Anatoli Loutsiouk | Searcharxiv

arXiv2018

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

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.

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arXiv2011

Realizations of Weyl groups on ellipsoids

For any finite-dimensional complex semisimple Lie algebra two ellipsoids (primary and secondary) are considered. The equations of these ellipsoids are Diophantine equations and the Weyl group acts on the sets of all their Diophantine solutions. This provides two realizations (primary and secondary) of the Weyl group. The primary realization suggests an order on the Weyl group, which is useful for an exploration of the Bruhat order. The secondary realization is useful for finding for any element of the Weyl group all of its reduced expressions, which implies another realization of the Bruhat order.

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