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

1, 0, 0, 100, 450, 960, 2800, 6600, 12300, 22400, 30690, 63000, 93150, 144000, 203100, 236080, 392850, 550800, 708350, 961800, 972780, 1581600, 1937250, 2495400, 2977400, 3063360, 4469400, 5547700, 6477600, 7963200, 7344920, 11094000, 12627000, 15127200, 17091900, 16459440, 22670850, 26899200

Offset

0,4

Comments

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)).

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

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.

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

Links

Robert Coquereaux, Table of n, a(n) for n = 0..48

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

Example

G.f. = 1 + 100*x^3 + 450*x^4 + 960*x^5 + ...
G.f. = 1 + 100*q^6 + 450*q^8 + 960*q^10 + ...

Prog

(Magma)
order:=48; // Example
H := DirichletGroup(25, CyclotomicField(EulerPhi(25)));
chars := Elements(H); eps := chars[11];
M := ModularForms([eps], 6);
Eltseq(PowerSeries(M![1, 0, 0, 100, 450, 960, 2800, 6600, 12300, 22400, 30690, 63000, 93150, 144000, 203100, 236080], order));

Crossrefs

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

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

Sequence in context: A105089 A334707 A017510 * A017642 A093004 A003588

Adjacent sequences: A290651 A290652 A290653 * A290655 A290656 A290657

Keyword

nonn

Author

Robert Coquereaux, Aug 08 2017

Status

approved