Minors and plucker relations
imported>Pythagoras0님의 2020년 11월 13일 (금) 05:35 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