A288779 Theta series of the 24-dimensional lattice of hyper-roots E_9(SU(3)).

1, 0, 756, 5760, 98928, 1092096, 8435760, 45142272, 202712400, 715373568, 2350118808, 6501914496, 17469036096, 40850459136, 95266994400, 197161655040, 413591044176, 781142621184, 1511741623812, 2655160539264, 4815051144480, 7984019699712, 13744582363152

Offset: 0

Author

Robert Coquereaux, Sep 01 2017

Keywords

• nonn

Comments

This lattice is associated with the exceptional module-category E_9(SU(3)) over the fusion (monoidal) category A_9(SU(3)).

The Grothendieck group of the former, a finite abelian category, is a Z+ - module over the Grothendieck ring of the latter, with a basis given by isomorphism classes of simple objects.

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

The classification of module-categories over A_k(SU(3)) was done, using another terminology, by P. Di Francesco and J.-B Zuber, and by A. Ocneanu (see refs below): it contains several infinite families that exist for all values of the positive integer k (among others one finds the A_k(SU(3)) themselves and the orbifold series D_k(SU(3))), and several exceptional cases for special values of k.

To every such module-category one can associate a set of hyper-roots (see refs below) and consider the corresponding lattice, denoted by the same symbol.

E_k(SU(3)), with k=9, is one of the exceptional cases; other exceptional cases exist for k=5 and k=21. It is also special because it has self-fusion (it is flat, in operator algebra parlance).

E_9(SU(3)) has r=12 simple objects. The rank of the lattice is 2r=24. Det =2^24. This lattice, using k=9, is defined by 2*r*(k+3)^2/3=1152 hyper-roots of norm 6. The first shell is made of vectors of norm 4, they are not hyper-roots, and the second shell, of norm 6, contains not only the hyper-roots, but other vectors as well. Note: for lattices of type A_k(SU(3)), vectors of shortest length and hyper-roots coincide, here this is not so.

The lattice is rescaled (q --> q^2): its theta function starts as 1 + 756*q^4 + 5760*q^6 +... See example.

This theta series is an element of the space of modular forms on Gamma_0(8) of weight 12 and dimension 13. - Andy Huchala, May 14 2023

Examples

			G.f. = 1 + 756*x^2 + 5760*x^3 + 98928*x^4 + ...
G.f. = 1 + 756*q^4 + 5760*q^6 + 98928*q^8 + ...
		

Links

• Robert Coquereaux, Theta functions for lattices of SU(3) hyper-roots, arXiv:1708.00560 [math.QA], 2017.
• P. Di Francesco and J.-B. Zuber, SU(N) lattice integrable models associated with graphs, Nucl. Phys., B 338, pp 602--646, (1990). See also.
• A. Ocneanu, The Classification of subgroups of quantum SU(N), 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.

Crossrefs

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

Cf. A290654 is A_2(SU(3)). Cf. A290655 is A_3(SU(3)). Cf. A287329 is A_4(SU(3)). Cf. A287944 is A_5(SU(3)).

Cf. A288488, A288489, A288776, A288909.

Programs

Magma

prec := 20;
gram := [[6,0,0,0,2,0,0,0,2,0,0,0,-2,0,0,1,-2,0,0,2,-2,0,0,2],[0,6,0,0,0,2,0,0,0,2,0,0,0,-2,0,1,0,-2,0,2,0,-2,0,2],[0,0,6,0,0,0,2,0,0,0,2,0,0,0,-2,1,0,0,-2,2,0,0,-2,2],[0,0,0,6,2,2,2,4,2,2,2,4,1,1,1,4,2,2,2,2,2,2,2,2],[2,0,0,2,6,0,0,0,2,0,0,2,2,0,0,2,-1,1,1,2,2,0,0,2],[0,2,0,2,0,6,0,0,0,2,0,2,0,2,0,2,1,-1,1,2,0,2,0,2],[0,0,2,2,0,0,6,0,0,0,2,2,0,0,2,2,1,1,-1,2,0,0,2,2],[0,0,0,4,0,0,0,6,2,2,2,2,0,0,0,4,2,2,2,1,2,2,2,2],[2,0,0,2,2,0,0,2,6,0,0,0,2,0,0,2,2,0,0,2,-1,1,1,2],[0,2,0,2,0,2,0,2,0,6,0,0,0,2,0,2,0,2,0,2,1,-1,1,2],[0,0,2,2,0,0,2,2,0,0,6,0,0,0,2,2,0,0,2,2,1,1,-1,2],[0,0,0,4,2,2,2,2,0,0,0,6,0,0,0,4,2,2,2,2,2,2,2,1],[-2,0,0,1,2,0,0,0,2,0,0,0,6,0,0,0,2,0,0,0,2,0,0,0],[0,-2,0,1,0,2,0,0,0,2,0,0,0,6,0,0,0,2,0,0,0,2,0,0],[0,0,-2,1,0,0,2,0,0,0,2,0,0,0,6,0,0,0,2,0,0,0,2,0],[1,1,1,4,2,2,2,4,2,2,2,4,0,0,0,6,2,2,2,4,2,2,2,4],[-2,0,0,2,-1,1,1,2,2,0,0,2,2,0,0,2,6,0,0,0,0,2,2,0],[0,-2,0,2,1,-1,1,2,0,2,0,2,0,2,0,2,0,6,0,0,2,0,2,0],[0,0,-2,2,1,1,-1,2,0,0,2,2,0,0,2,2,0,0,6,0,2,2,0,0],[2,2,2,2,2,2,2,1,2,2,2,2,0,0,0,4,0,0,0,6,0,0,0,4],[-2,0,0,2,2,0,0,2,-1,1,1,2,2,0,0,2,0,2,2,0,6,0,0,0],[0,-2,0,2,0,2,0,2,1,-1,1,2,0,2,0,2,2,0,2,0,0,6,0,0],[0,0,-2,2,0,0,2,2,1,1,-1,2,0,0,2,2,2,2,0,0,0,0,6,0],[2,2,2,2,2,2,2,2,2,2,2,1,0,0,0,4,0,0,0,4,0,0,0,6]];
S := Matrix(gram);
L := LatticeWithGram(S);
T := ThetaSeriesModularForm(L);
Coefficients(PowerSeries(T,prec)); // Andy Huchala, May 14 2023

Extensions

More terms from Andy Huchala, May 14 2023