Some Topics on Coxeter Groups and Weyl Groups

Abstract

Summary Starting from a brief introduction to Coxeter groups and Weyl groups, we shall describe several new results concerning these groups which are recently found in Japan. They are as follows: (i) finite Coxeter groups W of rank >2 is characterized by the lattice L(W) consisting of the subgroups of W, (ii) the finiteness of the order of a group given in a certain presentation is shown to be equivalent to the finiteness of the order of a Coxeter group associated with the given presentation in an explicit manner, (iii) the generating function of B. Kostant's partition function (relative to positive roots) is characterized by several functional properties. Also a remarkable property of this generating function is given.