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%I A288779 #26 Nov 23 2024 05:44:12
%S A288779 1,0,756,5760,98928,1092096,8435760,45142272,202712400,715373568,
%T A288779 2350118808,6501914496,17469036096,40850459136,95266994400,
%U A288779 197161655040,413591044176,781142621184,1511741623812,2655160539264,4815051144480,7984019699712,13744582363152
%N A288779 Theta series of the 24-dimensional lattice of hyper-roots E_9(SU(3)).
%C A288779 This lattice is associated with the exceptional module-category E_9(SU(3)) over the fusion (monoidal) category A_9(SU(3)).
%C A288779 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 A288779 Simple objects of A_k(SU(3)) are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.
%C A288779 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 A288779 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 A288779 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).
%C A288779 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.
%C A288779 The lattice is rescaled (q --> q^2): its theta function starts as 1 + 756*q^4 + 5760*q^6 +... See example.
%C A288779 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
%H A288779 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 A288779 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 A288779 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.
%e A288779 G.f. = 1 + 756*x^2 + 5760*x^3 + 98928*x^4 + ...
%e A288779 G.f. = 1 + 756*q^4 + 5760*q^6 + 98928*q^8 + ...
%o A288779 (Magma)
%o A288779 prec := 20;
%o A288779 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]];
%o A288779 S := Matrix(gram);
%o A288779 L := LatticeWithGram(S);
%o A288779 T := ThetaSeriesModularForm(L);
%o A288779 Coefficients(PowerSeries(T,prec)); // _Andy Huchala_, May 14 2023
%Y A288779 Cf. A008434. {D_6}^{+} lattice is rescaled A_1(SU(3)).
%Y A288779 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 A288779 Cf. A288488, A288489, A288776, A288909.
%K A288779 nonn
%O A288779 0,3
%A A288779 _Robert Coquereaux_, Sep 01 2017
%E A288779 More terms from _Andy Huchala_, May 14 2023