The Lattice A2 x D4

An entry from the Catalogue of Lattices, which is a joint project of
Gabriele Nebe, University of Ulm (nebe@mathematik.uni-ulm.de) and
Neil J. A. Sloane, AT&T Shannon Labs (njas@research.att.com)

Last modified Thu Jun 28 23:45:34 EDT 2001

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

• NAME
  A2 x D4
  
• DIMENSION
  8
  
• 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
  
• DIVISORS
  1 1 1 1 6 6 6 6
  
• DET
  1296
  
• MINIMAL_NORM
  4
  
• KISSING_NUMBER
  72
  
• GROUP_ORDER
  6912
  
• GROUP_NAME
  S3 x W(F4) = W(A2) x W(F4)
  
• 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
  
• PROPERTIES
  INTEGRAL=1
  MODULAR=6
  
• 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.
  
• 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);
  4 9 16
  1 + 2 q + 2 q + 2 q + O(q )
  
  > series(eta,q,10);
  25 49 121
  -- -- ---
  1/12 12 12 12
  q - q - q + O(q )
  
  Define eta^{(3)} as
  
  eta(q)eta(q^3), and call it "e3":
  
  > series(e3,q,10);
  1/3 7/3 13/3 19/3 25/3 31/3
  q - q - q - q + q + O(q )
  
  
  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);
  2 3 4 5
  1 + 2 q + 2 q + 6 q + 6 q + O(q )
  
  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);
  2 3 4 5
  q - 2 q + q - 4 q + O(q )
  
  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);
  4 6 8 10
  1 + 72 q + 192 q + 504 q + O(q )
  
  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://www.research.att.com/~njas/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);
  
  
• LAST_LINE
  

INDEX FILE | ABBREVIATIONS

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