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A290654 Theta series of the 12-dimensional lattice of hyper-roots A_2(SU(3)).

%I A290654 #30 Apr 24 2023 11:50:07
%S A290654 1,0,0,100,450,960,2800,6600,12300,22400,30690,63000,93150,144000,
%T A290654 203100,236080,392850,550800,708350,961800,972780,1581600,1937250,
%U A290654 2495400,2977400,3063360,4469400,5547700,6477600,7963200,7344920,11094000,12627000,15127200,17091900,16459440,22670850,26899200
%N A290654 Theta series of the 12-dimensional lattice of hyper-roots A_2(SU(3)).
%C A290654 This lattice is the k=2 member of the family of lattices of SU(3) hyper-roots associated with the fusion category A_k(SU(3)).
%C A290654 Simple objects of the latter are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.
%C A290654 With k=2 there are r = (k+1)*(k+2)/2 = 6 simple objects. The lattice is defined by 2 * r * (k+3)^2/3=100 hyper-roots of norm 6 which are also the vectors of shortest length. Minimal norm is 6. Det = 5^9.
%C A290654 The lattice is rescaled: its theta function starts as 1 + 100*q^6 + 450*q^8 + ... See example.
%H A290654 Robert Coquereaux, <a href="/A290654/b290654.txt">Table of n, a(n) for n = 0..48</a>
%H A290654 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 A290654 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. Coquereaux R., Garcia A. and Trinchero R., AMS Contemporary Mathematics, 294, pp. 133-160, (2000). End of Sec 2.5.
%e A290654 G.f. = 1 + 100*x^3 + 450*x^4 + 960*x^5 + ...
%e A290654 G.f. = 1 + 100*q^6 + 450*q^8 + 960*q^10 + ...
%o A290654 (Magma)
%o A290654 order:=48; // Example
%o A290654 H := DirichletGroup(25,CyclotomicField(EulerPhi(25)));
%o A290654 chars := Elements(H); eps := chars[11];
%o A290654 M := ModularForms([eps],6);
%o A290654 Eltseq(PowerSeries(M![1, 0, 0,100, 450, 960, 2800, 6600, 12300, 22400, 30690, 63000, 93150, 144000, 203100, 236080], order));
%Y A290654 Cf. A008434. {D_6}^{+} lattice is rescaled A_1(SU(3)).
%Y A290654 Cf. A290655, A287329, A287944, A288488, A288489, A288776, A288779, A288909.
%K A290654 nonn
%O A290654 0,4
%A A290654 _Robert Coquereaux_, Aug 08 2017