A290655 Theta series of the 20-dimensional lattice of hyper-roots A_3(SU(3)).

1, 0, 0, 240, 1782, 9072, 59328, 216432, 810000, 2059152, 6080832, 12349584, 31045596, 57036960, 122715648, 204193872, 418822650, 622067040, 1193611392, 1734272208, 3043596384, 4217152080, 7354100160, 9446435136, 15901091892, 20507712192, 32268036096, 40493364288, 64454759856

Offset: 0

Author

Robert Coquereaux, Aug 08 2017

Keywords

• nonn

Comments

This lattice is the k=3 member of the family of lattices of SU(3) hyper-roots associated with the fusion category A_k(SU(3)).

Simple objects of the latter are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.

With k=3 there are r=(k+1)(k+2)/2=10 simple objects. The lattice is defined by 2 * r * (k+3)^2/3=240 hyper-roots of norm 6 which are also the vectors of shortest length. Minimal norm is 6. Det = 6^12.

The lattice is rescaled (q --> q^2): its theta function starts as 1 + 240*q^6 + 1782*q^8 +... See example.

Examples

			G.f. = 1 + 240*x^3 + 1782*x^4 + 9072*x^5 + ...
G.f. = 1 + 240*q^6 + 1782*q^8 + 9072*q^10 + ...
		

Links

• Robert Coquereaux, Table of n, a(n) for n = 0..59
• Robert Coquereaux, Theta functions for lattices of SU(3) hyper-roots, arXiv:1708.00560 [math.QA], 2017.
• A. Ocneanu, The Classification of subgroups of quantum SU(N), in "Quantum symmetries in theoretical physics and mathematics", Bariloche 2000, Eds. Coquereaux R., Garcia A. and Trinchero R., AMS Contemporary Mathematics, 294, pp. 133-160, (2000). End of Sec 2.5.

Crossrefs

Cf. A008434. {D_6}^{+} lattice is rescaled A_1(SU(3)).

A290654 is A_2(SU(3)). Cf. A287329, A287944, A288488, A288489, A288776, A288779, A288909.

Programs

Magma

order:=60; // Example
L:=LatticeWithGram(20,[6,0,0,0,2,0,2,2,2,0,0,2,0,2,1,-1,1,2,0,2,0,6,0,2,2,0,0,2,0,2,2,2,0,0,1,1,-1,0,2,2,0,0,\
6,0,2,2,0,0,2,2,0,2,2,0,-1,1,1,2,2,0,0,2,0,6,0,0,0,0,0,2,-2,1,0,0,0,0,2,0,2,0,2,2,2,0,6,0,0,2,2,2,1,0,1,1,2,2,2,\
2,2,2,0,0,2,0,0,6,0,0,2,0,0,1,-2,0,2,0,0,2,0,0,2,0,0,0,0,0,6,2,0,0,0,1,0,-2,0,2,0,0,0,2,2,2,0,0,2,0,2,6,0,0,2,0,\
0,-2,2,0,-2,1,1,-1,2,0,2,0,2,2,0,0,6,0,0,0,-2,2,0,-2,2,-1,1,1,0,2,2,2,2,0,0,0,0,6,-2,0,2,0,-2,2,0,1,-1,1,0,2,0,-\
2,1,0,0,2,0,-2,6,0,0,0,2,0,-2,0,2,0,2,2,2,1,0,1,1,0,0,0,0,6,0,0,0,0,0,2,2,2,0,0,2,0,1,-2,0,0,-2,2,0,0,6,0,-2,2,0\
,2,0,0,2,0,0,0,1,0,-2,-2,2,0,0,0,0,6,0,-2,2,0,0,2,1,1,-1,0,2,2,0,2,0,-2,2,0,-2,0,6,0,0,2,2,0,-1,1,1,0,2,0,2,0,-2\
,2,0,0,2,-2,0,6,0,2,0,2,1,-1,1,2,2,0,0,-2,2,0,-2,0,0,2,0,0,6,0,2,2,2,0,2,0,2,2,0,1,-1,1,0,2,2,0,2,2,0,6,0,0,0,2,\
2,2,2,0,0,1,1,-1,2,2,0,0,2,0,2,0,6,0,2,2,0,0,2,0,2,-1,1,1,0,2,0,2,0,2,2,0,0,6]);
theta:=ThetaSeriesModularForm(L); PowerSeries(theta,order);