A288776 Theta series of the 24-dimensional lattice of hyper-roots E_5(SU(3)).

1, 0, 0, 512, 11232, 145920, 1055616, 5618688, 25330128, 89127936, 295067136, 810542592, 2185379968, 5109275136, 11899724544, 24646120448, 51701896272, 97674279936, 188911940608, 331864693248, 602050989120, 997987350528, 1717717782144, 2714582258688

Offset: 0

Author

Robert Coquereaux, Sep 01 2017

Keywords

• nonn

Comments

This lattice is associated with the exceptional module-category E_5(SU(3)) over the fusion (monoidal) category A_5(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=5, is one of the exceptional cases; other exceptional cases exist for k=9 and k=21. It is also special because it has self-fusion (it is flat, in operator algebra parlance).

E_5(SU(3)) has r=12 simple objects. The rank of the lattice is 2r=24. Det =2^30. This lattice, with k=5, is defined by 2 * r * (k+3)^2/3=512 hyper-roots of norm 6. They are also the vectors of shortest length (so, vectors of shortest length and hyper-roots coincide, like for lattices of type A_k(SU(3))). Minimal norm is 6.

The lattice is rescaled (q --> q^2): its theta function starts as 1 + 512*q^6 + 11232*q^8 +... See example.

This theta series is an element of the space of modular forms on Gamma_0(16) of weight 12 and dimension 25. - Andy Huchala, May 14 2023

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, A288779, A288909.

Programs

Magma

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

Formula

G.f. = 1 + 512*x^3 + 11232*x^4 + 145920*x^5 + ...

G.f. = 1 + 512*q^6 + 11232*q^8 + 145920*q^10 + ...

Extensions

More terms from Andy Huchala, May 14 2023