A new method of weight multiplicities evaluation for semi-simple Lie algebras

Abstract: In most applications of semi-simple Lie groups and algebras representation theory, calculating weight multiplicities is one of the most often used and effort consuming operations. The existing tools were created many years ago by Kostant and Freudenthal. The celebrated Kostant weight multiplicity formula uses summation over the Weyl group of values of Kostant partition function, and the Freudenthal formula is recurrent. In this paper, a new way for calculating weight multiplicities is presented. The method does not employ the Weyl group and is direct, not recurrent. The algorithm realized in accordance with this method is much faster than those realized with the previously employed techniques. Many examples of programs realized in Fortran language are given. They are ready for compilation and execution on desktop and laptop computers.