Weight q-multiplicities for representations of the exceptional Lie algebra \mathfrakg₂

2020/03/17 by Cockerham, Jerrell, González, Melissa Gutiérrez, Harris, Pamela E. +4
#17B10 #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2003.07814

Abstract

Given a simple Lie algebra \mathfrakg, Kostant's weight q-multiplicity formula is an alternating sum over the Weyl group whose terms involve the q-analog of Kostant's partition function. For ξ (a weight of \mathfrakg), the q-analog of Kostant's partition function is a polynomial-valued function defined by \wp_q(ξ)=∑ c_i qⁱ where c_i is the number of ways ξ can be written as a sum of i positive roots of \mathfrakg. In this way, the evaluation of Kostant's weight q-multiplicity formula at q = 1 recovers the multiplicity of a weight in a highest weight representation of \mathfrakg. In this paper, we give closed formulas for computing weight q-multiplicities in a highest weight representation of the exceptional Lie algebra \mathfrakg₂.

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