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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 \(E_8\).

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