### Equivalence of Carter diagrams

#### Abstract

We introduce the equivalence relation \(\rho\) on the set of Carter diagrams and construct an explicit transformation of any Carter diagram containing \(l\)-cycles with \(l > 4\) to an equivalent Carter diagram containing only \(4\)-cycles. Transforming one Carter diagram \(\Gamma_1\) to another Carter diagram \(\Gamma_2\) we can get a certain intermediate diagram \(\Gamma'\) which is not necessarily a Carter diagram. Such an intermediate diagram is called a connection diagram. The relation \(\rho\) is the equivalence relation on the set of Carter diagrams and connection diagrams. The properties of connection and Carter diagrams are studied in this paper. The paper contains an alternative proof of Carter's classification of admissible diagrams.

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R. W. Carter, Conjugacy classes in the Weyl group. 1970 Seminar on Algebraic Groups and Related Finite Groups (The Institute for Advanced Study, Princeton, N.J., 1968/69) pp . 297-318 Springer, Berlin

R. W. Carter, Conjugacy classes in the Weyl group. Compositio Math. 25 (1972), 1-59

R. W. Carter, G. B. Elkington, A Note on the Parametrization of Conjugacy Classes. J. Algebra 20 (1972), 350-354

V. Kac, Infinite root systems, representations of graphs and invariant theory. Invent. Math. 56 (1980), no. 1, 57-92

D. Madore, The E8 root system, 2010, http://www.madore.org/~david/math/e8rotate.html.

R. Stekolshchik, Notes on Coxeter Transformations and the McKay Correspondence, Springer Monographs in

Mathematics, 2008, XX, 240 p.

R. Stekolshchik, Root systems and diagram calculus. I. Regular extensions of Carter diagrams and the uniqueness of conjugacy classes, arXiv:1005.2769v6.

J. Stembridge, Coxeter Planes, 2007, http://www.math.lsa.umich.edu/~jrs/coxplane.html

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