The "rootdatum" command prints out the root datum defining the complex
reductive group G.
The software uses a basis of the character lattice X^*(T) that depends
on the way the group was specified by the user. If G is defined using
the "sc" (respectively "ad") option to be simply connected (resp.
adjoint) this basis is the basis of fundamental weights (resp. simple
roots), plus a basis of a direct factor Z^r if G is non-semisimple with
a central torus of rank r. In general (notably when the user specifies
kernel generators explicitly) this basis may be fairly arbitrary
and difficult to understand; in particular in the non-semisimple case
there might not be any basis vectors on which all coroots vanish.
The software uses a basis of the co-character lattice X_*(T), that
is the dual basis of the basis for the character lattice. In
particular it consists of the simple coroots in the case "sc" was
specified, and of fundamental coweights in the case "ad" was specified,
in both cases plus a copy of Z^n if G is not semisimple.
The output consists of:
1) cartan matrix
See the help file for the cmatrix command
2) root basis
The columnns give the simple roots in the given basis, as in the
output of the roots and posroots commands.
3) coroot basis
The columnns give the simple coroots in the given basis, as in the
output of the coroots and poscoroots commands.
4) radical basis
The columns gives a basis of X_*(S) in the given basis (S is the maximal
central torus); X_*(S) is the sublattice of the co-character lattice
X_*(T) on which all roots vanish.
5) coradical basis
The columns gives a basis of the sublattice of the character lattice
X^*(T) on which all coroots vanish. This is not the dual basis of the
radical basis, nor does the sublattice correspond to X^*(S) in general.
6) positive roots
The list of positive roots (printed as row vectors) in the given basis.
See the posroots command.
7) positive coroots
The list of positive coroots (printed as row vectors) in the given
basis. See the poscoroots command.
These matrices satisfy the following identity. Let R be the "root
basis" matrix, C the "coroot basis" matrix, and M the Cartan
matrix. Then
R^t*C=M
An abstract root datum can be defined to be a pair of m by n integral
matrices (R,C) satisfying R^t*C=M where M is a Cartan matrix.
Equivalence is by the action (R,C) -> (gR,transpose(g^{-1})C) for g in
GL(m,Z). Such a root datum is the root datum of a reductive algebraic
group, where the Cartan matrix of the derived group is M.
For the numbering of the simple roots see the help file for the
cmatrix command.
Example: root datum of Sp(4)
Since G is simply connected the basis of X^*(T) is the fundamental
weights, and the basis of X_*(T) is the simple coroots, so the
"coroot basis" matrix is the identity.
main: type
Lie type: C2 sc s
main: rootdatum
Name an output file (return for stdout, ? to abandon):
cartan matrix :
2 -1
-2 2
root basis :
2 -2
-1 2
coroot basis :
1 0
0 1
positive roots :
[2,-1]
[-2,2]
[0,1]
[2,0]
positive coroots :
[1,0]
[0,1]
[1,2]
[1,1]
Example: root datum of GL(3)
Since G is neither simply connected nor adjoint the bases of X^*(T)
and X_*(T) are not partcularly nice. Both the root and coroot basis
matrices have 3 rows (rank 3) and 2 columns (semisimple rank 2).
main: type
Lie type: A2.T1
elements of finite order in the center of the simply connected group:
Z/3.Q/Z
enter kernel generators, one per line
(ad for adjoint, ? to abort):
1/3,1/3
enter inner class(es): ss
main: rootdatum
Name an output file (return for stdout, ? to abandon):
cartan matrix :
2 -1
-1 2
root basis :
2 -1
1 0
0 0
coroot basis :
1 -2
0 3
0 -1
radical basis :
0
0
1
coradical basis :
0
1
3
positive roots :
[2,1,0]
[-1,0,0]
[1,1,0]
positive coroots :
[1,0,0]
[-2,3,-1]
[-1,3,-1]