Relationship to Classical Invariant Theory

In this chapter, we describe a connection between classical invariant theory and the Grassmannian variety. Namely, for G = SL d (K), X the space of n × d matrices (n > d), and R = K[x ij ∣1 ≤ i ≤ n, 1 ≤ j ≤ d] (X = Spec(R)), we have a G action on X by right multiplication, and hence a G action on R. We will show that the categorical quotient Open image in new window is isomorphic to the cone over the Grassmannian, and thus obtain a K-basis for R G , the ring of invariants, consisting of standard monomials. In this chapter, we shall work just with the closed points of an algebraic variety.