A288909 - OEIS

%I #29 May 15 2023 15:41:16

%S 1,0,144,64512,54181224,9051337728,600733473408,20812816594944,

%T 448918973204472,6740188251918336,76049259049861920,

%U 680967847813874688,5038062720867937080,31753526303307884544,174598186489865835840,853480923125492828160,3765776231556517654872

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

%C This lattice is associated with the exceptional module-category E_21(SU(3)) over the fusion (monoidal) category A_21(SU(3)).

%C 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 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 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 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 The lattice is rescaled (q --> q^2): its theta function starts as 1 +144*q^4 + 64512*q^6 +... See example.

%C This theta series is an element of Gamma_0(3) of weight 24 and dimension 9. - _Andy Huchala_, May 14 2023

%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="/A288909/b288909.txt">Table of n, a(n) for n = 0..10000</a>

%H 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 Andy Huchala, <a href="/A288909/a288909_1.txt">Magma Program</a>

%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 + 144*x^2 + 64512*x^3 + 54181224*x^4 + ...

%e G.f. = 1 + 144*q^4 + 64512*q^6 + 54181224*q^8 + ...

%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. A288488, A288489, A288776, A288779.

%K nonn

%O 0,3

%A _Robert Coquereaux_, Sep 01 2017

%E More terms from _Andy Huchala_, May 15 2023