Mathematics – Representation Theory
Scientific paper
2011-09-12
Mathematics
Representation Theory
4 pages, short note
Scientific paper
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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