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%I A288776 #25 Jun 02 2025 12:22:35
%S A288776 1,0,0,512,11232,145920,1055616,5618688,25330128,89127936,295067136,
%T A288776 810542592,2185379968,5109275136,11899724544,24646120448,51701896272,
%U A288776 97674279936,188911940608,331864693248,602050989120,997987350528,1717717782144,2714582258688
%N A288776 Theta series of the 24-dimensional lattice of hyper-roots E_5(SU(3)).
%C A288776 This lattice is associated with the exceptional module-category E_5(SU(3)) over the fusion (monoidal) category A_5(SU(3)).
%C A288776 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 A288776 Simple objects of A_k(SU(3)) are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.
%C A288776 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 A288776 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 A288776 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 A288776 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 A288776 The lattice is rescaled (q --> q^2): its theta function starts as 1 + 512*q^6 + 11232*q^8 +... See example.
%C A288776 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 A288776 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 A288776 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 A288776 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 A288776 G.f. = 1 + 512*x^3 + 11232*x^4 + 145920*x^5 + ...
%F A288776 G.f. = 1 + 512*q^6 + 11232*q^8 + 145920*q^10 + ...
%o A288776 (Magma)
%o A288776 prec := 20;
%o A288776 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 A288776 S := Matrix(gram);
%o A288776 L := LatticeWithGram(S);
%o A288776 T := ThetaSeriesModularForm(L);
%o A288776 Coefficients(PowerSeries(T,prec)); // _Andy Huchala_, May 14 2023
%Y A288776 Cf. A008434. {D_6}^{+} lattice is rescaled A_1(SU(3)).
%Y A288776 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 A288776 Cf. A288488, A288489, A288779, A288909.
%K A288776 nonn
%O A288776 0,4
%A A288776 _Robert Coquereaux_, Sep 01 2017
%E A288776 More terms from _Andy Huchala_, May 14 2023