cp's OEIS Frontend

A288489 Theta series of the 24-dimensional lattice of hyper-roots D_6(SU(3)).

%I A288489 #40 Nov 23 2024 05:42:38
%S A288489 1,0,162,2322,35478,273942,1771326,9680148,40813632,150043014,
%T A288489 484705782,1366155396,3583894788,8667408078,19470974076,41670759564,
%U A288489 84998113668,164677106052,309748771332,562229221500,985246266636,1687344227604,2821267240722,4582295154396
%N A288489 Theta series of the 24-dimensional lattice of hyper-roots D_6(SU(3)).
%C A288489 This lattice is the k=6 member of the family of lattices of SU(3) hyper-roots associated with the module-category D_k(SU(3)) over the fusion (monoidal) category A_k(SU(3)).
%C A288489 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 A288489 Simple objects of A_k(SU(3)) are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.
%C A288489 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 A288489 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 A288489 Members of the subfamily D_{3s} are special because they  have self-fusion (they are flat, in operator algebra parlance). D_6(SU(3)) is the second smallest member of the D_{3s} family (s=2).
%C A288489 With k=6 there are r = ((k+1)*(k+2)/2 - 1)/3 + 3 = 12 simple objects. The rank of the lattice is 2r=24. The lattice is defined by 2*r*(k+3)^2/3 = 648 hyper-roots of norm 6. Det = 3^18. 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. Note: for lattices of type A_k(SU(3)), vectors of shortest length and hyper-roots coincide, here this is not so.
%C A288489 The lattice is rescaled (q --> q^2): its theta function starts as 1 + 162*q^4 + 2322*q^6 + ... See example.
%C A288489 This theta series is an element of the space of modular forms on Gamma_0(27) of weight 12 and dimension 36. - _Andy Huchala_, May 14 2023
%H A288489 Andy Huchala, <a href="/A288489/b288489.txt">Table of n, a(n) for n = 0..20000</a>
%H A288489 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 A288489 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 A288489 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 A288489 G.f. = 1 + 162*x^2 + 2322*x^3 + 35478*x^4 + ...
%e A288489 G.f. = 1 + 162*q^4 + 2322*q^6 + 35478*q^8 + ...
%o A288489 (Magma)
%o A288489 prec := 10;
%o A288489 gram_matrix := [[6,0,0,0,0,0,0,0,2,0,2,0,-2,1,0,1,1,1,2,0,0,0,0,2],[0,6,0,0,0,0,0,0,2,0,2,0,1,-2,0,1,1,1,2,0,0,0,0,2],[0,0,6,0,0,0,0,2,0,2,0,0,0,0,-2,1,0,0,2,-2,0,-2,0,2],[0,0,0,6,0,0,2,2,2,2,2,2,1,1,1,0,1,2,0,0,2,0,2,0],[0,0,0,0,6,0,0,0,2,0,2,0,1,1,0,1,-2,1,2,0,0,0,0,2],[0,0,0,0,0,6,2,0,2,0,2,2,1,1,0,2,1,-1,-2,2,2,2,2,-2],[0,0,0,2,0,2,6,0,0,2,2,0,0,0,0,2,0,2,-1,1,2,2,2,0],[0,0,2,2,0,0,0,6,0,2,0,2,0,0,2,2,0,0,1,-1,1,2,0,2],[2,2,0,2,2,2,0,0,6,0,4,2,2,2,0,2,2,2,2,1,2,0,4,2],[0,0,2,2,0,0,2,2,0,6,0,0,0,0,2,2,0,0,2,2,0,-1,1,1],[2,2,0,2,2,2,2,0,4,0,6,0,2,2,0,2,2,2,2,0,4,1,2,2],[0,0,0,2,0,2,0,2,2,0,0,6,0,0,0,2,0,2,0,2,2,1,2,-1],[-2,1,0,1,1,1,0,0,2,0,2,0,6,0,0,0,0,0,0,0,2,0,2,0],[1,-2,0,1,1,1,0,0,2,0,2,0,0,6,0,0,0,0,0,0,2,0,2,0],[0,0,-2,1,0,0,0,2,0,2,0,0,0,0,6,0,0,0,0,2,0,2,0,0],[1,1,1,0,1,2,2,2,2,2,2,2,0,0,0,6,0,0,2,2,2,2,2,2],[1,1,0,1,-2,1,0,0,2,0,2,0,0,0,0,0,6,0,0,0,2,0,2,0],[1,1,0,2,1,-1,2,0,2,0,2,2,0,0,0,0,0,6,2,0,2,0,2,2],[2,2,2,0,2,-2,-1,1,2,2,2,0,0,0,0,2,0,2,6,0,0,-2,0,4],[0,0,-2,0,0,2,1,-1,1,2,0,2,0,0,2,2,0,0,0,6,0,0,2,-2],[0,0,0,2,0,2,2,1,2,0,4,2,2,2,0,2,2,2,0,0,6,2,2,0],[0,0,-2,0,0,2,2,2,0,-1,1,1,0,0,2,2,0,0,-2,0,2,6,0,0],[0,0,0,2,0,2,2,0,4,1,2,2,2,2,0,2,2,2,0,2,2,0,6,0],[2,2,2,0,2,-2,0,2,2,1,2,-1,0,0,0,2,0,2,4,-2,0,0,0,6]];
%o A288489 S := Matrix(gram_matrix);
%o A288489 L := LatticeWithGram(S);
%o A288489 T := ThetaSeriesModularForm(L);
%o A288489 Coefficients(PowerSeries(T,prec)); // _Andy Huchala_, May 14 2023
%Y A288489 Cf. A008434. {D_6}^{+} lattice is rescaled A_1(SU(3)).
%Y A288489 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 A288489 Cf. A288488 is D_3(SU(3)). Cf. A288776, A288779, A288909.
%K A288489 nonn
%O A288489 0,3
%A A288489 _Robert Coquereaux_, Sep 01 2017
%E A288489 More terms from _Andy Huchala_, May 14 2023