A multi-variate generating function for the Weyl Dimension Formula

2014/03/13 by Wayne Johnson, Johnson, Wayne
Mathematics · #17B10 #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B10

paper · pdf · doi:10.48550/arxiv.1403.3423

arxiv created 2014/03/13 · arxiv updated 2014/03/17

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 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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