public class OctonionE8Lattice extends HypercomplexLattice
CANONICAL
Constructor and Description |
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OctonionE8Lattice()
Constructor
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Modifier and Type | Method and Description |
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Octonion[] |
get16UnaryElements() |
Octonion[] |
getAllUnitElements() |
Octonion |
getBrandtTransformer(int nr)
// TODO explain //
|
Octonion[] |
getE7Roots() |
Octonion[] |
getOctavianUnits(int from,
int to) |
Octonion[] |
getSixthRootsOfUnity() |
Octonion[] |
getSO16Roots()
The 112 roots of SO(16), a maximal subgroup of E8 are generated.
|
Octonion[] |
getSO8Roots()
The 24 roots of SO(8) correspond with the 24 Hurwitz numbers of a given
quaternionic subalgebra.
|
SevenPointsSTS |
getSTS7() |
Octonion |
getUnitElement(int nr) |
boolean |
isUnitElement(Octonion o) |
void |
set128And112BasisHardCoded()
public void setDiadBasis () throws Exception
{
b[0] = new Octonion ("1,0,0,0,0,0,0,0");
b[1] = new Octonion ("0,1,0,0,0,0,0,0");
b[2] = new Octonion ("0,0,1,0,0,0,0,0");
b[3] = new Octonion ("0,0,0,1,0,0,0,0");
b[4] = new Octonion ("0,0,0,0,1,0,0,0");
b[5] = new Octonion ("0,0,0,0,0,1,0,0");
b[6] = new Octonion ("0,0,0,0,0,0,1,0");
b[7] = new Octonion ("0,0,0,0,0,0,0,1");
}
// TODO don't understand that any more !
|
void |
setOctavianUnits_Canonical()
This construction of the unit integral octonions serves as a reference.
|
void |
setOctavianUnits_D8Representation()
Set 128 elements +/-l0+/-l1+/-l2+/-l3+/-l4+/-l5+/-l6+/-l7 with an odd number of "-"-signs
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void |
setOctavianUnits_FanoPlaneRepresentation(int class_nr,
int nr)
Representation based on one of the 30 different Fano planes (= STS(7)).
112 basis vectors are constructed from the seven lines of the Fano plane considering all 16
sign combinations.
112 basis vectors are constructed from the dual vectors.
16 basis vectors are +/- one element base vectors.
|
void |
setOctavianUnits_SubalgebrasRepresentation()
The basis is constructed from a STS(7) defined by the 7 (quaternionic) subalgebras of order 4.
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getAllUnitsAsString, getBasis, getElements, getMutliplicativeOrderDistribution, getNumberOfUnitElement, isIntegralUnitElement, isUnitElement
public OctonionE8Lattice() throws java.lang.Exception
java.lang.Exception
public void setOctavianUnits_Canonical() throws java.lang.Exception
java.lang.Exception
public void setOctavianUnits_SubalgebrasRepresentation() throws java.lang.Exception
java.lang.Exception
public void setOctavianUnits_FanoPlaneRepresentation(int class_nr, int nr) throws java.lang.Exception
class_nr
- Class number of the Fano planee (1 or 2).nr
- Number of Fano plane within a class (1,...15).java.lang.Exception
public void setOctavianUnits_D8Representation() throws java.lang.Exception
java.lang.Exception
public void set128And112BasisHardCoded() throws java.lang.Exception
java.lang.Exception
public Octonion[] get16UnaryElements()
public Octonion[] getOctavianUnits(int from, int to) throws java.lang.Exception
java.lang.Exception
public Octonion[] getSO8Roots() throws java.lang.Exception
java.lang.Exception
public Octonion[] getSO16Roots() throws java.lang.Exception
java.lang.Exception
public Octonion[] getE7Roots() throws java.lang.Exception
java.lang.Exception
public SevenPointsSTS getSTS7()
public Octonion getBrandtTransformer(int nr) throws java.lang.Exception
nr
- Number of Brand transformerjava.lang.Exception
public Octonion[] getSixthRootsOfUnity() throws java.lang.Exception
java.lang.Exception
public Octonion[] getAllUnitElements()
getAllUnitElements
in class HypercomplexLattice
public Octonion getUnitElement(int nr) throws java.lang.Exception
getUnitElement
in class HypercomplexLattice
nr
- Number of the unit integral elements, counting starts with "1".java.lang.Exception
public boolean isUnitElement(Octonion o) throws java.lang.Exception
o
- Octonion.java.lang.Exception