A288776 - OEIS

%I #23 Nov 23 2024 05:42:45

%S 1,0,0,512,11232,145920,1055616,5618688,25330128,89127936,295067136,

%T 810542592,2185379968,5109275136,11899724544,24646120448,51701896272,

%U 97674279936,188911940608,331864693248,602050989120,997987350528,1717717782144,2714582258688

%N Theta series of the 24-dimensional lattice of hyper-roots E_5(SU(3))

%C This lattice is associated with the exceptional module-category E_5(SU(3)) over the fusion (monoidal) category A_5(SU(3)).

%C 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.

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

%C 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.

%C 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.

%C 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).

%C 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.

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

%C 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

%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 P. Di Francesco and J.-B. Zuber, <a href="https://doi.org/10.1016/0550-3213(90)90645-T">SU(N) lattice integrable models associated with graphs</a>, Nucl. Phys., B 338, pp 602--646, (1990). See <a href="https://www.lpthe.jussieu.fr/~zuber/MesPapiers/dfz_NP90.pdf">also</a>.

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

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

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

%o (Magma)

%o prec := 20;

%o 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]];

%o S := Matrix(gram);

%o L := LatticeWithGram(S);

%o T := ThetaSeriesModularForm(L);

%o Coefficients(PowerSeries(T,prec)); // _Andy Huchala_, May 14 2023

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

%Y 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)).

%Y Cf. A288488, A288489, A288779, A288909.

%K nonn

%O 0,4

%A _Robert Coquereaux_, Sep 01 2017

%E More terms from _Andy Huchala_, May 14 2023