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Search: a287944 -id:a287944
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A008434
Theta series of {D_6}^{+} lattice.
+10
10
1, 0, 0, 32, 60, 0, 0, 192, 252, 0, 0, 480, 544, 0, 0, 832, 1020, 0, 0, 1440, 1560, 0, 0, 2112, 2080, 0, 0, 2624, 3264, 0, 0, 3840, 4092, 0, 0, 4992, 4380, 0, 0, 5440, 6552, 0, 0, 7392, 8160, 0, 0, 8832, 8224
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
From Robert Coquereaux, Aug 05 2017: (Start)
Other avatars of {D_6}^{+} and its theta series:
The lattice L4 generated by cuts of the complete graph on a set of 4 vertices (rescaled by sqrt(2)).
The generalized laminated lattice Lambda_6[3] with minimal norm 3.
The first member (k=1) 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. This lattice has to be rescaled: q --> q^2 since its minimal norm is 6 whereas the minimal norm of {D_6}^{+} is 3.
The space of modular forms on Gamma_1(16) of weight 3, twisted by a Dirichlet character defined as the Kronecker character -4, has dimension 7 and basis b1,...b7, where bn has leading term q^(n-1).
The theta function of {D_6}^{+} is b1 + 32 b4 + 60 b5.
(End)
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 120.
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Robert Coquereaux, Theta functions for lattices of SU(3) hyper-roots, arXiv:1708.00560 [math.QA], 2017.
M. Deza and V. Grishukhin, Delaunay Polytopes of Cut Lattices, Linear Algebra and Its Applications, 226- 228:667-685 (1995).
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.
W. Plesken and M. Pohst, Constructing integral lattices with prescribed minimum, Mathematics of Computation, Vol 45, No 171, pp. 209-221, and supplement S5-S16 (1985).
FORMULA
Expansion of (theta_2(q)^6 + theta_3(q)^6 + theta_4(q)^6)/2. - Seiichi Manyama, Oct 21 2018
EXAMPLE
G.f. = 1 + 32*q^3 + 60*q^4 + 192*q^7 + 252*q^8 + 480*q^11 + 544*q^12 + ... - Michael Somos, Sep 09 2018
MATHEMATICA
order = 50; S = (1/2) Series[
EllipticTheta[2, 0, q^2]^6 + EllipticTheta[3, 0, q^2]^6 +
EllipticTheta[4, 0, q^2]^6, {q, 0, order}];
CoefficientList[Simplify[Normal[S], Assumptions -> q > 0], q] (* Robert Coquereaux, Aug 05 2017 *)
a[ n_] := With [{e1 = QPochhammer[ q^2]^12, e2 = QPochhammer[ q^4]^6, e3 = QPochhammer[ q^8]^12}, SeriesCoefficient[ (e2^6 + e1 e3 (e1 + 64 q^3 e3)) / (2 e1 e2 e3), {q, 0, n}]]; (* Michael Somos, Sep 09 2018 *)
PROG
(Magma)
order:=50; // Example
H := DirichletGroup(16, CyclotomicField(EulerPhi(16)));
chars := Elements(H); eps := chars[2];
M := ModularForms([eps], 3);
Eltseq(PowerSeries(M![1, 0, 0, 32, 60, 0, 0], order)); // Robert Coquereaux, Aug 05 2017
(Magma) A := Basis( ModularForms( Gamma1(16), 3), 50); A[1] + 32*A[4] + 60*A[5] + 192*A[8] + 252*A[9] + 480*A[12] + 544*A[13] + 832*A[16] + 1020*A[17] + 1440*A[20] + 1560*A[21]; /* Michael Somos, Sep 09 2018 */
(PARI) {a(n) = my(A, e1, e2, e3); if( n<0, 0, A = x * O(x^n); e1 = eta(x^2)^12; e2 = eta(x^4 + A)^6; e3 = eta(x^8 + A)^12; polcoeff( (e2^6 + e1*e3*(e1 + 64 * x^3 * e3)) / (2 * e1 * e2 * e3), n))}; /* Michael Somos, Sep 09 2018 */
CROSSREFS
Cf. A290654, A290655, A287329, A287944, A288488, A288489, A288776, A288779, A288909.
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane
STATUS
approved
A288488
Theta series of the 12-dimensional lattice of hyper-roots D_3(SU(3)).
+10
10
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
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
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.
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.
EXAMPLE
G.f. = 1 + 36*x^2 + 144*x^3 + 486*x^4 + ...
G.f. = 1 + 36*q^4 + 144*q^6 + 486*q^8 + ...
PROG
(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<q> := 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
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.
KEYWORD
nonn
AUTHOR
Robert Coquereaux, Sep 01 2017
EXTENSIONS
More terms from Andy Huchala, May 14 2023
STATUS
approved
A288489
Theta series of the 24-dimensional lattice of hyper-roots D_6(SU(3)).
+10
10
1, 0, 162, 2322, 35478, 273942, 1771326, 9680148, 40813632, 150043014, 484705782, 1366155396, 3583894788, 8667408078, 19470974076, 41670759564, 84998113668, 164677106052, 309748771332, 562229221500, 985246266636, 1687344227604, 2821267240722, 4582295154396
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
This lattice is the k=6 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 subfamily D_{3s} are special because they have self-fusion (they are flat, in operator algebra parlance). D_6(SU(3)) is the second smallest member of the D_{3s} family (s=2).
With k=6 there are r = ((k+1)*(k+2)/2 - 1)/3 + 3 = 12 simple objects. The rank of the lattice is 2r=24. The lattice is defined by 2*r*(k+3)^2/3 = 648 hyper-roots of norm 6. Det = 3^18. 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. 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 + 162*q^4 + 2322*q^6 + ... See example.
This theta series is an element of the space of modular forms on Gamma_0(27) of weight 12 and dimension 36. - Andy Huchala, May 14 2023
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
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. R. Coquereaux, A. Garcia. and R. Trinchero, AMS Contemporary Mathematics, 294, pp. 133-160, (2000). End of Sec 2.5.
EXAMPLE
G.f. = 1 + 162*x^2 + 2322*x^3 + 35478*x^4 + ...
G.f. = 1 + 162*q^4 + 2322*q^6 + 35478*q^8 + ...
PROG
(Magma)
prec := 10;
gram_matrix := [[6, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, -2, 1, 0, 1, 1, 1, 2, 0, 0, 0, 0, 2], [0, 6, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 1, -2, 0, 1, 1, 1, 2, 0, 0, 0, 0, 2], [0, 0, 6, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, -2, 1, 0, 0, 2, -2, 0, -2, 0, 2], [0, 0, 0, 6, 0, 0, 2, 2, 2, 2, 2, 2, 1, 1, 1, 0, 1, 2, 0, 0, 2, 0, 2, 0], [0, 0, 0, 0, 6, 0, 0, 0, 2, 0, 2, 0, 1, 1, 0, 1, -2, 1, 2, 0, 0, 0, 0, 2], [0, 0, 0, 0, 0, 6, 2, 0, 2, 0, 2, 2, 1, 1, 0, 2, 1, -1, -2, 2, 2, 2, 2, -2], [0, 0, 0, 2, 0, 2, 6, 0, 0, 2, 2, 0, 0, 0, 0, 2, 0, 2, -1, 1, 2, 2, 2, 0], [0, 0, 2, 2, 0, 0, 0, 6, 0, 2, 0, 2, 0, 0, 2, 2, 0, 0, 1, -1, 1, 2, 0, 2], [2, 2, 0, 2, 2, 2, 0, 0, 6, 0, 4, 2, 2, 2, 0, 2, 2, 2, 2, 1, 2, 0, 4, 2], [0, 0, 2, 2, 0, 0, 2, 2, 0, 6, 0, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, -1, 1, 1], [2, 2, 0, 2, 2, 2, 2, 0, 4, 0, 6, 0, 2, 2, 0, 2, 2, 2, 2, 0, 4, 1, 2, 2], [0, 0, 0, 2, 0, 2, 0, 2, 2, 0, 0, 6, 0, 0, 0, 2, 0, 2, 0, 2, 2, 1, 2, -1], [-2, 1, 0, 1, 1, 1, 0, 0, 2, 0, 2, 0, 6, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0], [1, -2, 0, 1, 1, 1, 0, 0, 2, 0, 2, 0, 0, 6, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0], [0, 0, -2, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 6, 0, 0, 0, 0, 2, 0, 2, 0, 0], [1, 1, 1, 0, 1, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 6, 0, 0, 2, 2, 2, 2, 2, 2], [1, 1, 0, 1, -2, 1, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 6, 0, 0, 0, 2, 0, 2, 0], [1, 1, 0, 2, 1, -1, 2, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 6, 2, 0, 2, 0, 2, 2], [2, 2, 2, 0, 2, -2, -1, 1, 2, 2, 2, 0, 0, 0, 0, 2, 0, 2, 6, 0, 0, -2, 0, 4], [0, 0, -2, 0, 0, 2, 1, -1, 1, 2, 0, 2, 0, 0, 2, 2, 0, 0, 0, 6, 0, 0, 2, -2], [0, 0, 0, 2, 0, 2, 2, 1, 2, 0, 4, 2, 2, 2, 0, 2, 2, 2, 0, 0, 6, 2, 2, 0], [0, 0, -2, 0, 0, 2, 2, 2, 0, -1, 1, 1, 0, 0, 2, 2, 0, 0, -2, 0, 2, 6, 0, 0], [0, 0, 0, 2, 0, 2, 2, 0, 4, 1, 2, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 0, 6, 0], [2, 2, 2, 0, 2, -2, 0, 2, 2, 1, 2, -1, 0, 0, 0, 2, 0, 2, 4, -2, 0, 0, 0, 6]];
S := Matrix(gram_matrix);
L := LatticeWithGram(S);
T := ThetaSeriesModularForm(L);
Coefficients(PowerSeries(T, prec)); // Andy Huchala, May 14 2023
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. A288488 is D_3(SU(3)). Cf. A288776, A288779, A288909.
KEYWORD
nonn
AUTHOR
Robert Coquereaux, Sep 01 2017
EXTENSIONS
More terms from Andy Huchala, May 14 2023
STATUS
approved
A288909
Theta series of the 48-dimensional lattice of hyper-roots E_21(SU(3)).
+10
10
1, 0, 144, 64512, 54181224, 9051337728, 600733473408, 20812816594944, 448918973204472, 6740188251918336, 76049259049861920, 680967847813874688, 5038062720867937080, 31753526303307884544, 174598186489865835840, 853480923125492828160, 3765776231556517654872
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
This lattice is associated with the exceptional module-category E_21(SU(3)) over the fusion (monoidal) category A_21(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.
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).
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.
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.
The lattice is rescaled (q --> q^2): its theta function starts as 1 +144*q^4 + 64512*q^6 +... See example.
This theta series is an element of Gamma_0(3) of weight 24 and dimension 9. - Andy Huchala, May 14 2023
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..10000
Robert Coquereaux, Theta functions for lattices of SU(3) hyper-roots, arXiv:1708.00560 [math.QA], 2017.
Andy Huchala, Magma Program
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.
EXAMPLE
G.f. = 1 + 144*x^2 + 64512*x^3 + 54181224*x^4 + ...
G.f. = 1 + 144*q^4 + 64512*q^6 + 54181224*q^8 + ...
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. A288488, A288489, A288776, A288779.
KEYWORD
nonn
AUTHOR
Robert Coquereaux, Sep 01 2017
EXTENSIONS
More terms from Andy Huchala, May 15 2023
STATUS
approved
A290654
Theta series of the 12-dimensional lattice of hyper-roots A_2(SU(3)).
+10
10
1, 0, 0, 100, 450, 960, 2800, 6600, 12300, 22400, 30690, 63000, 93150, 144000, 203100, 236080, 392850, 550800, 708350, 961800, 972780, 1581600, 1937250, 2495400, 2977400, 3063360, 4469400, 5547700, 6477600, 7963200, 7344920, 11094000, 12627000, 15127200, 17091900, 16459440, 22670850, 26899200
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
This lattice is the k=2 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=2 there are r = (k+1)*(k+2)/2 = 6 simple objects. The lattice is defined by 2 * r * (k+3)^2/3=100 hyper-roots of norm 6 which are also the vectors of shortest length. Minimal norm is 6. Det = 5^9.
The lattice is rescaled: its theta function starts as 1 + 100*q^6 + 450*q^8 + ... See example.
LINKS
Robert Coquereaux, Table of n, a(n) for n = 0..48
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 + 100*x^3 + 450*x^4 + 960*x^5 + ...
G.f. = 1 + 100*q^6 + 450*q^8 + 960*q^10 + ...
PROG
(Magma)
order:=48; // Example
H := DirichletGroup(25, CyclotomicField(EulerPhi(25)));
chars := Elements(H); eps := chars[11];
M := ModularForms([eps], 6);
Eltseq(PowerSeries(M![1, 0, 0, 100, 450, 960, 2800, 6600, 12300, 22400, 30690, 63000, 93150, 144000, 203100, 236080], order));
CROSSREFS
Cf. A008434. {D_6}^{+} lattice is rescaled A_1(SU(3)).
Cf. A290655, A287329, A287944, A288488, A288489, A288776, A288779, A288909.
KEYWORD
nonn
AUTHOR
Robert Coquereaux, Aug 08 2017
STATUS
approved
A290655
Theta series of the 20-dimensional lattice of hyper-roots A_3(SU(3)).
+10
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, Table of n, a(n) for n = 0..59
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)).
A290654 is A_2(SU(3)). Cf. A287329, A287944, A288488, A288489, A288776, A288779, A288909.
KEYWORD
nonn
AUTHOR
Robert Coquereaux, Aug 08 2017
STATUS
approved
A287329
Theta series of the 30-dimensional lattice of hyper-roots A_4(SU(3)).
+10
9
1, 0, 0, 490, 4998, 45864, 464422, 3429426, 21668094, 111678742, 492567012, 1876801038, 6352945942, 19484903508, 54935857326, 144330551050
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
This lattice is the k=4 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=4 there are r=(k+1)(k+2)/2=15 simple objects. The rank of the lattice is 2r=30.
The lattice is defined by 2r(k+3)^2/3=490 hyper-roots of norm 6 which are also the vectors of shortest length.
Minimal norm is 6. Det =(k+3)^(3(k+1))=7^15.
The lattice is rescaled (q --> q^2): its theta function starts as 1 + 490*q^6 + 4998*q^8 +... See example.
LINKS
Table of n, a(n) for n=0..15.
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. R. Coquereaux, A. Garcia. and R. Trinchero, AMS Contemporary Mathematics, 294, pp. 133-160, (2000). End of Sec 2.5.
EXAMPLE
G.f. = 1 + 490*x^3 + 4998*x^4 + 45864*x^5 + ...
G.f. = 1 + 490*q^6 + 4998*q^8 + 45864*q^10 + ...
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. A287944 is A_5(SU(3)). Cf. A288488, A288489, A288776, A288779, A288909.
KEYWORD
nonn,more
AUTHOR
Robert Coquereaux, Sep 01 2017
STATUS
approved
A288776
Theta series of the 24-dimensional lattice of hyper-roots E_5(SU(3))
+10
9
1, 0, 0, 512, 11232, 145920, 1055616, 5618688, 25330128, 89127936, 295067136, 810542592, 2185379968, 5109275136, 11899724544, 24646120448, 51701896272, 97674279936, 188911940608, 331864693248, 602050989120, 997987350528, 1717717782144, 2714582258688
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
This lattice is associated with the exceptional module-category E_5(SU(3)) over the fusion (monoidal) category A_5(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.
E_k(SU(3)), with k=5, is one of the exceptional cases; other exceptional cases exist for k=9 and k=21. It is also special because it has self-fusion (it is flat, in operator algebra parlance).
E_5(SU(3)) has r=12 simple objects. The rank of the lattice is 2r=24. Det =2^30. This lattice, with k=5, is defined by 2 * r * (k+3)^2/3=512 hyper-roots of norm 6. They are also the vectors of shortest length (so, vectors of shortest length and hyper-roots coincide, like for lattices of type A_k(SU(3))). Minimal norm is 6.
The lattice is rescaled (q --> q^2): its theta function starts as 1 + 512*q^6 + 11232*q^8 +... See example.
This theta series is an element of the space of modular forms on Gamma_0(16) of weight 12 and dimension 25. - Andy Huchala, May 14 2023
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
Table of n, a(n) for n=0..23.
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. R. Coquereaux, A. Garcia. and R. Trinchero, AMS Contemporary Mathematics, 294, pp. 133-160, (2000). End of Sec 2.5.
FORMULA
G.f. = 1 + 512*x^3 + 11232*x^4 + 145920*x^5 + ...
G.f. = 1 + 512*q^6 + 11232*q^8 + 145920*q^10 + ...
PROG
(Magma)
prec := 20;
gram := [[6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, -2, 0, 1, 1, 0, 2, -2, 2, 2, 0, -2, 2], [0, 6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, -2, 1, 1, 2, 0, 2, -2, 0, 2, 2, -2], [0, 0, 6, 0, 2, 0, 2, 2, 2, 0, 2, 2, 1, 1, 0, 2, -2, 2, 2, 0, -2, 2, 0, 2], [0, 0, 0, 6, 0, 2, 2, 2, 0, 2, 2, 2, 1, 1, 2, 0, 2, -2, 0, 2, 2, -2, 2, 0], [0, 0, 2, 0, 6, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, -2, 0, 1, 1, 0, 0, 0, 2], [0, 0, 0, 2, 0, 6, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, -2, 1, 1, 0, 0, 2, 0], [2, 0, 2, 2, 0, 0, 6, 0, 2, 0, 2, 2, 2, 0, 2, 2, 1, 1, 0, 2, 2, 0, 2, 2], [0, 2, 2, 2, 0, 0, 0, 6, 0, 2, 2, 2, 0, 2, 2, 2, 1, 1, 2, 0, 0, 2, 2, 2], [0, 0, 2, 0, 0, 0, 2, 0, 6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, -2, 0, 1, 1], [0, 0, 0, 2, 0, 0, 0, 2, 0, 6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, -2, 1, 1], [2, 0, 2, 2, 0, 2, 2, 2, 0, 0, 6, 0, 2, 0, 2, 2, 0, 2, 2, 2, 1, 1, 0, 2], [0, 2, 2, 2, 2, 0, 2, 2, 0, 0, 0, 6, 0, 2, 2, 2, 2, 0, 2, 2, 1, 1, 2, 0], [-2, 0, 1, 1, 0, 0, 2, 0, 0, 0, 2, 0, 6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0], [0, -2, 1, 1, 0, 0, 0, 2, 0, 0, 0, 2, 0, 6, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2], [1, 1, 0, 2, 2, 0, 2, 2, 2, 0, 2, 2, 0, 0, 6, 0, 2, 0, 2, 2, 2, 0, 2, 2], [1, 1, 2, 0, 0, 2, 2, 2, 0, 2, 2, 2, 0, 0, 0, 6, 0, 2, 2, 2, 0, 2, 2, 2], [0, 2, -2, 2, -2, 0, 1, 1, 0, 0, 0, 2, 0, 0, 2, 0, 6, 0, 0, 0, 2, 0, 2, -2], [2, 0, 2, -2, 0, -2, 1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 6, 0, 0, 0, 2, -2, 2], [-2, 2, 2, 0, 1, 1, 0, 2, 2, 0, 2, 2, 2, 0, 2, 2, 0, 0, 6, 0, -2, 2, 2, 0], [2, -2, 0, 2, 1, 1, 2, 0, 0, 2, 2, 2, 0, 2, 2, 2, 0, 0, 0, 6, 2, -2, 0, 2], [2, 0, -2, 2, 0, 0, 2, 0, -2, 0, 1, 1, 0, 0, 2, 0, 2, 0, -2, 2, 6, 0, 0, 0], [0, 2, 2, -2, 0, 0, 0, 2, 0, -2, 1, 1, 0, 0, 0, 2, 0, 2, 2, -2, 0, 6, 0, 0], [-2, 2, 0, 2, 0, 2, 2, 2, 1, 1, 0, 2, 2, 0, 2, 2, 2, -2, 2, 0, 0, 0, 6, 0], [2, -2, 2, 0, 2, 0, 2, 2, 1, 1, 2, 0, 0, 2, 2, 2, -2, 2, 0, 2, 0, 0, 0, 6]];
S := Matrix(gram);
L := LatticeWithGram(S);
T := ThetaSeriesModularForm(L);
Coefficients(PowerSeries(T, prec)); // Andy Huchala, May 14 2023
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. A288488, A288489, A288779, A288909.
KEYWORD
nonn
AUTHOR
Robert Coquereaux, Sep 01 2017
EXTENSIONS
More terms from Andy Huchala, May 14 2023
STATUS
approved
A288779
Theta series of the 24-dimensional lattice of hyper-roots E_9(SU(3)).
+10
9
1, 0, 756, 5760, 98928, 1092096, 8435760, 45142272, 202712400, 715373568, 2350118808, 6501914496, 17469036096, 40850459136, 95266994400, 197161655040, 413591044176, 781142621184, 1511741623812, 2655160539264, 4815051144480, 7984019699712, 13744582363152
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
This lattice is associated with the exceptional module-category E_9(SU(3)) over the fusion (monoidal) category A_9(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.
E_k(SU(3)), with k=9, is one of the exceptional cases; other exceptional cases exist for k=5 and k=21. It is also special because it has self-fusion (it is flat, in operator algebra parlance).
E_9(SU(3)) has r=12 simple objects. The rank of the lattice is 2r=24. Det =2^24. This lattice, using k=9, is defined by 2*r*(k+3)^2/3=1152 hyper-roots of norm 6. 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.
The lattice is rescaled (q --> q^2): its theta function starts as 1 + 756*q^4 + 5760*q^6 +... See example.
This theta series is an element of the space of modular forms on Gamma_0(8) of weight 12 and dimension 13. - Andy Huchala, May 14 2023
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
Table of n, a(n) for n=0..22.
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. R. Coquereaux, A. Garcia. and R. Trinchero, AMS Contemporary Mathematics, 294, pp. 133-160, (2000). End of Sec 2.5.
EXAMPLE
G.f. = 1 + 756*x^2 + 5760*x^3 + 98928*x^4 + ...
G.f. = 1 + 756*q^4 + 5760*q^6 + 98928*q^8 + ...
PROG
(Magma)
prec := 20;
gram := [[6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, -2, 0, 0, 1, -2, 0, 0, 2, -2, 0, 0, 2], [0, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, -2, 0, 1, 0, -2, 0, 2, 0, -2, 0, 2], [0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, -2, 1, 0, 0, -2, 2, 0, 0, -2, 2], [0, 0, 0, 6, 2, 2, 2, 4, 2, 2, 2, 4, 1, 1, 1, 4, 2, 2, 2, 2, 2, 2, 2, 2], [2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, -1, 1, 1, 2, 2, 0, 0, 2], [0, 2, 0, 2, 0, 6, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 1, -1, 1, 2, 0, 2, 0, 2], [0, 0, 2, 2, 0, 0, 6, 0, 0, 0, 2, 2, 0, 0, 2, 2, 1, 1, -1, 2, 0, 0, 2, 2], [0, 0, 0, 4, 0, 0, 0, 6, 2, 2, 2, 2, 0, 0, 0, 4, 2, 2, 2, 1, 2, 2, 2, 2], [2, 0, 0, 2, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, -1, 1, 1, 2], [0, 2, 0, 2, 0, 2, 0, 2, 0, 6, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 1, -1, 1, 2], [0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 6, 0, 0, 0, 2, 2, 0, 0, 2, 2, 1, 1, -1, 2], [0, 0, 0, 4, 2, 2, 2, 2, 0, 0, 0, 6, 0, 0, 0, 4, 2, 2, 2, 2, 2, 2, 2, 1], [-2, 0, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0], [0, -2, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0], [0, 0, -2, 1, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0], [1, 1, 1, 4, 2, 2, 2, 4, 2, 2, 2, 4, 0, 0, 0, 6, 2, 2, 2, 4, 2, 2, 2, 4], [-2, 0, 0, 2, -1, 1, 1, 2, 2, 0, 0, 2, 2, 0, 0, 2, 6, 0, 0, 0, 0, 2, 2, 0], [0, -2, 0, 2, 1, -1, 1, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 6, 0, 0, 2, 0, 2, 0], [0, 0, -2, 2, 1, 1, -1, 2, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 6, 0, 2, 2, 0, 0], [2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 0, 4], [-2, 0, 0, 2, 2, 0, 0, 2, -1, 1, 1, 2, 2, 0, 0, 2, 0, 2, 2, 0, 6, 0, 0, 0], [0, -2, 0, 2, 0, 2, 0, 2, 1, -1, 1, 2, 0, 2, 0, 2, 2, 0, 2, 0, 0, 6, 0, 0], [0, 0, -2, 2, 0, 0, 2, 2, 1, 1, -1, 2, 0, 0, 2, 2, 2, 2, 0, 0, 0, 0, 6, 0], [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 6]];
S := Matrix(gram);
L := LatticeWithGram(S);
T := ThetaSeriesModularForm(L);
Coefficients(PowerSeries(T, prec)); // Andy Huchala, May 14 2023
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. A288488, A288489, A288776, A288909.
KEYWORD
nonn
AUTHOR
Robert Coquereaux, Sep 01 2017
EXTENSIONS
More terms from Andy Huchala, May 14 2023
STATUS
approved
page 1

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