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

Author

Roger L. Bagula, May 11 2007

Keywords

• nonn
• fini
• full
• uned

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

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

Crossrefs

Cf. A128894, A109161, A129024, A129025.

Programs

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]}]]

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]}]]