Γɼw0 ∈ W ͸࠷௕‫͋Ͱݩ‬ΔɽαΠζ͕ |W |
w
ͷߦྻ I Λ Iuv = (μX(Φ) , [Y u ][Xv ]) ͰఆΊΔͱɼߏ଄ఆ਺ cw
uv ͱަࠥ਺ Iuv ͷؒʹ࣍ͷؔ܎
͕͋Δ͜ͱ͕ै͏ɽ
w
Iuv
=
Iww cw
uv
w ∈W
࣮͸ߦྻ I ͸ Z ্ͷՄ‫͋Ͱྻߦٯ‬Γɼߏ଄ఆ਺͸ަࠥ਺ʹΑͬͯ‫ه‬ड़͞ΕΔɽ
cw
uv =
w
(I −1 )ww Iuv
w ∈W
w
4
͜ͷΑ͏ʹͯ͠ߏ଄ఆ਺ cw
uv ͱަࠥ਺ Iuv ͸Մ‫ͳٯ‬ઢ‫ܗ‬ม‫͓͍ͯͭͼ݁Ͱ׵‬Γ ɼ͜ͷҙຯ
Ͱ྆ऀ͸౳ՁͳྔͰ͋Δͱ‫͑ݴ‬Δɽ
ίϗϞϩδʔ‫؀‬ͷߏ଄ఆ਺ cw
uv
ަࠥߦྻ I
w
Xu , Xv , Y w ͷަࠥ਺ Iuv
w
ϠϯάਤΛ༻͍ͯަࠥ਺ Iuv
Λ‫͢ࢉܭ‬Δํ๏ʹ͍ͭͯ࣍અͰड़΂Δɽ
5
ϠϯάਤʹΑΔަࠥ਺ͷ‫ࢉܭ‬
͜ͷઆͰ͸ϠϯάਤΛ༻͍ͨަࠥ਺ͷ‫͍ͯͭʹࢉܭ‬આ໌͢ΔɽAn ‫͍ͯͭʹܕ‬ͷΈղઆ
͢Δ͕ɼଞͷ‫ݹ‬య‫ܕ‬ϧʔτ‫ ܥ‬B, C, D ‫ܕ‬ͷ৔߹΋ಉ༷ͳٞ࿦ΛਐΊ͍ͯ͘͜ͱ͕Ͱ͖Δɽ
ஔ‫ ׵‬u ∈ Sn+1 ʹରͯ͠ Xu ⊂ X(An ) ͷϙΞϯΧϨ૒ର [Xu ] ͷ‫ه‬ड़͸ (2) Ͱ༩͕͑ͨɼ
uαi ∈ Φ− ͱ͍͏৚݅͸ u(i) > u(i + 1) ͱಉ஋ͳͷͰɼAn ‫ܕ‬ͷ৔߹ʹ͸࣍ͷΑ͏ʹͳΔɽ
[Xu ] =
τuωi
∈ H ∗ (X(An ); Z)
i∈[n]
u(i)>u(i+1)
ಛʹ [Xu ] ͷ࣍਺͸ 2d(u) Ͱ༩͑ΒΕΔɽ͜͜Ͱ
d(u) := |{i ∈ [n] | u(i) > u(i + 1)}|ɽ
֤ [Xu ] ͸ͲͷΑ͏ͳ τuωi ͷੵʹͳ͍ͬͯΔ͔ͱ͍͏৘ใͰܾ·͓ͬͯΓɼͦͷ৘ใΛϠ
ϯάਤͰද‫͢ݱ‬Δ͜ͱΛߟ͑ΔɽAn ‫ܕ‬ͷͱ͖͸ί΢ΣΠτͷू߹ Φ∗ = {uωi | 1 ≤ i ≤
n, u ∈ Sn+1 } ͱ [n + 1](= {1, 2, · · · , n + 1}) ͷۭͰͳ͍ਅ෦෼ू߹ͷͳ͢ू߹ΛରԠ uωi →
{u(1), · · · , u(i)} ʹΑͬͯಉҰࢹ͢Δ͜ͱ͕Ͱ͖Δɽ͜ΕΛҙ্ࣝͨ͠Ͱɼ࣍ͷΑ͏ʹϠϯ
άਤΛ࡞Δɽ·ͣɼu(i) > u(i + 1) Λຬ֤ͨ͢ i ͕֤ߦͷശͷ਺Ͱ͋ΔΑ͏ͳϠϯάਤΛ
࡞Δɽͦͯͦ͠ͷϠϯάਤͷ্ʹ u(1), u(2), · · · ୡΛ͜ͷॱͰฒ΂Δɽஔ‫ ׵‬u Λͦͷ஋ͷ
ྻ u(1)u(2) · · · u(n) Ͱॻ͘ͱ͖ɼྫ͑͹
4153
u = 41532 ͳΒ͹ɺ[Xu ] = τuω1 τuω3 τuω4 ͸
4
Ͱද͞ΕΔɽ
ߦྻ I ͸্ࡾ֯ߦྻͱࢥ͏͜ͱ͕Ͱ͖ɼඞཁͳΒ‫ྻߦٯ‬Λ I ͷ‫ݴ‬༿Ͱ۩ମతʹॻ͖Լ͢͜ͱ΋Ͱ͖Δɽ
第６１回トポロジーシンポジウム講演集２０１４年７月於東北大学
ͭ·ΓɼϠϯάਤͷ֤ߦ͕ 1 ͭͷ τuωi Λද͠ɼͦͷఴ͑ࣈ uωi ʹରԠ͢Δ {u(1), · · · , u(i)}
͕ͪΐ͏Ͳͦͷߦͷ্ʹฒͿΑ͏ʹ͍ͯ͠ΔͷͰ͋Δɽ
ಉ༷ʹͯ͠ɼ[Y w ] = [w0 Xw0 w ] ͸ u(i) < u(i + 1) Λຬ֤ͨ͢ i Λσʔλͱͯ͠࡞ͬͨϠ
ϯάਤʹΑͬͯදࣔ͢Δ͜ͱ͕Ͱ͖Δʢ͙͢Լͷྫࢀরʣɽ
ஔ‫ ׵‬u, v, w ∈ Sn+1 ʹର͠ɼϠϯάਤ λw
uv Λ࣍ͷΑ͏ʹͯ͠ఆΊΔɽ΋͠ d(u) + d(v) =
d(w) Ͱɼ͔ͭ Xu , Xv , Y w Λද͢Ϡϯάਤͷ֤ߦͷ্ʹฒͿ਺ࣈͷू߹͕ʢॱংΛద౰ʹ
ೖΕସ͑ͯʣแ‫ྻؚ‬Λͳ͍ͯ͠ΔͳΒ͹ɼͦΕͧΕͷू߹ͷҐ਺Λ޿ٛ୯ௐ‫ݮ‬গʹͳΔΑ
w
͏ʹฒ΂ͯಘΒΕΔϠϯάਤΛ λw
uv ͱ͠ɼͦΕҎ֎ͷ৔߹͸ λuv = ∅ ͱఆΊΔɽ
ྫ 5.1. n = 4 ͷͱ͖ɼu = 31254, v = 12354, w = 35421 ͱ͢Δͱ
3125
X31254 =
,
X12354 =
1235
,
Y 35421 =
3
แ‫ྻؚ‬Λͳ͍ͯ͠Δ͔Ͳ͏͔͔֬ΊΔू߹ྻ͸ {3}, {3, 1, 2, 5}, {1, 2, 3, 5}, {3} Ͱ͋Δ͕ɼ໌
Β͔ʹ {3} = {3} ⊂ {3, 1, 2, 5} = {1, 2, 3, 5} ͱ͍͏แ‫ྻؚ‬Λͳ͢ʢΘ͟Θ͟ू߹Λॻ͖Լ͞
ͳͯ͘΋ϠϯάਤΛ‫ݟ‬Ε͹͙͢ʹ͔֬ΊΒΕΔʣɽΑͬͯ u = 31254, v = 12354, w = 35421
ʹରͯ͠ఆ·ΔϠϯάਤ λw
uv ͸ԼਤͷΑ͏ʹͳΔɽ
λ35421
31254,12354 =
ަࠥ਺ͷ‫ࢉܭ‬ͷͨΊʹɼϠϯάਤʹ஋ΛͱΔؔ਺Λߟ͑Δɽࠓ λ Λ n ߦ͔ΒͳΔϠϯ
άਤͱ͠ɼ֤ߦͷശͷ‫਺ݸ‬͸ n ҎԼͰ͋Δͱ͢Δɽλ ͷತͳ֯ͷ਺Λ s ͱஔ͘ɽ͢ͳΘͪ
s = |{i ∈ [n] | λi > λi+1 }| + 1ɽತͳ֯Λ༩͑Δߦͷ൪߸ͷू߹Λ࣍ͷΑ͏ʹॻ͘ɽ
{i ∈ [n] | λi > λi+1 } ∪ {n} = {i1 , · · · , is }
ͨͩ͠ i1 < i2 < · · · < is ͱ͢Δɽಛʹ is = nɽλ ͷ r ൪໨ͷತͳ֯Λ༩͑Δߦʹ͓͚Δശͷ
਺Λ λr := λir Ͱද͢ʢ1 ≤ r ≤ sʣɽ͜͜Ͱ λs+1 := 0ɽ͜ͷ‫߸ه‬Λ༻͍ͯɼ֤ r = 1, · · · , s
ʹର͠ɼ࣍ͷΑ͏ʹఆΊΔɽ
ar := ir − ir−1 − 1,
br := λr − λr+1 − 1,
cr := λr + ir − n − 1
͜͜Ͱɼ0 ≤ y ≤ x Ͱͳ͚Ε͹ xy = 0 ͱ໿ଋ͢Δɽ͜ΕΒͷ਺ͷਤ‫ܗ‬తͳҙຯ͸ൺֱత୯
७Ͱ͋Δʢਤ 2ʣɽ࣮ࡍɼϠϯάਤ λ ͷʢӈ্͔Βʣr ൪໨ͷತͳ֯ͷശΛࠇ͘ృ͓ͬͯ͘
ͱɼar ͸ࠇശͱͦͷ্ଆʹ͋ΔԜͳ֯·Ͱͷؒʹ͋Δശͷ਺Ͱ͋Γɼbr ͸ࠇശͱͦͷࠨଆ
ʹ͋ΔԜͳ֯·Ͱͷؒʹ͋Δശͷ਺Ͱ͋Γɼcr ͸൓ର֯ઢͱͷަ఺ͱࠇശͷؒʹ͋Δശͷ
਺Ͱ͋Δɽ͜ΕΒͷ਺͸Ϡϯάਤͱ൓ର֯ઢΛ࣮ࡍʹॻ͚͹؆୯ʹ਺͑Δ͜ͱ͕Ͱ͖Δɽ
第６１回トポロジーシンポジウム講演集２０１４年７月於東北大学
ar
λ
cr
br
cr
s = 4 ͸Ϡϯάਤ λ ͷತͳ֯ͷ਺.
ਤ 2: ar , br , cr ͷਤ‫ܗ‬తͳҙຯ
ࠓɼ֤ r = 1, · · · , s ʹରͯ͠
br
ar
yr :=
cr
cr
ͱஔ͖ɼI(λ) Λ࣍ͷΑ͏ʹఆΊΔɽ
I(λ) := (−1)n−s y1 · · · ys
ఆཧ 5.2. ([1]) ஔ‫ ׵‬u, v, w ∈ Sn+1 ʹର͠ɼX(An ) ʹ͓͚Δ Y w , Xu , Xv ͷަࠥ਺͸࣍Ͱ༩
͑ΒΕΔɽ
μX(An ) , [Y w ][Xu ][Xv ] = I(λw
uv )
͜͜ͰɼμX(An ) ͸ X(An ) ͷ‫ج‬ຊྨͰ͋Δɽ
ྫ 5.3. ྫ͑͹ A4 ‫ܕ‬ͷͱ͖ʢϫΠϧ‫܈‬͸ 5 ࣍ͷஔ‫ ܈׵‬S5 ʣɼY 35421 , X12354 , X31254 ͔Βఆ
·ΔϠϯάਤ͸ྫ 5.1 Ͱ͢Ͱʹ༩͓͑ͯΓɼఆཧ͔Βަࠥ਺͕‫͞ࢉܭ‬ΕΔɽ
μX(A4 ) , [Y 35421 ][X12354 ][X31254 ] = 2
λ35421
12354,31254 =
ࢀߟจ‫ݙ‬
[1] H. Abe, Young diagrams and intersection numbers for toric manifolds associated
with Weyl chambers, preprint, arXiv:1404.3805.
[2] V. Batyrev and Mark Blume, The functor for toric varieties associated with Weyl
chambers and Losev-Manin moduli spaces, Tohoku Math. J. 63 (2011), 581-604.
第６１回トポロジーシンポジウム講演集２０１４年７月於東北大学
[3] F. De Mari, C. Procesi and M. A. Shayman, Hessenberg varieties, Trans. Amer.
Math. Soc. 332 (1992), no. 2, 529-534.
[4] V. Dolgachev and V. Lunts, A character formula for the representation of a Wyel
group in the cohomology of the associated toric variety, J. Algebra 168 (1994),
741-772.
[5] W. Fulton, An Introduction to Toric Varieties, Ann. of Math. Stud., vol. 113,
Princeton Univ. Press, Princeton NJ, 1993.
[6] A. Klyachko, Orbits of a maximal torus on a ﬂag space, Functional Anal. Appl.
19 (1985), no. 2, 65-66.
[7] A. Losev and Y. Manin, New moduli spaces of pointed curves and pencils of ﬂat
connections, Michigan Math. J. 48 (2000), 443-472.
[8] C. Procesi, The toric variety associated to Weyl chambers, Mots, 153-161, Lang.
Raison. Calc., Herm`es, Paris, 1990.
[9] J. Stembridge, Some permutation representations of Weyl groups associated with
the cohomology of toric varieties, Adv. Math. 106 (1994), 244-301.

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