Partial fraction decompositions and an algorithm for computing the vector partition function

Abstract

This paper gives an exposition of well known results on vector partition functions. The exposition is based on works of M. Brion, A. Szenes and M. Vergne and is geared toward explicit computer realizations. In particular, the paper presents two algorithms for computing the vector partition function with respect to a finite set of vectors $I$ as a quasipolynomial over a finite set of pointed polyhedral cones. We use the developed techniques to relate a result of P. Tumarkin and A. Felikson (and present an independent proof in the particular case of finite-dimensional root systems) to give bounds for the periods of the Kostant partition functions of $E_6$, $E_7$, $E_8$, $F_4$, $G_2$ (the periods are divisors of respectively 6, 12, 60, 12, 6). The first of the described algorithms has been realized and is publicly available under the Library General Public License v3.0 at {http://vectorpartition.sourceforge.net/}. We include (non-unique) partial fraction decompositions for the generating functions of the Kostant partition function for $A_2$, $A_3$, $A_4$, $B_2$, $B_3$, $C_2$, $C_3$, $G_2$ in the appendix.