Sylvester Waves in the Coxeter Groups

Details Sylvester Waves in the Coxeter Groups Sylvester Waves in the Coxeter Groups

2000-05-17

arxiv.org/abs/math/0005174v1

Mathematics

Number Theory

One figure, 21 pages, submitted to Quart. J. Math

Scientific paper

A new recursive procedure of the calculation of partition numbers function $W(s,{\bf d}^m)$ is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of $W(s,{\bf d}^m)$ and their periods $\tau(G)$ for the irreducible Coxeter groups and a list for the first ten symmetric group ${\cal S}_m$ are presented. A {\it least common multiple} ${\cal L}(m)$ of the series of the natural numbers 1,2,..,$m$ plays a role of the period $\tau({\cal S}_m)$ of $W(s,{\bf d}^m)$ in ${\cal S}_m$. An asymptotic behaviour of ${\cal L}(m)$ with $m \to \infty$ is found.

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