%I #25 Apr 24 2023 11:20:20

%S 1,0,0,896,11856,154368,2331648,27065088,281311128

%N Theta series of the 42-dimensional lattice of hyper-roots A_5(SU(3)).

%C This lattice is the k=5 member of the family of lattices of SU(3) hyper-roots associated with the fusion category A_k(SU(3)).

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

%C With k=5 there are r=(k+1)(k+2)/2=21 simple objects. The rank of the lattice is 2r=42.

%C The lattice is defined by 2r(k+3)^2/3=896 hyper-roots of norm 6 which are also the vectors of shortest length. Minimal norm is 6. Det =(k+3)^(3(k+1)) = 8^18.

%C The lattice is rescaled (q --> q^2): its theta function starts as 1 + 896*q^6 + 11856*q^8 +... See example.

%H Robert Coquereaux, <a href="https://arxiv.org/abs/1708.00560">Theta functions for lattices of SU(3) hyper-roots</a>, arXiv:1708.00560 [math.QA], 2017.

%H A. Ocneanu, <a href="https://cel.archives-ouvertes.fr/cel-00374414">The Classification of subgroups of quantum SU(N)</a>, 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.

%e G.f. = 1 + 896*x^3 + 11856*x^4 + 154368*x^5 + ...

%e G.f. = 1 + 896*q^6 + 11856*q^8 + 154368*q^10 + ...

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

%Y Cf. A290654 is A_2(SU(3)). Cf. A290655 is A_3(SU(3)). Cf. A287329 is A_4(SU(3). Cf. A288488, A288489, A288776, A288779, A288909.

%K nonn,more

%O 0,4

%A _Robert Coquereaux_, Sep 01 2017