A Coxeter system is defined by the numbers mij∈{2, 3, ..., ∞} for i, j=1, ... n and i<j, as in the previous section. Setting mji=mij and mii=1, yields a matrix M=(mij)i, j=1n that is called the Coxeter matrix.
Since ∞ is not an integer in Magma, it will be represented by 0 in Coxeter matrices.
Returns true if, and only if, the matrix M is the Coxeter matrix of some Coxeter group.
The Coxeter matrix corresponding to a Coxeter graph G, Cartan matrix C, or Dynkin digraph D.
> M := SymmetricMatrix([1, 3,1, 2,3,1]); > M; [1 3 2] [3 1 3] [2 3 1] > IsCoxeterMatrix(M); true
Returns true if and only if the Coxeter matrices M1 and M2 give rise to isomorphic Coxeter systems. If so, a sequence giving the permutation of the underlying basis which takes M1 to M2 is also returned.
The (factored) order of the Coxeter group with Coxeter matrix M.
> 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
Returns true if, and only if, the matrix 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 mij=2 whenever i∈I, j∉I.
Returns true if, and only if, the Coxeter matrix M is simply laced, i.e. all its entries are 1, 2, or 3.
> M := SymmetricMatrix([1, 3,1, 2,3,1]);
> IsCoxeterIrreducible(M);
true
> M := SymmetricMatrix([1, 2,1, 2,3,1]);
> IsCoxeterIrreducible(M);
false { 1 }