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The $$q$$-analog of Kostant’s partition function and the highest root of the simple Lie algebras. (English) Zbl 1459.17021
Summary: For a given weight of a complex simple Lie algebra, the $$q$$-analog of Kostant’s partition function is a polynomial valued function in the variable $$q$$, where the coefficient of $$q^k$$ is the number of ways the weight can be written as a nonnegative integral sum of exactly $$k$$ positive roots. In this paper we determine generating functions for the $$q$$-analog of Kostant’s partition function when the weight in question is the highest root of the classical Lie algebras of types $$B$$, $$C$$, and $$D$$, and the exceptional Lie algebras of type $$G_2$$, $$F_4$$, $$E_6$$, $$E_7$$, and $$E_8$$.

##### MSC:
 17B20 Simple, semisimple, reductive (super)algebras 17B25 Exceptional (super)algebras 05A17 Combinatorial aspects of partitions of integers
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