Barnes-Wall Lattice

Author

Eric W. Weisstein
July 27, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Lattices/Barnes-WallLattice.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/Barnes-WallLattice.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Properties

Select[LatticeData[],LatticeData[#,"Dimension"]==16&]

LatticeData["BarnesWall"]

Select[Range[1000],Head[LatticeData[{"BarnesWall",#}]]=!=LatticeData&]//Quiet

TableForm[Table[{2^n,LatticeData[{"BarnesWall",2^n},"StandardName"],LatticeData[{"BarnesWall",2^n},"Notation"]},{n,6}],TableDepth->2]

LatticeData[{"RootLattice",{"D",4}},"Notation"]

LatticeData[{"RootLattice",{"E",8}},"Notation"]

LatticeData["Root*"]

AutomorphismCount

Table[LatticeData[{"BarnesWall",2^n},"AutomorphismGroupOrder"],{n,10}]

(2^((n^2)+n+1))((2^n)-1)Product[(2^(2i))-1,{i,n-1}]

Table[Product[(2^(2i))-1,{i,n-1}],{n,5}]

Table[If[n==3,696729600,(2^((n^2)+n+1))((2^n)-1)Product[(2^(2i))-1,{i,n-1}]],{n,10}]

Theta Series

Table[{n,LatticeData[{"BarnesWall",2^n},"Dimension"]},{n,10}]

LatticeData[{"BarnesWall",2},"ThetaSeriesFunction"][Log[q]/(I π)]//TraditionalForm

Series[EllipticTheta[3,0,q]^2,{q,0,20}]

CoefficientList[%,q]

LatticeData[{"BarnesWall",4},"ThetaSeriesFunction"][Log[q]/(I π)]//TraditionalForm

Series[(1/2)(EllipticTheta[3,0,q]^4+EllipticTheta[4,0,q]^4),{q,0,20}]

CoefficientList[%,q][[1;;-1;;2]]

LatticeData[{"BarnesWall",8},"ThetaSeriesFunction"][Log[q]/(I π)]//TraditionalForm

Series[(1/2)(EllipticTheta[2,0,q]^8+EllipticTheta[3,0,q]^8+EllipticTheta[4,0,q]^8),{q,0,10}]

CoefficientList[%,q][[1;;-1;;2]]

LatticeData[{"BarnesWall",16},"ThetaSeriesFunction"]

(1/2)(((EllipticTheta[2,0,q])^16)+((EllipticTheta[3,0,q])^16)+((EllipticTheta[4,0,q])^16)+30((EllipticTheta[2,0,q])^8)((EllipticTheta[3,0,q])^8))//TraditionalForm

1/2Series[((EllipticTheta[2,0,q])^16)+((EllipticTheta[3,0,q])^16)+((EllipticTheta[4,0,q])^16)+30((EllipticTheta[2,0,q])^8)((EllipticTheta[3,0,q])^8),{q,0,10}]

CoefficientList[%,q]