Accessing the root datum

Many of the basic operations for Coxeter groups are shortcuts for obtaining information about the underlying root datum (Chapter ROOT DATA FOR LIE THEORY). We list these functions here; see Sections Operators on root data, Properties of root data and Roots, coroots and weights for more details and examples of their use.

Subsections

• Operations
• Properties
• Roots, coroots and weights

RootDatum( W ) : GrpCox -> RootDtm

The underlying root datum of a Coxeter group.

RootDatum( F ) : GrpCox -> RootDtm

The underlying root datum of the FP Coxeter group F; cf. Section Finitely presented Coxeter groups.

Example GrpCox_RDFromCox (H36E4)

> W := CoxeterGroup( "B5" );
> RootDatum( W );
Adjoint root datum of type B5 

Operations

RootSpace( W ) : GrpCox -> .

CorootSpace( W ) : GrpCox -> .

The (co)root space of the Coxeter group W. This can be a vector space over a field of characteristic zero (Chapter VECTOR SPACES), or an integer lattice in the crystallographic case (Chapter LATTICES). The (co)reflection group of W acts on the (co)root space.

SimpleRoots( W ) : GrpCox -> Mtrx

SimpleCoroots( W ) : GrpCox -> Mtrx

The simple (co)roots of the Coxeter group W as the rows of a matrix.

CartanMatrix( W ) : GrpCox -> AlgMatElt

The Cartan matrix of the Coxeter group W.

Rank( W ) : GrpCox -> RngIntElt

The rank of the Coxeter group W, ie. the number of simple (co)roots.

Dimension( W ) : GrpCox -> RngIntElt

The dimension of the Coxeter group W, ie. the dimension of the matrices in the reflection group.

IsogenyType( W ) : GrpCox -> List

The isogeny type of the Coxeter group W.

DynkinDiagram( W ) : GrpCox ->

Print the Dynkin diagram of the Coxeter group W.

Properties

IsIrreducible( W ) : GrpCox -> BoolElt

True if the Coxeter group W is irreducible.

IsCrystallographic( W ) : GrpCox -> BoolElt

True if the Coxeter group W is crystallographic (ie. if it is a Weyl group).

Roots, coroots and weights

NumberOfPositiveRoots( W ) : GrpCox -> RngIntElt

NumPosRoots( W ) : GrpCox -> RngIntElt

The number of positive roots of the Coxeter group W.

Roots( W ) : GrpCox -> {@@}

Coroots( W ) : GrpCox -> {@@}

    basis: MonStgElt                    Default: "standard"

An indexed set containing the (co)roots of the Coxeter group W.

PositiveRoots( W ) : GrpCox -> {@@}

PositiveCoroots( W ) : GrpCox -> {@@}

    basis: MonStgElt                    Default: "standard"

An indexed set containing the positive (co)roots of the Coxeter group W.

Root( W, r ) : GrpCox, RngIntElt -> {@@}

Coroot( W, r ) : GrpCox, RngIntElt -> {@@}

    basis: MonStgElt                    Default: "standard"

The rth (co)root of the Coxeter group W.

RootPosition( W, v ) : GrpCox, . -> {@@}

CorootPosition( W, v ) : GrpCox, . -> {@@}

    basis: MonStgElt                    Default: "standard"

If v is a (co)root of the Coxeter group W, this returns its position; otherwise it returns 0.

WeightLattice( W ) : RootDtm -> Lat

CoweightLattice( W ) : RootDtm -> Lat

The (co)weight lattice of the Coxeter group W.

FundamentalWeights( W ) : GrpCox -> SeqEnum

FundamentalCoweights( W ) : GrpCox -> SeqEnum

    basis: MonStgElt                    Default: "standard"

The fundamental (co)weights of the Coxeter group W.