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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.
CoxeterMatrix(C) : AlgMatElt -> AlgMatElt
CoxeterMatrix(D) : GrphDir -> AlgMatElt
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.
CoxeterGroupFactoredOrder(M) : AlgMatElt -> SeqEnum
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 }
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