Constructing Root Systems

Constructing Root Systems#

We first describe some optional parameters that are common to many functions described in this section.

RealInjection : Any : 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).

Nonreduced : SetEnum : {}

The optional argument Nonreduced is used to distinguish the reducedness of a root system in case the input doesn’t uniquely determine it.

Symmetric : BoolElt : false

If the Symmetric flag is set true, the symmetric Cartan matrix is used. For types \(I_2(m)\), \(H_3\), \(H_4\) the symmetric Cartan matrix is always used, since the root system is nonreduced otherwise.

BaseField : MonStgElt : “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 FldNum:main.

2. "Cyclotomic" or "SparseCyclotomic": A cyclotomic field with the sparse representation for elements. See Chapter ChapFldCyc.

3. "DenseCyclotomic": A cyclotomic field with the dense representation for elements. See Chapter ChapFldCyc.

\name{IntrRootSystemN}{RootSystem}(N)(): MonStgElt → RootSys#
BaseField : MonStgElt                    Default: ``NumberField"

Symmetric : BoolElt                      Default: \texttt{false}

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

Example: Creating Root Systems Name#
> RootSystem("H3 E6");
Root system of type H3 E6
> RootSystem("A2 T1 I2(5)");
Root system of type A2 I2(5)
RootSystem(M): AlgMatElt → RootSys#
RootSystem(G): GrphUnd → RootSys#
BaseField : MonStgElt                    Default: ``NumberField"

Symmetric : BoolElt                      Default: \texttt{false}

Nonreduced : SetEnum                      Default: \{\}

The semisimple root system with Coxeter matrix \(M\) or Coxeter graph \(G\) (see Chapter ChapCartan). 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.

RootSystem(C): AlgMatElt → RootSys#
Nonreduced : SetEnum                      Default: \{\}

RealInjection : Any                          Default: \texttt{false}

The semisimple root system with Cartan matrix \(C\) (see Chapter ChapCartan). 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.

RootSystem(D): GrphDir → RootSys#
Nonreduced : SetEnum                      Default: \{\}

The semisimple crystallographic root system with Cartan matrix \(C\), or Dynkin diagram \(D\) (see Chapter ChapCartan). If the corresponding Coxeter group is infinite, an error is flagged. For a description of the parameter Nonreduced see the beginning of this section.

Example: Creating Root Systems Matrix#
> 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 ]
RootSystem(A, B): Mtrx, Mtrx → RootSys#
Nonreduced : SetEnum                      Default: \{\}

RealInjection : Any                          Default: \texttt{false}

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;

  1. the number of columns must be at least the number of rows; and

  2. \(AB^t\) must be the Cartan matrix of a finite Coxeter group.

For descriptions of the parameters RealInjection and Nonreduced see the beginning of this section.

Example: G2Root System#

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
IrreducibleRootSystem(X, n): MonStgElt, RngIntElt → RootSys#
BaseField : MonStgElt                    Default: ``NumberField"

Symmetric : BoolElt                      Default: \texttt{false}

The irreducible root system with Cartan name \(X_n\) (or \(I_2(n)\) 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.

\name{IntrStandardRootSystem}{StandardRootSystem}(X, n)(): MonStgElt, RngIntElt → RootSys#

The standard root system with Cartan name \(X_n\) (or \(I_2(n)\) 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 \(A_n\), the standard root system is not semisimple.

Example: Irreducible Root System#
> 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)
}
ToralRootSystem(n): RngIntElt → RootSys#

The toral root system of dimension \(n\), i.e., the \(n\)-dimensional root system with no roots or coroots.

TrivialRootSystem() → RootSys#

The trivial root system of dimension \(0\).