"Minors and plucker relations"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) (→메타데이터: 새 문단) |
Pythagoras0 (토론 | 기여) |
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| 2번째 줄: | 2번째 줄: | ||
# (mat = Array[Subscript[a, ##] &, {3, 3}]) // MatrixForm Minors[mat] // MatrixForm Minors[mat, 1] // MatrixForm Minors[mat, 2] // MatrixForm Minors[mat, 3] // MatrixForm | # (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, | + | # 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])] |
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==3-term Plucker relation (Ptolemy relation)== | ==3-term Plucker relation (Ptolemy relation)== | ||
| 17번째 줄: | 17번째 줄: | ||
* <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], | + | # 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]] | # Simplify[Minor[1, 2] Minor[3, 4] + Minor[1, 4] Minor[2, 3]] Simplify[Minor[1, 3] Minor[2, 4]] | ||
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==Plucker relations== | ==Plucker relations== | ||
| 30번째 줄: | 30번째 줄: | ||
# \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} | ||
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==Plucker coordinates of a Grassmannian== | ==Plucker coordinates of a Grassmannian== | ||
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==memo== | ==memo== | ||
2020년 12월 28일 (월) 04:14 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
memo
- 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
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