Root systems for asymmetric geometric representations of Coxeter groups

Results are obtained concerning the roots of asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a Coxeter group in such a way as to include all Kac--Moody Weyl groups. In particular, a characterization of when a non-trivial multiple of a root may also be a root is given in the general context. Characterizations of when the number of such multiples of a root is finite and when the number of positive roots sent to negative roots by a group element is finite are also given. These characterizations are stated in terms of combinatorial conditions on a graph closely related to the Coxeter graph for the group. Other finiteness results for the symmetric case which are connected to the Tits cone and to a natural partial order on positive roots are extended to this asymmetric setting.Comment: References updated; connections to the literature sharpened; some applications further developed. 15 pages, 1 figur

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This study traces the development and growth of dance on the Broadway stage and the parallel growth of the effectiveness of choreography in enhancing the musical Theatre libretto. The study surveys the origins and early evolution of stage dance in the United States from 1775 to the introduction of ballet choreography in 1922. It concludes with an examination of Selected scripts which use choreography to dramatize the musical Theatre libretto, 1922 to 1990

Hudson River-Black River Regulating District and CSEA Local 120, Hudson-Black River District

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