Minors and plucker relations

introduction

1. (mat = Array[Subscript[a, ##] &, {3, 3}]) // MatrixForm Minors[mat] // MatrixForm Minors[mat, 1] // MatrixForm Minors[mat, 2] // MatrixForm Minors[mat, 3] // MatrixForm
2. 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)

• \(\Delta _{i,k} \Delta _{j,l}=\Delta _{i,j} \Delta _{k,l}+\Delta _{i,l} \Delta _{j,k}\)
• \(\Delta _{1,2}\Delta _{3,4}+\Delta _{1,4}\Delta _{2,3}=\Delta _{1,3}\Delta _{2,4}\)

1. 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]
2. Simplify[Minor[1, 2] Minor[3, 4] + Minor[1, 4] Minor[2, 3]] Simplify[Minor[1, 3] Minor[2, 4]]

 

 

Plucker relations

• \(\Delta _{1,2}\Delta _{12,13}=\Delta _{1,3}\Delta _{12,12}+\Delta _{1,1}\Delta _{12,23}\)

1. \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}

 

 

 

 

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