A system of quadrics describing the orbit of the highest weight vector

Abstract

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a complex semisimple Lie group acting irreducibly on a finite dimensional vector space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper V"> <mml:semantics> <mml:mi>V</mml:mi> <mml:annotation encoding="application/x-tex">V</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. A simple method is given for constructing a system of quadratic equations which defines the orbit of the highest weight vector in the projective space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P upper V"> <mml:semantics> <mml:mrow> <mml:mi>P</mml:mi> <mml:mi>V</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">PV</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.