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A272696 Coxeter number for the reflection group E_n. 1
6, 5, 8, 12, 18, 30 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,1
COMMENTS
A good definition of E_n is to take (-3,1,...,1)^perp in Z^(1,n) (and change the sign). This is the correct definition when one relates E_n to the blowup of P^2 at n points, and gives the sequence E_8, E_7, E_6, D_5, A_4, A_2 X A_1.
For n>8, the Coxeter number is infinity.
REFERENCES
J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.2, page 80.
LINKS
Benedict H. Gross, Eriko Hironaka, and Curtis T. McMullen, Cyclotomic factors of Coxeter polynomials, Journal of Number Theory (2009) 129(5): 1034-1043. See also.
EXAMPLE
Starting with the Coxeter-Dynkin diagram for E_8, one repeatedly chops off nodes from one end, getting the sequence E_8, E_7, E_6, D_5, A_4, A_2 X A_1, whose Coxeter numbers are 30, 18, 12, 8, 5, 3 X 2=6. - N. J. A. Sloane, May 05 2016
CROSSREFS
Cf. A272764.
Sequence in context: A080799 A262512 A348909 * A048236 A193719 A364931
KEYWORD
nonn,fini,full
AUTHOR
Curtis T. McMullen, May 04 2016
STATUS
approved

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Last modified April 19 19:02 EDT 2024. Contains 371798 sequences. (Running on oeis4.)