A272696 Coxeter number for the reflection group E_n.

6, 5, 8, 12, 18, 30

Offset: 3

Author

Curtis T. McMullen, May 04 2016

Keywords

• nonn
• fini
• full

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.

Examples

			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
		

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.

Crossrefs

Cf. A272764.