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A290655 Theta series of the 20-dimensional lattice of hyper-roots A_3(SU(3)). 10
1, 0, 0, 240, 1782, 9072, 59328, 216432, 810000, 2059152, 6080832, 12349584, 31045596, 57036960, 122715648, 204193872, 418822650, 622067040, 1193611392, 1734272208, 3043596384, 4217152080, 7354100160, 9446435136, 15901091892, 20507712192, 32268036096, 40493364288, 64454759856 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
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)).
Simple objects of the latter are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.
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.
The lattice is rescaled (q --> q^2): its theta function starts as 1 + 240*q^6 + 1782*q^8 +... See example.
LINKS
Robert Coquereaux, Theta functions for lattices of SU(3) hyper-roots, arXiv:1708.00560 [math.QA], 2017.
A. Ocneanu, The Classification of subgroups of quantum SU(N), 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.
EXAMPLE
G.f. = 1 + 240*x^3 + 1782*x^4 + 9072*x^5 + ...
G.f. = 1 + 240*q^6 + 1782*q^8 + 9072*q^10 + ...
PROG
(Magma)
order:=60; // Example
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, \
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, \
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, \
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, -\
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\
, 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\
, 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, \
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]);
theta:=ThetaSeriesModularForm(L); PowerSeries(theta, order);
CROSSREFS
Cf. A008434. {D_6}^{+} lattice is rescaled A_1(SU(3)).
Sequence in context: A268906 A234763 A268796 * A212781 A205264 A205257
KEYWORD
nonn
AUTHOR
Robert Coquereaux, Aug 08 2017
STATUS
approved

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Last modified July 20 06:10 EDT 2024. Contains 374441 sequences. (Running on oeis4.)