A129068 A128894[n,k] for k=1 : Coxeter numbers as defined by Bulgadaev for exceptional group sequence using critical exponent solution.

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

Offset

1,1

Comments

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.

References

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

Links

Table of n, a(n) for n=1..10.

S. A. Bulgadaev, BKT phase transition in two-dimensional systems with internal symmetries, arXiv:hep-th/9906091 (1999).

Formula

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]}]]

Mathematica

(*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]}]]

Crossrefs

Cf. A128894, A109161, A129024, A129025.

Sequence in context: A074717 A218137 A320002 * A079888 A165257 A368223

Adjacent sequences: A129065 A129066 A129067 * A129069 A129070 A129071

Keyword

nonn,fini,full,uned

Author

Roger L. Bagula, May 11 2007

Status

approved