On the Hilbert polynomials and Hilbert series of homogeneous projective varieties

Among all complex projective varieties X →֒ P(V ), the equivarient embeddings of homogeneous varieties—those admitting a transitive action of a semi-simple complex algebraic group G—are the easiest to study. These include projective spaces, Grassmannians, non-singular quadrics, Segre varieties, and Veronese varieties. In Joe Harris’ book “Algebraic Geometry: A First Course” [H], he computes the dimension d = dim(X) and degree deg(X) of X →֒ P(V ) for many homogeneous varieties, in a geometric fashion. In this expository paper we redo this calculation using some representation theory of G. We determine the Hilbert polynomial h(t) and Hilbert series of the homogeneous coordinate ring of X →֒ P(V ). Since