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We first describe some optional parameters that are common to many
functions described in this section.
var RealInjection: Any Default: false
Number field elements and cyclotomic field elements do not have a natural
identification with real numbers. The RealInjection flag allows the user
to provide one. If the base field of the Cartan matrix C is a number field,
the flag should be an injection into the real field; if the base field is cyclotomic,
the flag should be an injection into the complex field taking real values on the
entries of C (see more in Section Cartan Matrices).
var Nonreduced: SetEnum Default:
The optional argument Nonreduced is used to distinguish the
reducedness of a root system in case the input doesn't uniquely determine it.
var Symmetric: BoolElt Default: false
If the Symmetric flag is set true, the symmetric Cartan matrix is
used.
For types I2(m), H3, H4 the symmetric Cartan matrix is always
used, since the root system is nonreduced otherwise.
var BaseField: MonStgElt Default: "NumberField"
The BaseField flag determines the field over which the Cartan matrix is defined.
The possible values are:
- 1.
- "NumberField": An algebraic number field. This is the
default. See Chapter NUMBER FIELDS.
- 2.
- "Cyclotomic" or "SparseCyclotomic": A cyclotomic field
with the sparse representation for elements. See Chapter CYCLOTOMIC FIELDS.
- 3.
- "DenseCyclotomic": A cyclotomic field with the dense
representation for elements. See Chapter CYCLOTOMIC FIELDS.
RootSystem(N) : MonStgElt -> RootSys
Symmetric: BoolElt Default: false
BaseField: MonStgElt Default: "NumberField"
The root system with Cartan name given by the string N.
In addition to the Cartan names in Section Finite and Affine Coxeter Groups,
we allow "BCn" for the irreducible nonreduced system, and
"Tn" for the n-dimensional toral subsystem.
Note that "Tn" is used for input only and does not appear
in the string returned by CartanName when applied to the
resulting root system (see example below).
For descriptions of the parameters
Symmetric and BaseField see the beginning of this section
> RootSystem("H3 E6");
Root system of type H3 E6
> RootSystem("A2 T1 I2(5)");
Root system of type A2 I2(5)
RootSystem(G) : GrphUnd -> RootSys
Nonreduced: SetEnum Default: {}
Symmetric: BoolElt Default: false
BaseField: MonStgElt Default: "NumberField"
The semisimple root system with Coxeter matrix M or Coxeter graph G
(see Chapter COXETER SYSTEMS).
If the corresponding Coxeter group is infinite, an error is flagged.
For descriptions of the parameters Nonreduced, Symmetric, and
BaseField see the beginning of this section.
RealInjection: Any Default: false
Nonreduced: SetEnum Default: {}
The semisimple root system with
Cartan matrix C (see Chapter COXETER SYSTEMS).
If the corresponding Coxeter group is infinite, an error is flagged.
For descriptions of the parameters RealInjection and Nonreduced
see the beginning of this section.
Nonreduced: SetEnum Default: {}
The semisimple crystallographic root system with
Cartan matrix C, or Dynkin diagram D (see Chapter COXETER SYSTEMS).
If the corresponding Coxeter group is infinite, an error is flagged.
For a description of the parameter Nonreduced
see the beginning of this section.
> M := SymmetricMatrix([1, 3,1, 2,3,1]);
> RootSystem(M);
Root system of type A3
> M := SymmetricMatrix([1, 3,1, 3,3,1]);
> RootSystem(M);
>> RootSystem(M);
^
Runtime error in 'RootSystem': Not a finite root system in rows/columns
[ 1, 2, 3 ]
RealInjection: Any Default: false
Nonreduced: SetEnum Default: {}
The root system with simple roots given by the rows of the matrix
A and simple coroots given by the rows of the matrix B. The
matrices A and B must have the following properties:
- 1.
- A and B must have the same number of rows and the same number
of columns; they must be defined over the same ring, which must be the integers,
the rational
field, a number field, or a cyclotomic field;
- 2.
- the number of columns must be at least the number of rows;
and
- 3.
- ABt must be the Cartan matrix of a finite Coxeter group.
For descriptions of the parameters
RealInjection and Nonreduced
see the beginning of this section.
The following code creates a nonsemisimple root system of type G 2.
> A := Matrix(2,3, [1,-1,0, -1,1,-1]);
> B := Matrix(2,3, [1,-1,1, 0,1,-1]);
> RootSystem(A, B);
Root system of type G2
Symmetric: BoolElt Default: false
BaseField: MonStgElt Default: "NumberField"
The irreducible root system with Cartan name Xn (or I2(n)
if X="I") given by the string X and integer n.
In addition to the Cartan names in Section Finite and Affine Coxeter Groups,
we allow "BCn" for the irreducible nonreduced system.
For descriptions of the parameters
Symmetric and BaseField
see the beginning of this section.
StandardRootSystem(X, n) : MonStgElt, RngIntElt -> RootSys
The standard root system with Cartan name Xn (or I2(n) if X="I")
given by the string X and integer n,
i.e. the root system whose Coxeter form is the same as the standard inner product.
In addition to the Cartan names in Section Finite and Affine Coxeter Groups,
we allow "BCn" for the irreducible nonreduced system.
For type An, the standard root system is not semisimple.
> Rs := { IrreducibleRootSystem("I", n) : n in [3..20] };
> { R : R in Rs | IsCrystallographic(R) };
{
Root system of type I2(3) ,
Root system of type I2(4) ,
Root system of type I2(6)
}
The toral root system of dimension n, i.e., the n-dimensional root system with
no roots or coroots.
The trivial root system of dimension 0.
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