A272696 - OEIS

%I #31 Mar 13 2019 12:28:09

%S 6,5,8,12,18,30

%N Coxeter number for the reflection group E_n.

%C 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.

%C For n>8, the Coxeter number is infinity.

%D J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge, 1990. See Table 3.2, page 80.

%H Benedict H. Gross, Eriko Hironaka, and Curtis T. McMullen, <a href="https://doi.org/10.1016/j.jnt.2008.09.021">Cyclotomic factors of Coxeter polynomials</a>, Journal of Number Theory (2009) 129(5): 1034-1043. See <a href="http://nrs.harvard.edu/urn-3:HUL.InstRepos:3446011">also</a>.

%e 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

%Y Cf. A272764.

%K nonn,fini,full

%O 3,1

%A _Curtis T. McMullen_, May 04 2016