"Minors and plucker relations"의 두 판 사이의 차이
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(피타고라스님이 이 페이지의 위치를 <a href="/pages/7642905">Grassmannian variety</a>페이지로 이동하였습니다.) |
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| 14번째 줄: | 14번째 줄: | ||
<h5>3-term Plucker relation (Ptolemy relation)</h5> | <h5>3-term Plucker relation (Ptolemy relation)</h5> | ||
| − | <math>\Delta _{i,k} \Delta _{j,l}=\Delta _{i,j} \Delta _{k,l}+\Delta _{i,l} \Delta _{j,k}</math> | + | * <math>\Delta _{i,k} \Delta _{j,l}=\Delta _{i,j} \Delta _{k,l}+\Delta _{i,l} \Delta _{j,k}</math> |
| − | + | * <math>\Delta _{1,2}\Delta _{3,4}+\Delta _{1,4}\Delta _{2,3}=\Delta _{1,3}\Delta _{2,4}</math> | |
| − | <math>\Delta _{1,2}\Delta _{3,4}+\Delta _{1,4}\Delta _{2,3}=\Delta _{1,3}\Delta _{2,4}</math> | ||
# T := {{Subscript[a, 1, 1], Subscript[a, 1, 2], Subscript[a, 1, 3],<br> Subscript[a, 1, 4]}, {Subscript[a, 2, 1], Subscript[a, 2, 2],<br> Subscript[a, 2, 3], Subscript[a, 2, 4]}}<br> Minor[i_, j_] := Det[{Transpose[T][[i]], Transpose[T][[j]]}]<br> Minor[1, 2] | # T := {{Subscript[a, 1, 1], Subscript[a, 1, 2], Subscript[a, 1, 3],<br> Subscript[a, 1, 4]}, {Subscript[a, 2, 1], Subscript[a, 2, 2],<br> Subscript[a, 2, 3], Subscript[a, 2, 4]}}<br> Minor[i_, j_] := Det[{Transpose[T][[i]], Transpose[T][[j]]}]<br> Minor[1, 2] | ||
| 27번째 줄: | 26번째 줄: | ||
<h5>Plucker relations</h5> | <h5>Plucker relations</h5> | ||
| − | <math>\Delta _{1,2}\Delta _{12,13}=\Delta _{1,3}\Delta _{12,12}+\Delta _{1,1}\Delta _{12,23}</math> | + | * <math>\Delta _{1,2}\Delta _{12,13}=\Delta _{1,3}\Delta _{12,12}+\Delta _{1,1}\Delta _{12,23}</math> |
# \Delta _{12,12}\text{:=}-a_{1,2} a_{2,1}+a_{1,1} a_{2,2}\Delta _{12,23}\text{:=}-a_{1,3} a_{2,2}+a_{1,2} a_{2,3}\Delta _{1,3}\text{:=}a_{1,1}\Delta _{1,3}\text{:=}a_{1,3}\Delta _{1,3}\Delta _{12,12}+\Delta _{1,1}\Delta _{12,23} | # \Delta _{12,12}\text{:=}-a_{1,2} a_{2,1}+a_{1,1} a_{2,2}\Delta _{12,23}\text{:=}-a_{1,3} a_{2,2}+a_{1,2} a_{2,3}\Delta _{1,3}\text{:=}a_{1,1}\Delta _{1,3}\text{:=}a_{1,3}\Delta _{1,3}\Delta _{12,12}+\Delta _{1,1}\Delta _{12,23} | ||
| 36번째 줄: | 35번째 줄: | ||
<h5>Plucker coordinates of a Grassmannian</h5> | <h5>Plucker coordinates of a Grassmannian</h5> | ||
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| 45번째 줄: | 42번째 줄: | ||
<h5>memo</h5> | <h5>memo</h5> | ||
| − | [http://www.math.msu.edu/%7Emagyar/papers/MinorIdentities.pdf http://www.math.msu.edu/~magyar/papers/MinorIdentities.pdf] | + | * [http://www.math.msu.edu/%7Emagyar/papers/MinorIdentities.pdf http://www.math.msu.edu/~magyar/papers/MinorIdentities.pdf] |
| − | + | * http://www.ams.org/journals/proc/2008-136-01/S0002-9939-07-09122-8/S0002-9939-07-09122-8.pdf | |
| − | http://www.ams.org/journals/proc/2008-136-01/S0002-9939-07-09122-8/S0002-9939-07-09122-8.pdf | ||
2012년 8월 26일 (일) 12:44 판
introduction
- (mat = Array[Subscript[a, ##] &, {3, 3}]) // MatrixForm
Minors[mat] // MatrixForm
Minors[mat, 1] // MatrixForm
Minors[mat, 2] // MatrixForm
Minors[mat, 3] // MatrixForm - Simplify[Subscript[a, 1,
3]*(-Subscript[a, 1, 2] Subscript[a, 2, 1] +
Subscript[a, 1, 1] Subscript[a, 2, 2]) +
Subscript[a, 1,
1]*(-Subscript[a, 1, 3] Subscript[a, 2, 2] +
Subscript[a, 1, 2] Subscript[a, 2, 3])]
3-term Plucker relation (Ptolemy relation)
- \(\Delta _{i,k} \Delta _{j,l}=\Delta _{i,j} \Delta _{k,l}+\Delta _{i,l} \Delta _{j,k}\)
- \(\Delta _{1,2}\Delta _{3,4}+\Delta _{1,4}\Delta _{2,3}=\Delta _{1,3}\Delta _{2,4}\)
- T := {{Subscript[a, 1, 1], Subscript[a, 1, 2], Subscript[a, 1, 3],
Subscript[a, 1, 4]}, {Subscript[a, 2, 1], Subscript[a, 2, 2],
Subscript[a, 2, 3], Subscript[a, 2, 4]}}
Minor[i_, j_] := Det[{Transpose[T]i, Transpose[T]j}]
Minor[1, 2] - Simplify[Minor[1, 2] Minor[3, 4] + Minor[1, 4] Minor[2, 3]]
Simplify[Minor[1, 3] Minor[2, 4]]
Plucker relations
- \(\Delta _{1,2}\Delta _{12,13}=\Delta _{1,3}\Delta _{12,12}+\Delta _{1,1}\Delta _{12,23}\)
- \Delta _{12,12}\text{:=}-a_{1,2} a_{2,1}+a_{1,1} a_{2,2}\Delta _{12,23}\text{:=}-a_{1,3} a_{2,2}+a_{1,2} a_{2,3}\Delta _{1,3}\text{:=}a_{1,1}\Delta _{1,3}\text{:=}a_{1,3}\Delta _{1,3}\Delta _{12,12}+\Delta _{1,1}\Delta _{12,23}
Plucker coordinates of a Grassmannian