Root systems and diagram calculus. I. Cycles in the Carter diagrams

Details Root systems and diagram calculus. I. Cycles in the Carter diagrams Root systems and diagram calculus. I. Cycles in the Carter diagrams

2010-05-16

arxiv.org/abs/1005.2769v5

Mathematics

Representation Theory

38 pages, 78 figures. Edited English, updated figures 3.28, 3.29

Scientific paper

We consider admissible diagrams (a.k.a. Carter diagrams) introduced by R. Carter in 1972 for the classification of conjugacy classes in a finite Weyl group $W$. Cycles in the Carter diagrams are the focus of this paper. We show that the 4-cycles determine outcome. The explicit transformations of any Carter diagram containing long cycles ($l > 4$) to another Carter diagram containing only 4-cycles are constructed. Thus, all Carter diagrams containing long cycles can be discarded from the classification list. Conjugate elements of $W$ give rise to the same diagram $\Gamma$. The converse is not true, as there exist diagrams determining two conjugacy classes in $W$. It is shown that the connected Carter diagram $\Gamma$ containing at least one 4-cycle determines the single conjugacy class. We study a generalization of the Carter diagrams called connection diagrams.

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