login
Theta series of the 12-dimensional lattice of hyper-roots A_2(SU(3)).
10

%I #30 Apr 24 2023 11:50:07

%S 1,0,0,100,450,960,2800,6600,12300,22400,30690,63000,93150,144000,

%T 203100,236080,392850,550800,708350,961800,972780,1581600,1937250,

%U 2495400,2977400,3063360,4469400,5547700,6477600,7963200,7344920,11094000,12627000,15127200,17091900,16459440,22670850,26899200

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

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

%C 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 The lattice is rescaled: its theta function starts as 1 + 100*q^6 + 450*q^8 + ... See example.

%H Robert Coquereaux, <a href="/A290654/b290654.txt">Table of n, a(n) for n = 0..48</a>

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

%e G.f. = 1 + 100*x^3 + 450*x^4 + 960*x^5 + ...

%e G.f. = 1 + 100*q^6 + 450*q^8 + 960*q^10 + ...

%o (Magma)

%o order:=48; // Example

%o H := DirichletGroup(25,CyclotomicField(EulerPhi(25)));

%o chars := Elements(H); eps := chars[11];

%o M := ModularForms([eps],6);

%o Eltseq(PowerSeries(M![1, 0, 0,100, 450, 960, 2800, 6600, 12300, 22400, 30690, 63000, 93150, 144000, 203100, 236080], order));

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

%Y Cf. A290655, A287329, A287944, A288488, A288489, A288776, A288779, A288909.

%K nonn

%O 0,4

%A _Robert Coquereaux_, Aug 08 2017