cp's OEIS Frontend

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

%I A287329 #31 Apr 24 2023 11:49:58
%S A287329 1,0,0,490,4998,45864,464422,3429426,21668094,111678742,492567012,
%T A287329 1876801038,6352945942,19484903508,54935857326,144330551050
%N A287329 Theta series of the 30-dimensional lattice of hyper-roots A_4(SU(3)).
%C A287329 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)).
%C A287329 Simple objects of the latter are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.
%C A287329 With k=4 there are r=(k+1)(k+2)/2=15 simple objects. The rank of the lattice is 2r=30.
%C A287329 The lattice is defined by 2r(k+3)^2/3=490 hyper-roots of norm 6 which are also the vectors of shortest length.
%C A287329 Minimal norm is 6. Det =(k+3)^(3(k+1))=7^15.
%C A287329 The lattice is rescaled (q --> q^2): its theta function starts as 1 + 490*q^6 + 4998*q^8 +... See example.
%H A287329 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 A287329 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 A287329 G.f. = 1 + 490*x^3 + 4998*x^4 + 45864*x^5 + ...
%e A287329 G.f. = 1 + 490*q^6 + 4998*q^8 + 45864*q^10 + ...
%Y A287329 Cf. A008434. {D_6}^{+} lattice is rescaled A_1(SU(3)).
%Y A287329 Cf. A290654 is A_2(SU(3)). Cf. A290655 is A_3(SU(3)).
%Y A287329 Cf. A287944 is A_5(SU(3)). Cf. A288488, A288489, A288776, A288779, A288909.
%K A287329 nonn,more
%O A287329 0,4
%A A287329 _Robert Coquereaux_, Sep 01 2017