cp's OEIS Frontend

A290655 Theta series of the 20-dimensional lattice of hyper-roots A_3(SU(3)).

%I A290655 #21 Apr 24 2023 11:49:37
%S A290655 1,0,0,240,1782,9072,59328,216432,810000,2059152,6080832,12349584,
%T A290655 31045596,57036960,122715648,204193872,418822650,622067040,1193611392,
%U A290655 1734272208,3043596384,4217152080,7354100160,9446435136,15901091892,20507712192,32268036096,40493364288,64454759856
%N A290655 Theta series of the 20-dimensional lattice of hyper-roots A_3(SU(3)).
%C A290655 This lattice is the k=3 member of the family of lattices of SU(3) hyper-roots associated with the fusion category A_k(SU(3)).
%C A290655 Simple objects of the latter are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.
%C A290655 With k=3 there are r=(k+1)(k+2)/2=10 simple objects. The lattice is defined by 2 * r * (k+3)^2/3=240 hyper-roots of norm 6 which are also the vectors of shortest length. Minimal norm is 6.  Det = 6^12.
%C A290655 The lattice is rescaled (q --> q^2): its theta function starts as 1 + 240*q^6 + 1782*q^8 +... See example.
%H A290655 Robert Coquereaux, <a href="/A290655/b290655.txt">Table of n, a(n) for n = 0..59</a>
%H A290655 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 A290655 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. Coquereaux R., Garcia A. and Trinchero R., AMS Contemporary Mathematics, 294, pp. 133-160, (2000). End of Sec 2.5.
%e A290655 G.f. = 1 + 240*x^3 + 1782*x^4 + 9072*x^5 + ...
%e A290655 G.f. = 1 + 240*q^6 + 1782*q^8 + 9072*q^10 + ...
%o A290655 (Magma)
%o A290655 order:=60; // Example
%o A290655 L:=LatticeWithGram(20,[6,0,0,0,2,0,2,2,2,0,0,2,0,2,1,-1,1,2,0,2,0,6,0,2,2,0,0,2,0,2,2,2,0,0,1,1,-1,0,2,2,0,0,\
%o A290655 6,0,2,2,0,0,2,2,0,2,2,0,-1,1,1,2,2,0,0,2,0,6,0,0,0,0,0,2,-2,1,0,0,0,0,2,0,2,0,2,2,2,0,6,0,0,2,2,2,1,0,1,1,2,2,2,\
%o A290655 2,2,2,0,0,2,0,0,6,0,0,2,0,0,1,-2,0,2,0,0,2,0,0,2,0,0,0,0,0,6,2,0,0,0,1,0,-2,0,2,0,0,0,2,2,2,0,0,2,0,2,6,0,0,2,0,\
%o A290655 0,-2,2,0,-2,1,1,-1,2,0,2,0,2,2,0,0,6,0,0,0,-2,2,0,-2,2,-1,1,1,0,2,2,2,2,0,0,0,0,6,-2,0,2,0,-2,2,0,1,-1,1,0,2,0,-\
%o A290655 2,1,0,0,2,0,-2,6,0,0,0,2,0,-2,0,2,0,2,2,2,1,0,1,1,0,0,0,0,6,0,0,0,0,0,2,2,2,0,0,2,0,1,-2,0,0,-2,2,0,0,6,0,-2,2,0\
%o A290655 ,2,0,0,2,0,0,0,1,0,-2,-2,2,0,0,0,0,6,0,-2,2,0,0,2,1,1,-1,0,2,2,0,2,0,-2,2,0,-2,0,6,0,0,2,2,0,-1,1,1,0,2,0,2,0,-2\
%o A290655 ,2,0,0,2,-2,0,6,0,2,0,2,1,-1,1,2,2,0,0,-2,2,0,-2,0,0,2,0,0,6,0,2,2,2,0,2,0,2,2,0,1,-1,1,0,2,2,0,2,2,0,6,0,0,0,2,\
%o A290655 2,2,2,0,0,1,1,-1,2,2,0,0,2,0,2,0,6,0,2,2,0,0,2,0,2,-1,1,1,0,2,0,2,0,2,2,0,0,6]);
%o A290655 theta:=ThetaSeriesModularForm(L); PowerSeries(theta,order);
%Y A290655 Cf. A008434. {D_6}^{+} lattice is rescaled A_1(SU(3)).
%Y A290655 A290654 is A_2(SU(3)). Cf. A287329, A287944, A288488, A288489, A288776, A288779, A288909.
%K A290655 nonn
%O A290655 0,4
%A A290655 _Robert Coquereaux_, Aug 08 2017