A287329 Theta series of the 30-dimensional lattice of hyper-roots A_4(SU(3)).

1, 0, 0, 490, 4998, 45864, 464422, 3429426, 21668094, 111678742, 492567012, 1876801038, 6352945942, 19484903508, 54935857326, 144330551050

Offset

0,4

Comments

This lattice is the k=4 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=4 there are r=(k+1)(k+2)/2=15 simple objects. The rank of the lattice is 2r=30.

The lattice is defined by 2r(k+3)^2/3=490 hyper-roots of norm 6 which are also the vectors of shortest length.

Minimal norm is 6. Det =(k+3)^(3(k+1))=7^15.

The lattice is rescaled (q --> q^2): its theta function starts as 1 + 490*q^6 + 4998*q^8 +... See example.

Links

Table of n, a(n) for n=0..15.

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. R. Coquereaux, A. Garcia. and R. Trinchero, AMS Contemporary Mathematics, 294, pp. 133-160, (2000). End of Sec 2.5.

Example

G.f. = 1 + 490*x^3 + 4998*x^4 + 45864*x^5 + ...
G.f. = 1 + 490*q^6 + 4998*q^8 + 45864*q^10 + ...

Crossrefs

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

Cf. A290654 is A_2(SU(3)). Cf. A290655 is A_3(SU(3)).

Cf. A287944 is A_5(SU(3)). Cf. A288488, A288489, A288776, A288779, A288909.

Sequence in context: A172951 A392912 A392913 * A303556 A056936 A056052

Adjacent sequences: A287326 A287327 A287328 * A287330 A287331 A287332

Keyword

nonn,more

Author

Robert Coquereaux, Sep 01 2017

Status

approved