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arXiv · 1403.3423

A multi-variate generating function for the Weyl Dimension Formula

Wayne Johnson
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Abstract

We present a closed form for a multi-variate generating function for the dimensions of the irreducible representations of a semisimple, simply connected linear algebraic group over $\mathbb{C}$ whose highest weights lie in a finitely generated lattice cone in the dominant chamber. This result generalizes the formula for the Hilbert series of an equivariant embedding of a homogeneous projective variety. As a special case, we show how the multi-variate series can be used to compute the Hilbert series of the determinantal varieties.

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Wayne Johnson. 2014-03-13. A multi-variate generating function for the Weyl Dimension Formula. https://arxiv.org/abs/1403.3423

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