Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
LieART`
RootSystem |
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Details and Options
Examples
(3)
Basic Examples
(3)
The roots are computed in the so called ω-basis, also known as the basis of fundamental weights, indicated by the head RootOmega:
In[1]:=
RootSystem[SU5]//InputForm
Out[1]//InputForm=
{RootOmega[A][1, 0, 0, 1], RootOmega[A][-1, 1, 0, 1], RootOmega[A][1, 0, 1, -1], RootOmega[A][-1, 1, 1, -1], RootOmega[A][0, -1, 1, 1],
RootOmega[A][1, 1, -1, 0], RootOmega[A][-1, 2, -1, 0], RootOmega[A][0, -1, 2, -1], RootOmega[A][0, 0, -1, 2], RootOmega[A][2, -1, 0, 0],
RootOmega[A][0, 0, 0, 0], RootOmega[A][0, 0, 0, 0], RootOmega[A][0, 0, 0, 0], RootOmega[A][0, 0, 0, 0], RootOmega[A][-2, 1, 0, 0],
RootOmega[A][0, 0, 1, -2], RootOmega[A][0, 1, -2, 1], RootOmega[A][1, -2, 1, 0], RootOmega[A][-1, -1, 1, 0], RootOmega[A][0, 1, -1, -1],
RootOmega[A][1, -1, -1, 1], RootOmega[A][-1, 0, -1, 1], RootOmega[A][1, -1, 0, -1], RootOmega[A][-1, 0, 0, -1]}
RootOmega[A][1, 1, -1, 0], RootOmega[A][-1, 2, -1, 0], RootOmega[A][0, -1, 2, -1], RootOmega[A][0, 0, -1, 2], RootOmega[A][2, -1, 0, 0],
RootOmega[A][0, 0, 0, 0], RootOmega[A][0, 0, 0, 0], RootOmega[A][0, 0, 0, 0], RootOmega[A][0, 0, 0, 0], RootOmega[A][-2, 1, 0, 0],
RootOmega[A][0, 0, 1, -2], RootOmega[A][0, 1, -2, 1], RootOmega[A][1, -2, 1, 0], RootOmega[A][-1, -1, 1, 0], RootOmega[A][0, 1, -1, -1],
RootOmega[A][1, -1, -1, 1], RootOmega[A][-1, 0, -1, 1], RootOmega[A][1, -1, 0, -1], RootOmega[A][-1, 0, 0, -1]}
Roots are sorted by root level. (Roots are displayed in similar to Dynkin labels but with the entries separated by commas since they can be negative.)
In[2]:=
RootSystem[SU5]//StandardForm
Out[2]//StandardForm=
{(1, 0, 0, 1),(-1, 1, 0, 1),(1, 0, 1, -1),(-1, 1, 1, -1),(0, -1, 1, 1),(1, 1, -1, 0),(-1, 2, -1, 0),(0, -1, 2, -1),(0, 0, -1, 2),(2, -1, 0, 0),(0, 0, 0, 0),(0, 0, 0, 0),(0, 0, 0, 0),(0, 0, 0, 0),(-2, 1, 0, 0),(0, 0, 1, -2),(0, 1, -2, 1),(1, -2, 1, 0),(-1, -1, 1, 0),(0, 1, -1, -1),(1, -1, -1, 1),(-1, 0, -1, 1),(1, -1, 0, -1),(-1, 0, 0, -1)}
In[3]:=
%//TraditionalForm
Out[3]//TraditionalForm=
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1 | 0 | 0 | 1 |
-1 | 1 | 0 | 1 |
1 | 0 | 1 | -1 |
-1 | 1 | 1 | -1 |
0 | -1 | 1 | 1 |
1 | 1 | -1 | 0 |
-1 | 2 | -1 | 0 |
0 | -1 | 2 | -1 |
0 | 0 | -1 | 2 |
2 | -1 | 0 | 0 |
0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 |
-2 | 1 | 0 | 0 |
0 | 0 | 1 | -2 |
0 | 1 | -2 | 1 |
1 | -2 | 1 | 0 |
-1 | -1 | 1 | 0 |
0 | 1 | -1 | -1 |
1 | -1 | -1 | 1 |
-1 | 0 | -1 | 1 |
1 | -1 | 0 | -1 |
-1 | 0 | 0 | -1 |
RootLevel computes the level of a root, which is also used to sort the roots:
In[4]:=
TableForm[{#,RootLevel[#]}&/@RootSystem[SU5],TableHeadings{None,{"Root","Level"}},TableAlignmentsCenter]
Out[4]//TableForm=
Root | Level | ||||
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| 1 | ||||
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| -4 |
Spindle shape formatting with the option SpindleShape:
In[5]:=
RootSystem[SU5,SpindleShapeTrue]
Out[5]=
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The command AlphaBasis transforms the roots to the so called α-basis with the head RootAlpha, where the roots are expressed in terms of their components along the simple roots. The α-basis exhibits the definition of the positive roots as those roots with only positive entries:
In[1]:=
AlphaBasis[RootSystem[SU5]]//InputForm
Out[1]//InputForm=
{RootAlpha[A][1, 1, 1, 1], RootAlpha[A][0, 1, 1, 1], RootAlpha[A][1, 1, 1, 0], RootAlpha[A][0, 1, 1, 0], RootAlpha[A][0, 0, 1, 1],
RootAlpha[A][1, 1, 0, 0], RootAlpha[A][0, 1, 0, 0], RootAlpha[A][0, 0, 1, 0], RootAlpha[A][0, 0, 0, 1], RootAlpha[A][1, 0, 0, 0],
RootAlpha[A][0, 0, 0, 0], RootAlpha[A][0, 0, 0, 0], RootAlpha[A][0, 0, 0, 0], RootAlpha[A][0, 0, 0, 0], RootAlpha[A][-1, 0, 0, 0],
RootAlpha[A][0, 0, 0, -1], RootAlpha[A][0, 0, -1, 0], RootAlpha[A][0, -1, 0, 0], RootAlpha[A][-1, -1, 0, 0], RootAlpha[A][0, 0, -1, -1],
RootAlpha[A][0, -1, -1, 0], RootAlpha[A][-1, -1, -1, 0], RootAlpha[A][0, -1, -1, -1], RootAlpha[A][-1, -1, -1, -1]}
RootAlpha[A][1, 1, 0, 0], RootAlpha[A][0, 1, 0, 0], RootAlpha[A][0, 0, 1, 0], RootAlpha[A][0, 0, 0, 1], RootAlpha[A][1, 0, 0, 0],
RootAlpha[A][0, 0, 0, 0], RootAlpha[A][0, 0, 0, 0], RootAlpha[A][0, 0, 0, 0], RootAlpha[A][0, 0, 0, 0], RootAlpha[A][-1, 0, 0, 0],
RootAlpha[A][0, 0, 0, -1], RootAlpha[A][0, 0, -1, 0], RootAlpha[A][0, -1, 0, 0], RootAlpha[A][-1, -1, 0, 0], RootAlpha[A][0, 0, -1, -1],
RootAlpha[A][0, -1, -1, 0], RootAlpha[A][-1, -1, -1, 0], RootAlpha[A][0, -1, -1, -1], RootAlpha[A][-1, -1, -1, -1]}
The command OrthogonalBasis transforms the roots into the orthogonal basis with the head RootOrthogonal, where the basis is orthogonal. The orthogonal basis is useful for the construction of Weyl orbits.
In[2]:=
OrthogonalBasis[RootSystem[SU5]]//InputForm
Out[2]//InputForm=
{RootOrthogonal[A][1, 0, 0, 0, -1], RootOrthogonal[A][0, 1, 0, 0, -1], RootOrthogonal[A][1, 0, 0, -1, 0], RootOrthogonal[A][0, 1, 0, -1, 0],
RootOrthogonal[A][0, 0, 1, 0, -1], RootOrthogonal[A][1, 0, -1, 0, 0], RootOrthogonal[A][0, 1, -1, 0, 0], RootOrthogonal[A][0, 0, 1, -1, 0],
RootOrthogonal[A][0, 0, 0, 1, -1], RootOrthogonal[A][1, -1, 0, 0, 0], RootOrthogonal[A][0, 0, 0, 0, 0], RootOrthogonal[A][0, 0, 0, 0, 0],
RootOrthogonal[A][0, 0, 0, 0, 0], RootOrthogonal[A][0, 0, 0, 0, 0], RootOrthogonal[A][-1, 1, 0, 0, 0], RootOrthogonal[A][0, 0, 0, -1, 1],
RootOrthogonal[A][0, 0, -1, 1, 0], RootOrthogonal[A][0, -1, 1, 0, 0], RootOrthogonal[A][-1, 0, 1, 0, 0], RootOrthogonal[A][0, 0, -1, 0, 1],
RootOrthogonal[A][0, -1, 0, 1, 0], RootOrthogonal[A][-1, 0, 0, 1, 0], RootOrthogonal[A][0, -1, 0, 0, 1], RootOrthogonal[A][-1, 0, 0, 0, 1]}
RootOrthogonal[A][0, 0, 1, 0, -1], RootOrthogonal[A][1, 0, -1, 0, 0], RootOrthogonal[A][0, 1, -1, 0, 0], RootOrthogonal[A][0, 0, 1, -1, 0],
RootOrthogonal[A][0, 0, 0, 1, -1], RootOrthogonal[A][1, -1, 0, 0, 0], RootOrthogonal[A][0, 0, 0, 0, 0], RootOrthogonal[A][0, 0, 0, 0, 0],
RootOrthogonal[A][0, 0, 0, 0, 0], RootOrthogonal[A][0, 0, 0, 0, 0], RootOrthogonal[A][-1, 1, 0, 0, 0], RootOrthogonal[A][0, 0, 0, -1, 1],
RootOrthogonal[A][0, 0, -1, 1, 0], RootOrthogonal[A][0, -1, 1, 0, 0], RootOrthogonal[A][-1, 0, 1, 0, 0], RootOrthogonal[A][0, 0, -1, 0, 1],
RootOrthogonal[A][0, -1, 0, 1, 0], RootOrthogonal[A][-1, 0, 0, 1, 0], RootOrthogonal[A][0, -1, 0, 0, 1], RootOrthogonal[A][-1, 0, 0, 0, 1]}
Root system of SO(10) in spindle shape:
In[1]:=
RootSystem[SO10,SpindleShapeTrue]
Out[1]=
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Root system of the exceptional Lie algebra :
G
2
In[2]:=
RootSystem[G2,SpindleShapeTrue]
Out[2]=
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