Universality of Rank 6 Plücker relations and Grassmann cone preserving maps

Abstract

The Plücker relations define a projective embedding of the Grassmann variety <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G r left-parenthesis p comma n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mi>r</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Gr(p,n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We give another finite set of quadratic equations which defines the same embedding, and whose elements all have rank 6. This is achieved by constructing a certain finite set of linear maps <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="logical-and Overscript p Endscripts k Superscript n Baseline right-arrow logical-and Overscript 2 Endscripts k Superscript 4"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mo> ⋀ </mml:mo> <mml:mi>p</mml:mi> </mml:mover> <mml:msup> <mml:mi>k</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false"> → </mml:mo> <mml:mover> <mml:mo> ⋀ </mml:mo> <mml:mn>2</mml:mn> </mml:mover> <mml:msup> <mml:mi>k</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\bigwedge ^pk^n\to \bigwedge ^2k^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and pulling back the unique Plücker relation on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="logical-and Overscript 2 Endscripts k Superscript 4"> <mml:semantics> <mml:mrow> <mml:mover> <mml:mo> ⋀ </mml:mo> <mml:mn>2</mml:mn> </mml:mover> <mml:msup> <mml:mi>k</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\bigwedge ^2k^4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also give a quadratic equation depending on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis p plus 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(p+2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> parameters having the same properties.