The Lattice A2 x D4
An entry from the Catalogue of Lattices, which is a joint project of
Gabriele Nebe, RWTH Aachen University
(nebe@math.rwth-aachen.de)
and
Neil J. A. Sloane
(njasloane@gmail.com)
Last modified Fri Jul 18 13:16:10 CEST 2014
INDEX FILE |
ABBREVIATIONS
Contents of this file
NAME
DIMENSION
GRAM
DIVISORS
DET
MINIMAL_NORM
KISSING_NUMBER
GROUP_ORDER
GROUP_NAME
GROUP_GENERATORS
PROPERTIES
REFERENCES
THETA_SERIES
LAST_LINE
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NAME
A2 x D4
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DIMENSION
8
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GRAM
8 8
4 2 0 0 -2 -1 0 0
2 4 2 2 -1 -2 -1 -1
0 2 4 0 0 -1 -2 0
0 2 0 4 0 -1 0 -2
-2 -1 0 0 4 2 0 0
-1 -2 -1 -1 2 4 2 2
0 -1 -2 0 0 2 4 0
0 -1 0 -2 0 2 0 4
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DIVISORS
1 1 1 1 6 6 6 6
-
DET
1296
-
MINIMAL_NORM
4
-
KISSING_NUMBER
72
-
GROUP_ORDER
6912
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GROUP_NAME
S3 x W(F4) = W(A2) x W(F4)
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GROUP_GENERATORS
3
8
0 0 0 0 0 -1 1 1
0 0 0 0 0 0 0 1
0 0 0 0 1 -1 0 1
0 0 0 0 0 1 0 0
0 1 -1 -1 0 1 -1 -1
0 0 0 -1 0 0 0 -1
-1 1 0 -1 -1 1 0 -1
0 -1 0 0 0 -1 0 0
8
0 0 0 0 -1 1 -1 -1
0 0 0 0 -1 0 0 0
0 0 0 0 0 -1 1 0
0 0 0 0 0 -1 0 1
-1 1 -1 -1 0 0 0 0
-1 0 0 0 0 0 0 0
0 -1 1 0 0 0 0 0
0 -1 0 1 0 0 0 0
8
-1 1 -1 -1 0 0 0 0
-1 0 0 0 0 0 0 0
0 -1 1 0 0 0 0 0
0 -1 0 1 0 0 0 0
0 0 0 0 -1 1 -1 -1
0 0 0 0 -1 0 0 0
0 0 0 0 0 -1 1 0
0 0 0 0 0 -1 0 1
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PROPERTIES
INTEGRAL=1
MODULAR=6
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REFERENCES
Plesken & Pohst, Math. Comp. 31 (1977), 552-573.
Conway-Sloane, "Low-Dimensional Lattices II: Subgroups of GL(n,Z)",
Proc. Royal Soc. London, A 419 (1988), 29-68.
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THETA_SERIES
Theta series of A2 X D4 ( = G2 X F4):
Define th3 (theta_3) and eta as usual, or more precisely so that
> series(th3,q,10);
1 + 2*q + 2*q^4 + 2*q^9 + O(q^{16} )
> series(eta,q,10);
q^(1/12)- q^(25/12)- q^(49/12) + O(q^(121/12))
Define eta^{(3)} as
eta(q)eta(q^3), and call it "e3":
> series(e3,q,10);
q^(1/3)- q^(7/3)- q^(13/3)- q^(19/3)+ q^(25/3)+ O(q^(13/3) )
Define g1 and g2 as follows:
g1:=th3*subs(q=q^2,th3)*subs(q=q^3,th3)*subs(q=q^6,th3);
> series(g1,q,5);
1 + 2* q + 2*q^2 + 6*q^3 + 6*q^4 + O(q^5 )
t1:=subs(q=sqrt(q),e3);
t2:=subs(q=q^4,e3);
t3:=e3;
t4:=subs(q=q^2,e3);
g2:=series( (t1*t2/(t3*t4))^2,q, m1);
> series(g2,q,5);
q - 2*q^2 + q^3 - 4*q^4 + O(q^5 )
By Quebbemann JNT Oct 1995 or Rains-Sloane 1998,
we know that the theta series we want is of the form:
t5:=series( g1^2*sum( c[i]*g2^i, i=0..4),q,40);
Take these coefiicients:
t6:=subs({c[0]=1,c[1]=-4,c[2]=0,c[3]=-16,c[4]=16},t5);
and we get the desired answer :
> series(t6,q,10);
1 + 72*q^4 + 192*q^6 + 504*q^8 + O(q^{10} )
or with more terms:
1+72*q^4+192*q^6+504*q^8+576*q^10+2280*q^12+1728*q^14+4248*q^16+4800*q^18+7920*q^20+6336*q^22+19416*q^24+10368*
q^26+21312*q^28+22464*q^30+33624*q^32+24192*q^34+63048*q^36+32832*q^38+65808*q^40+60864*q^42+83232*q^44+57600*q^46+
155640*q^48+76032*q^50+137520*q^52+130944*q^54+180288*q^56+116928*q^58+290736*q^60+O(q^61),q,61)
The Rains-Sloane reference is:
E. M. Rains and N. J. A. Sloane,
The Shadow Theory of Modular and Unimodular Lattices,
J. Number Theory, 73 (1998), pp. 359-389;
http://neilsloane.com/doc/shad.pdf or shad.ps
Here are the maple commands that NJAS uses::
maxd:=101:
kernelopts(printbytes=false);
# get th2, th3, th4 = Jacobi theta constants out to degree maxd
temp0:=trunc(evalf(sqrt(maxd)))+2:
a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od:
th2:=series(a,q,maxd);
a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od:
th3:=series(a,q,maxd);
th4:=series(subs(q=-q,th3),q,maxd);
a:=q^(1/24) : for m from 1 to maxd do a:=a*(1-q^m); od:
eta:=a;
m1:=100;
g1:=th3*subs(q=q^2,th3)*subs(q=q^3,th3)*subs(q=q^6,th3);
g1:=series(g1,q,m1);
eta:=subs(q=q^2,eta);
e3:=eta*subs(q=q^3,eta);
e3:=series(e3,q,m1);
t1:=subs(q=sqrt(q),e3);
t2:=subs(q=q^4,e3);
t3:=e3;
t4:=subs(q=q^2,e3);
g2:=series( (t1*t2/(t3*t4))^2,q, m1);
i:='i';
t5:=series( g1^2*sum( c[i]*g2^i, i=0..4),q,m1);
t6:=subs({c[0]=1,c[1]=-4,c[2]=0,c[3]=-16,c[4]=16},t5);
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LAST_LINE
Haftungsausschluss/Disclaimer
Gabriele Nebe