Constructing Root Systems

RootSystem(N) : MonStgElt -> RootSys

    Symmetric: BoolElt                  Default: false

    BaseField: MonStgElt                Default: "NumberField"

The semisimple root system with Cartan name N (see Section Finite and Affine Coxeter Groups). If the corresponding Coxeter group is infinite affine, an error is flagged.

If the Symmetric flag is set true, the symmetric Cartan matrix is used.

The BaseField flag determines which field the Cartan matrix is defined over. The possible values are:

1.

"NumberField": An algebraic number field. This is the default. See Chapter ORDERS AND ALGEBRAIC 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.

Example RootSys_CreatingRootSystemsName (H79E1)

> RootSystem( "H3E6" );
Root system of type H3 E6

RootSystem(M) : AlgMatElt -> RootSys

RootSystem(G) : GrphUnd -> RootSys

RootSystem(C) : AlgMatElt -> RootSys

RootSystem(D) : GrphDir -> RootSys

The semisimple root system with Coxeter matrix M, Coxeter graph G, Cartan matrix C, or Dynkin diagram D (see Chapter COXETER SYSTEMS). If the corresponding Coxeter group is infinite, an error is flagged.

Example RootSys_CreatingRootSystemsMatrix (H79E2)

> 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 the Coxeter or Cartan matrix of a finite group

RootSystem(A, B) : AlgMatElt, AlgMatElt -> RootSys

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.

AB^t must be the Cartan matrix of a finite Coxeter group.

Example RootSys_G2RootSystem (H79E3)

> 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

The irreducible root system with Cartan name X_n (or I_2(n) if X=()"I").

StandardRootSystem(X, n) : MonStgElt, RngIntElt -> RootSys

The standard root system with Cartan name X_n (or I_2(n) if X=()"I"), i.e. the root system with the Coxeter form the same as the standard inner product. For type A_n, the standard root system is not semisimple.

Example RootSys_IrreducibleRootSystem (H79E4)

> 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)
}