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%I #20 May 14 2023 20:10:01
%S 1,0,36,144,486,2880,5724,7776,31068,40320,47628,149184,178452,171072,
%T 511776,527904,500094,1309824,1339308,1143072,3049992,2840256,2451384,
%U 5942016,5709636,4510080,11313720,9849744,8199792,18929088,17426664,13211424,31971132
%N Theta series of the 12-dimensional lattice of hyper-roots D_3(SU(3)).
%C This lattice is the k=3 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)).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 Members of the sub-family D_{3s} are special because they have self-fusion (they are flat, in operator algebra parlance). D_3(SU(3)) is the smallest member of the D_{3s} family (s=1).
%C With k=3 there are r=((k+1)(k+2)/2 -1)/3+3=6 simple objects. The rank of the lattice is 2r=12. The lattice is defined by 2r(k+3)^2/3=144 hyper-roots of norm 6. Det =3^12. The first shell is made of vectors of norm 4, they are not hyper-roots, and the only vectors of the lattice that belong to the second shell, of norm 6, are precisely the hyper-roots. Note: for lattices of type A_k(SU(3)), vectors of shortest length and hyper-roots coincide, here this is not so.
%C The lattice is rescaled (q --> q^2): its theta function starts as 1 + 36*q^4 + 144*q^6 +... See example.
%D P. Di Francesco and J.-B. Zuber, SU(N) lattice integrable models associated with graphs, Nucl. Phys., B 338, pp 602--646, (1990).
%H Andy Huchala, <a href="/A288488/b288488.txt">Table of n, a(n) for n = 0..20000</a>
%H R. 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 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 G.f. = 1 + 36*x^2 + 144*x^3 + 486*x^4 + ...
%e G.f. = 1 + 36*q^4 + 144*q^6 + 486*q^8 + ...
%o (Magma)
%o prec := 20;
%o gram := [[6,0,0,0,2,2,-2,1,1,1,0,0],[0,6,0,0,2,2,1,-2,1,1,0,0],[0,0,6,0,2,2,1,1,-2,1,0,0],[0,0,0,6,2,2,1,1,1,-2,0,0],[2,2,2,2,6,4,2,2,2,2,1,4],[2,2,2,2,4,6,2,2,2,2,4,1],[-2,1,1,1,2,2,6,0,0,0,2,2],[1,-2,1,1,2,2,0,6,0,0,2,2],[1,1,-2,1,2,2,0,0,6,0,2,2],[1,1,1,-2,2,2,0,0,0,6,2,2],[0,0,0,0,1,4,2,2,2,2,6,0],[0,0,0,0,4,1,2,2,2,2,0,6]];
%o S := Matrix(gram);
%o L := LatticeWithGram(S);
%o T<q> := ThetaSeries(L, 14);
%o M := ThetaSeriesModularFormSpace(L);
%o B := Basis(M,prec);
%o Coefficients(&+[Coefficients(T)[2*i-1]*B[i] :i in [1..7]]); // _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. A288489, A288776, A288779, A288909.
%K nonn
%O 0,3
%A _Robert Coquereaux_, Sep 01 2017
%E More terms from _Andy Huchala_, May 14 2023
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