JP Journal of Algebra, Number Theory and Applications
Volume 3, Issue 1, Pages 157 - 167
(April 2003)
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ON
LENGTH AND LEVEL IN THE CLASSICAL LIE ALGEBRAS
Barbara A. Shipman (U. S. A.)
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Abstract: The Weyl group of a complex
semisimple Lie algebra
G
acts transitively and with trivial stabilizer on
the vertices of the weight polytope of the
representation of G
whose highest weight d
is the sum of the fundamental weights. We define
the length of a vertex of this polytope to be
the length of the element of the Weyl group that
takes d to that
vertex. Each vertex of the polytope also has a
level; this is the number of times a simple
negative root must be added to the highest
weight to obtain the given vertex. This paper
investigates the relationship between length and
level and finds some of their common symmetries. |
Keywords and phrases: Lie algebra, weight polytope, Weyl group. |
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