A288488 Theta series of the 12-dimensional lattice of hyper-roots D_3(SU(3)).

1, 0, 36, 144, 486, 2880, 5724, 7776, 31068, 40320, 47628, 149184, 178452, 171072, 511776, 527904, 500094, 1309824, 1339308, 1143072, 3049992, 2840256, 2451384, 5942016, 5709636, 4510080, 11313720, 9849744, 8199792, 18929088, 17426664, 13211424, 31971132

Offset: 0

Author

Robert Coquereaux, Sep 01 2017

Keywords

• nonn

Comments

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.

Simple objects of A_k(SU(3)) are irreducible integrable representations of the affine Lie algebra of SU(3) at level k.

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.

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.

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).

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.

The lattice is rescaled (q --> q^2): its theta function starts as 1 + 36*q^4 + 144*q^6 +... See example.

Examples

			G.f. = 1 + 36*x^2 + 144*x^3 + 486*x^4 + ...
G.f. = 1 + 36*q^4 + 144*q^6 + 486*q^8 + ...
		

References

• P. Di Francesco and J.-B. Zuber, SU(N) lattice integrable models associated with graphs, Nucl. Phys., B 338, pp 602--646, (1990).

Links

• Andy Huchala, Table of n, a(n) for n = 0..20000
• R. 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. R. Coquereaux, A. Garcia. and R. Trinchero, AMS Contemporary Mathematics, 294, pp. 133-160, (2000). End of Sec 2.5.

Crossrefs

Cf. A008434. {D_6}^{+} lattice is rescaled A_1(SU(3)).

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)).

Cf. A288489, A288776, A288779, A288909.

Programs

Magma

prec := 20;
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]];
S := Matrix(gram);
L := LatticeWithGram(S);
T := ThetaSeries(L, 14);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M,prec);
Coefficients(&+[Coefficients(T)[2*i-1]*B[i] :i in [1..7]]); // Andy Huchala, May 14 2023

Extensions

More terms from Andy Huchala, May 14 2023