%I #9 Apr 23 2015 13:24:23

%S 2,3,3,6,9,12,18,24,30,50

%N A128894[n,k] for k=1 : Coxeter numbers as defined by Bulgadaev for exceptional group sequence using critical exponent solution.

%C The building exceptional group symmetry sequence in Cartan notation is ( Deligne-Landsberg): {A1,A2,G2,D4,F4,E6,E7,E7.5,E8,E9} The Coxeter number seem to be related to the total powers in the elliptical invariants for exceptional groups. I have used 2/11 for the F4 critical exponent instead of Bulgadaev's 1/4 because 2/11 fits the linearity of the groups better.

%D J. M. Landsberg, The sextonions and E_{7 1/2} (with L.Manivel) (Advances in Math 201(2006) p143 - 179) page 22

%H S. A. Bulgadaev, <a href="http://arxiv.org/abs/hep-th/9906091">BKT phase transition in two-dimensional systems with internal symmetries</a>, arXiv:hep-th/9906091 (1999).

%F Criticalexponent=k/(k+hg)={2/(1 + 3), 2/(2 + 3), 2/5, 1/4, 2/11, 1/7, 1/10, 1/13, 1/16, 1/26}; hg=Coxeter number=(number of roots)/(rank of group) hg = Flatten[Table[x /. Solve[2/(2 + x) - b[[n]] == 0, x], {n, 1, Length[b]}]]

%t (*S.A Bulgadaev, arXiv : hep - th/9906091v1 12 Jun 1999*) b = {2/(1 + 3), 2/(2 + 3), 2/5, 1/4, 2/11, 1/7, 1/10, 1/13, 1/16, 1/26}; hg = Flatten[Table[x /. Solve[2/(2 + x) - b[[n]] == 0, x], {n, 1, Length[b]}]]

%Y Cf. A128894, A109161, A129024, A129025.

%K nonn,fini,full,uned

%O 1,1

%A _Roger L. Bagula_, May 11 2007