See Chapter MATRIX GROUPS OVER GENERAL RINGS for general functions for matrix groups. Note that most of the functions in this section only work for real reflection groups.
Returns true if and only if the real reflection groups W1 and W2 are isomorphic as Coxeter systems.
Returns true if and only if the crystallographic real reflection groups W1 and W2 have Cartan equivalent Cartan matrices.
> W1 := ReflectionGroup("B3");
> W2 := ReflectionGroup("C3");
> IsCoxeterIsomorphic(W1, W2);
true [ 1, 2, 3 ]
> IsCartanEquivalent(W1, W2);
false
The Cartan name of the finite or affine real reflection group W (Section Finite and Affine Coxeter Groups).
A display of the Coxeter diagram of the real reflection group W (Section Finite and Affine Coxeter Groups). If W is not affine or finite, an error is flagged.
A display of the Coxeter diagram of the real reflection group W (Section Finite and Affine Coxeter Groups). If W is not affine or finite, or if W is not crystallographic, an error is flagged.
> G := CompleteGraph(3);
> W := ReflectionGroup(G);
> CartanName(W);
A~2
> CoxeterDiagram(W);
A~2 1 - 2
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- 3 -
The root system of the finite real reflection group W (Chapter ROOT SYSTEMS). If W is infinite, an error is flagged.
The root datum of the finite real reflection group W (Chapter ROOT DATA). The roots and coroots of W must have integral components, and W must be finite.
The Coxeter matrix of the real reflection group W (Section Coxeter Matrices).
The Coxeter graph of the real reflection group W (Section Coxeter Graphs).
The Cartan matrix of the real reflection group W (Section Cartan Matrices).
The Dynkin digraph of the real reflection group W (Section Dynkin Digraphs).
The rank of the reflection group W.
> R := StandardRootSystem("A", 4);
> W := ReflectionGroup(R);
> Rank(W);
4
> Dimension(W);
5
The fundamental groupof the real reflection group W (Subsection Isogeny of Split Reduced Root Data). The roots and coroots of W must have integral components.
The isogeny groupof the real reflection group W, together with the injection into the fundamental group (Subsection Isogeny of Split Reduced Root Data). The roots and coroots of W must have integral components.
The fundamental groupof the real reflection group W together with the projection onto the fundamental group (Subsection Isogeny of Split Reduced Root Data). The roots and coroots of W must have integral components.
The degrees of the basic invariant polynomials of the reflection group W. These are computed using the table in [Car72, page 155] if the group is real, and using the algorithm of [LT09] in other cases. If W is infinite, an error is flagged.
The basic codegrees of the reflection group W. These are computed using the algorithm of [LT09]. If W is infinite, an error is flagged.
> W := ReflectionGroup("E6");
> degs := BasicDegrees(W);
> degs;
[ 2, 5, 6, 8, 9, 12 ]
> &*degs eq #W;
true
> &+degs eq NumPosRoots(W) + Rank(W);
true
The unique longest element in the finite real reflection group W.
The Coxeter element in the reflection group W, ie. the product of the generators.
The order of the Coxeter element in the real reflection group W.
> W := ReflectionGroup("A4");
> LongestElement(W);
[ 0 0 0 -1]
[ 0 0 -1 0]
[ 0 -1 0 0]
[-1 0 0 0]
> CoxeterElement(W);
[-1 -1 -1 -1]
[ 1 0 0 0]
[ 0 1 0 0]
[ 0 0 1 0]
The set of indices r of simple roots of the finite real reflection group W such that the length of the product srw is less than that of the element w.
The set of indices r of simple roots of the finite real reflection group W such that the length of the product wsr is less than that of the element w.
> W := ReflectionGroup("A5");
> x := W.1*W.2*W.4*W.5;
> LeftDescentSet(W, x);
{ 1, 4 }
> RightDescentSet(W, x);
{ 2, 5 }