cp's OEIS Frontend

A288909 Theta series of the 48-dimensional lattice of hyper-roots E_21(SU(3)).

%I A288909 #34 Nov 23 2024 05:42:50
%S A288909 1,0,144,64512,54181224,9051337728,600733473408,20812816594944,
%T A288909 448918973204472,6740188251918336,76049259049861920,
%U A288909 680967847813874688,5038062720867937080,31753526303307884544,174598186489865835840,853480923125492828160,3765776231556517654872
%N A288909 Theta series of the 48-dimensional lattice of hyper-roots E_21(SU(3)).
%C A288909 This lattice is associated with the exceptional module-category E_21(SU(3)) over the fusion (monoidal) category A_21(SU(3)).
%C A288909 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 A288909 Simple objects of A_k(SU(3)) are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.
%C A288909 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 A288909 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 A288909 E_k(SU(3)), with k=21, is one of the exceptional cases; other exceptional cases exist for k=5 and k=9. It is also special because it has self-fusion (it is flat, in operator algebra parlance).
%C A288909 E_21(SU(3)) has r=24 simple objects. The rank of the lattice is 2r=48. Det =3^12. This lattice, using k=21, is defined by 2r(k+3)^2/3=9216 hyper-roots of norm 6.
%C A288909 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 A288909 The lattice is rescaled (q --> q^2): its theta function starts as 1 +144*q^4 + 64512*q^6 +... See example.
%C A288909 This theta series is an element of Gamma_0(3) of weight 24 and dimension 9. - _Andy Huchala_, May 14 2023
%H A288909 Andy Huchala, <a href="/A288909/b288909.txt">Table of n, a(n) for n = 0..10000</a>
%H A288909 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 A288909 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 A288909 Andy Huchala, <a href="/A288909/a288909_1.txt">Magma Program</a>
%H A288909 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 A288909 G.f. = 1 + 144*x^2 + 64512*x^3 + 54181224*x^4 + ...
%e A288909 G.f. = 1 + 144*q^4 + 64512*q^6 + 54181224*q^8 + ...
%Y A288909 Cf. A008434. {D_6}^{+} lattice is rescaled A_1(SU(3)).
%Y A288909 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 A288909 Cf. A288488, A288489, A288776, A288779.
%K A288909 nonn
%O A288909 0,3
%A A288909 _Robert Coquereaux_, Sep 01 2017
%E A288909 More terms from _Andy Huchala_, May 15 2023