Wolfram Language Paclet Repository
Community-contributed installable additions to the Wolfram Language
LieART`
PositiveRoots |
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Examples
(3)
Basic Examples
(3)
Positive roots of SU(5) in the ω-basis:
In[1]:=
PositiveRoots[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][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]}
In[2]:=
%//TraditionalForm
Out[2]//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 |
The RootLevel of positive roots is positive:
In[1]:=
TableForm[{#,RootLevel[#]}&/@PositiveRoots[SU5],TableHeadings{None,{"Root","Level"}},TableAlignmentsCenter]
Out[1]//TableForm=
Root | Level | ||||
| 4 | ||||
| 3 | ||||
| 3 | ||||
| 2 | ||||
| 2 | ||||
| 2 | ||||
| 1 | ||||
| 1 | ||||
| 1 | ||||
| 1 |
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[PositiveRoots[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][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]}
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""