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We present a closed form for a multi-variate generating function for thedimensions of the irreducible representations of a semisimple, simply connectedlinear algebraic group over $\mathbb{C}$ whose highest weights lie in afinitely generated lattice cone in the dominant chamber. This resultgeneralizes the formula for the Hilbert series of an equivariant embedding of ahomogeneous projective variety. As a special case, we show how themulti-variate series can be used to compute the Hilbert series of thedeterminantal varieties.

A multi-variate generating function for the Weyl Dimension Formula