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