Kostant partition function

introduction

• Kostant’s partition function counts the number of ways to represent a particular vector (weight) as a nonnegative integral sum of positive roots of a Lie algebra.
• For a given weight the q-analog of Kostant’s partition function is a polynomial 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.

history

• Kostant’s partition function was introduced and studied by F.A. Berezin and I.M. Gelfand (Proc. Moscow Math. Soc. 5 (1956), 311-351) for the case $g=sl(n)$, and by B. Kostant (Trans. Amer. Math. Soc., 93 (1959), 53-73) for arbitrary semi–simple finite dimensional Lie algebra $g$

articles

• Flow polytopes and the Kostant partition function
• Harris, Pamela E., Erik Insko, and Mohamed Omar. “The $q$-Analog of Kostant’s Partition Function and the Highest Root of the Classical Lie Algebras.” arXiv:1508.07934 [math], August 31, 2015. http://arxiv.org/abs/1508.07934.