Grassmannian variety

introduction

• minors and plucker relations

 

 

\(Gr_{kn}(\mathbb{R})=GL_k\Mat(k,n)\)

 

 

 

Plücker embedding

\(N=\binom{n}{k}\)

\(Gr_{kn}(\mathbb{R}) \to \mathbb{P}^{N-1}\)

 

\(P_{I}(A)\) = determinant of submatrix of A  with column set I

\(\begin{array}{l} \Delta _{1,2}=a_{1,1} a_{2,2}-a_{1,2} a_{2,1} \\ \Delta _{1,3}=a_{1,1} a_{2,3}-a_{1,3} a_{2,1} \\ \Delta _{1,4}=a_{1,1} a_{2,4}-a_{1,4} a_{2,1} \\ \Delta _{2,3}=a_{1,2} a_{2,3}-a_{1,3} a_{2,2} \\ \Delta _{2,4}=a_{1,2} a_{2,4}-a_{1,4} a_{2,2} \\ \Delta _{3,4}=a_{1,3} a_{2,4}-a_{1,4} a_{2,3} \end{array}\)

 

 

expositions

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