Coxeter Matrices

A Coxeter system is defined by the numbers m_(ij) in {2, 3, ..., Infinity} for i, j=1, ... n and i<j, as in the previous section. Setting m_(ji)=m_(ij) and m_(ii)=1, we get a matrix M=(m_(ij))_(i, j=1)^n called the Coxeter matrix.

Since Infinity is not an integer, we represent it by 0 in our Coxeter matrices.

IsCoxeterMatrix( M ) : AlgMatElt -> BoolElt

Returns true if, and only if, M is the Coxeter matrix of some Coxeter group.

CoxeterMatrix( G ) : GrphUnd -> AlgMatElt

CoxeterMatrix( C ) : AlgMatElt -> AlgMatElt

CoxeterMatrix( D ) : GrphDir -> AlgMatElt

The Coxeter matrix corresponding to a Coxeter graph G, Cartan matrix C, or Dynkin digraph D.

Example Cartan_CoxeterMatrixConstruction (H82E1)

> M := SymmetricMatrix( [1, 3,1, 2,3,1] );
> M;
[1 3 2]
[3 1 3]
[2 3 1]
> IsCoxeterMatrix(M);
true

IsCoxeterIsomorphic( M1, M2 ) : AlgMatElt, AlgMatElt -> RngIntElt

Returns true if, and only if, the Coxeter matrices M_1 and M_2 give rise to Coxeter isomorphic groups. If true, we also return a sequence giving the permutation of the underlying basis which takes M_1 to M_2.

CoxeterGroupOrder( M ) : AlgMatElt -> RngIntElt

The order of the Coxeter group with Coxeter matrix M.

Example Cartan_CoxeterMatrixOperations (H82E2)

> M1 := SymmetricMatrix( [1, 3,1, 2,3,1] );
> M2 := SymmetricMatrix( [1, 3,1, 3,2,1] );
> IsCoxeterIsomorphic( M1, M2 );
true [ 2, 1, 3 ]
> 
> CoxeterGroupOrder( M1 );
24

IsCoxeterIrreducible( M ) : AlgMatElt -> BoolElt

Returns true if, and only if, M is the Coxeter matrix of an irreducible Coxeter system. If the Coxeter matrix is reducible, this function also returns a nontrivial subset I of {1, ..., n} such that m_(ij)=2 whenever i in I, j notin I.

IsSimplyLaced( M ) : AlgMatElt -> BoolElt

Returns true if, and only if, M is simply laced, i.e. all its entries are 1, 2, or 3.

Example Cartan_CoxeterMatrixProperties (H82E3)

> M := SymmetricMatrix( [1, 3,1, 2,3,1] );
> IsCoxeterIrreducible(M);
true
> M := SymmetricMatrix( [1, 2,1, 2,3,1] );
> IsCoxeterIrreducible(M);
false { 1 }