ochiai/orthogonal
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開始行:
Toshihiko Matsuki; Orthogonal multiple flag varieties of ...
Part I
-p01. 1. Introduction.
-p04. 2. Exclusion of multiple flag varieties of infinite...
-p04. 2.1. A technical lemma.
-p05. 2.2. The case of $n=1$.
-p05. 2.3. Proof of Proposition 1.2.
-p06. 2.4. A lemma for O$_5(\mathbb{F})$.
-p07. 2.5. Proof of Proposition 1.3.
-p07. 2.6. First lemma for O$_6(\mathbb{F})$.
-p09. 2.7. Second lemma for O$_6(\mathbb{F})$.
-p09. 2.8. Some conditions on a,b and c.
-p10. 2.9. A lemma for O$_7(\mathbb{F})$.
-p11. 2.10. Exclusion by Lemma 2.16.
-p11. 2.11. Conclusion.
-p11. 2.12. Proof of Proposition 1.4. *
-p12. 3. Orbits on $\mathcal{T}_{(\alpha),(\beta),(\gamma...
-p12. 3.1. Preliminaries.
-p15. 3.3. Structure of $R=...$.
-p16. 3.4. Invariants of the $R$-orbit of $V \in M = M_{(...
-p17. 3.5. Representative of the $R$-orbit of $V$.
-p23. 3.6. Proof of Theorem 3.15. (i)-(ix).
-p31. 3.7. Construction of elements in $R_V|_{U_+}$.
-p40. 4. Finiteness of $\mathcal{T}_{(\alpha_1,\alpha_2),...
-p40. 4.1. First reduction.
-p41. 4.2. Second reduction.
-p41. 4.3. Third reduction.
-p43. 4.4. Finiteness of orbits for $S_3$-part.
-p43. 4.5. A standards form of $S_4$.
-p45. 4.6. Normalization of $g_4 h_5(S_5\oplus S_6)$.
-p46. 4.7. Construction of a subgroup of $R'_V$.
-p47. 4.8. Finiteness of $S_5\oplus S_6$-part.
-p47. 5. Finiteness of $\mathcal{T}_{(\alpha),(1),(1^n)}$.
-p48. 5.1. Case of $a_-=1$.
-p48. 5.1.1. Case of $b_4=1$.
-p49. 5.1.2. Case of $b_{11}=1$.
-p49. 5.1.3. Case of $b_7=1$.
-p51. 5.2. Case of $a_1=1$.
-p51. 5.2.1. Case of $b_5=1$.
-p52. 5.2.2. Case of $b_6=1$.
-p52. 5.2.3. Case of $b_{15}=1$.
-p53. 5.2.4. Case of $b_8=1$.
-p54. 5.2.5. Case of $b_{13}=1$. *
-p55. 6. Appendix.
-p55. 6.1. An elementary lemma for GL$_n(\mathbb{F})$.
-p56. 6.2. Finiteness of some orbit decomposition on the ...
-p57. 6.3. An orbit decomposition of GL$_n(\mathbb{F})/B$.
-p60. References.
Part II
-p01. 1. Introduction
-p04. 2. Preliminaries
-p06. 3. Exclusion of multiple flag varieties of infinite...
-p06. 3.1. Proof of Proposition 1.2.
-p08. 3.2. Proof of Proposition 1.3.
-p10. 3.3. Proof of Proposition 1.4.
-p11. 3.4. Proof of Proposition 1.5.
-p14. 3.5. Case of a=(n), b=$(\beta)$ with $4\le \beta \l...
-p16. 3.5.1. Case of $r=3$ and $\gamma_1+\gamma_2+\gamma_...
-p17. 3.5.2. Case of $r=3$ and $\gamma_1+\gamma_2+\gamma_...
-p17. 3.5.3. Case of $r=4$ and $\gamma_1+\gamma_2+\gamma_...
-p18. 3.5.4. Case of $r=4$ and $\gamma_1+\gamma_2+\gamma_...
-p18. 3.5.5. Case of $r\ge 5$.
-p19. 3.6. Proof of Proposition 1.6(ii). *
-p20. 3.7. Proof of Proposition 1.6(iii). *
-p21. 4. Triple flag varieties of finite type.
-p22. 5. Review of technical results in [M15].
-p22. 5.1. Normalization of $U_+$ and $U_-$.
-p22. 5.2. Invariants of the $R$-orbit of $V\in M = M_{(n...
-p24. 5.3. Representative of the $R$-orbit of $V$.
-p25. 5.4. Construction of elements in $R_V$.
- p28. 6. Proof of Proposition 4.1.
- p28. 6.1. Reduction of flags in $U_+$.
- p31. 6.2. Proof of (i) and (ii).
-p33. 6.3. Proof of (iii). (A),(B),(C).
-p33. 6.4. Proof of (iv). (A),(B),(B.1),(B.2),(B.2.1), (B...
-p36. 6.5. Proof of (v). (A),(A.1), (A.2), (B),(C).
- p37. 7. Proof of Proposition 4.2 and 4.3.
- p37. 7.1. Proof of Proposition 4.2.
(A), (B), (C), (D), (D.1), (D.1.1), (D.1.2), (D.1.3), (D....
-p45. 7.2. Proof of Proposition 4.3.
(A),(B)*,(C),(C.0), (C.1),(C.2),(C.3).
- p47. 8. Proof of Proposition 4.4.
- p47. 8.1. Proof of the case of $\mathbf{b}=(\beta)=(n-1...
(A), (B),(C).
- p49. 8.2. Proof of the case of $\mathbf{b}=(1,n-1)$.
(A),(B),(C),(D),(E),(F).
- p54. 9. Proof of Proposition 4.5.
(A),(B),(C),(D),(E),(F)(i)*, (F)(ii), (G)(i)*, (G)(ii). p57
(A), (A.1),(A.2),(B),(B.1), (B.2), (B.3), (B.4), (B.5), (...
- p61. 10. Proof of Proposition 4.6.
(A),(A.1), (A.2), (A.3), (B), (B.1),(B.2),(B.3),(C),(D)(i...
- p69. 11. Proof of Proposition 4.7.
(A),(B),(C),(D),(E), (E.1),(E.2), (E.3) *, (E.4)
- p76. 12. Appendix
- p76. 12.1. Two lemmas.
- p77. 12.2. Extensions of Proposition 6.3 in [M15].
- p82. 12.3. Orbit decompositions of the full flag variety.
(A)-(K).
- p91. 12.4. A lemma.
- p92. 12.5. Lemmas related with SL$_2(\mathbb{F})$.
- p96. References
終了行:
Toshihiko Matsuki; Orthogonal multiple flag varieties of ...
Part I
-p01. 1. Introduction.
-p04. 2. Exclusion of multiple flag varieties of infinite...
-p04. 2.1. A technical lemma.
-p05. 2.2. The case of $n=1$.
-p05. 2.3. Proof of Proposition 1.2.
-p06. 2.4. A lemma for O$_5(\mathbb{F})$.
-p07. 2.5. Proof of Proposition 1.3.
-p07. 2.6. First lemma for O$_6(\mathbb{F})$.
-p09. 2.7. Second lemma for O$_6(\mathbb{F})$.
-p09. 2.8. Some conditions on a,b and c.
-p10. 2.9. A lemma for O$_7(\mathbb{F})$.
-p11. 2.10. Exclusion by Lemma 2.16.
-p11. 2.11. Conclusion.
-p11. 2.12. Proof of Proposition 1.4. *
-p12. 3. Orbits on $\mathcal{T}_{(\alpha),(\beta),(\gamma...
-p12. 3.1. Preliminaries.
-p15. 3.3. Structure of $R=...$.
-p16. 3.4. Invariants of the $R$-orbit of $V \in M = M_{(...
-p17. 3.5. Representative of the $R$-orbit of $V$.
-p23. 3.6. Proof of Theorem 3.15. (i)-(ix).
-p31. 3.7. Construction of elements in $R_V|_{U_+}$.
-p40. 4. Finiteness of $\mathcal{T}_{(\alpha_1,\alpha_2),...
-p40. 4.1. First reduction.
-p41. 4.2. Second reduction.
-p41. 4.3. Third reduction.
-p43. 4.4. Finiteness of orbits for $S_3$-part.
-p43. 4.5. A standards form of $S_4$.
-p45. 4.6. Normalization of $g_4 h_5(S_5\oplus S_6)$.
-p46. 4.7. Construction of a subgroup of $R'_V$.
-p47. 4.8. Finiteness of $S_5\oplus S_6$-part.
-p47. 5. Finiteness of $\mathcal{T}_{(\alpha),(1),(1^n)}$.
-p48. 5.1. Case of $a_-=1$.
-p48. 5.1.1. Case of $b_4=1$.
-p49. 5.1.2. Case of $b_{11}=1$.
-p49. 5.1.3. Case of $b_7=1$.
-p51. 5.2. Case of $a_1=1$.
-p51. 5.2.1. Case of $b_5=1$.
-p52. 5.2.2. Case of $b_6=1$.
-p52. 5.2.3. Case of $b_{15}=1$.
-p53. 5.2.4. Case of $b_8=1$.
-p54. 5.2.5. Case of $b_{13}=1$. *
-p55. 6. Appendix.
-p55. 6.1. An elementary lemma for GL$_n(\mathbb{F})$.
-p56. 6.2. Finiteness of some orbit decomposition on the ...
-p57. 6.3. An orbit decomposition of GL$_n(\mathbb{F})/B$.
-p60. References.
Part II
-p01. 1. Introduction
-p04. 2. Preliminaries
-p06. 3. Exclusion of multiple flag varieties of infinite...
-p06. 3.1. Proof of Proposition 1.2.
-p08. 3.2. Proof of Proposition 1.3.
-p10. 3.3. Proof of Proposition 1.4.
-p11. 3.4. Proof of Proposition 1.5.
-p14. 3.5. Case of a=(n), b=$(\beta)$ with $4\le \beta \l...
-p16. 3.5.1. Case of $r=3$ and $\gamma_1+\gamma_2+\gamma_...
-p17. 3.5.2. Case of $r=3$ and $\gamma_1+\gamma_2+\gamma_...
-p17. 3.5.3. Case of $r=4$ and $\gamma_1+\gamma_2+\gamma_...
-p18. 3.5.4. Case of $r=4$ and $\gamma_1+\gamma_2+\gamma_...
-p18. 3.5.5. Case of $r\ge 5$.
-p19. 3.6. Proof of Proposition 1.6(ii). *
-p20. 3.7. Proof of Proposition 1.6(iii). *
-p21. 4. Triple flag varieties of finite type.
-p22. 5. Review of technical results in [M15].
-p22. 5.1. Normalization of $U_+$ and $U_-$.
-p22. 5.2. Invariants of the $R$-orbit of $V\in M = M_{(n...
-p24. 5.3. Representative of the $R$-orbit of $V$.
-p25. 5.4. Construction of elements in $R_V$.
- p28. 6. Proof of Proposition 4.1.
- p28. 6.1. Reduction of flags in $U_+$.
- p31. 6.2. Proof of (i) and (ii).
-p33. 6.3. Proof of (iii). (A),(B),(C).
-p33. 6.4. Proof of (iv). (A),(B),(B.1),(B.2),(B.2.1), (B...
-p36. 6.5. Proof of (v). (A),(A.1), (A.2), (B),(C).
- p37. 7. Proof of Proposition 4.2 and 4.3.
- p37. 7.1. Proof of Proposition 4.2.
(A), (B), (C), (D), (D.1), (D.1.1), (D.1.2), (D.1.3), (D....
-p45. 7.2. Proof of Proposition 4.3.
(A),(B)*,(C),(C.0), (C.1),(C.2),(C.3).
- p47. 8. Proof of Proposition 4.4.
- p47. 8.1. Proof of the case of $\mathbf{b}=(\beta)=(n-1...
(A), (B),(C).
- p49. 8.2. Proof of the case of $\mathbf{b}=(1,n-1)$.
(A),(B),(C),(D),(E),(F).
- p54. 9. Proof of Proposition 4.5.
(A),(B),(C),(D),(E),(F)(i)*, (F)(ii), (G)(i)*, (G)(ii). p57
(A), (A.1),(A.2),(B),(B.1), (B.2), (B.3), (B.4), (B.5), (...
- p61. 10. Proof of Proposition 4.6.
(A),(A.1), (A.2), (A.3), (B), (B.1),(B.2),(B.3),(C),(D)(i...
- p69. 11. Proof of Proposition 4.7.
(A),(B),(C),(D),(E), (E.1),(E.2), (E.3) *, (E.4)
- p76. 12. Appendix
- p76. 12.1. Two lemmas.
- p77. 12.2. Extensions of Proposition 6.3 in [M15].
- p82. 12.3. Orbit decompositions of the full flag variety.
(A)-(K).
- p91. 12.4. A lemma.
- p92. 12.5. Lemmas related with SL$_2(\mathbb{F})$.
- p96. References
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