Operations on Reflection Groups

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.

IsCoxeterIsomorphic(W1, W2): GrpMat, GrpMat -> BoolElt

Returns true if and only if the real reflection groups \(W_1\) and \(W_2\) are isomorphic as Coxeter systems.

IsCartanEquivalent(W1, W2): GrpMat, GrpMat -> BoolElt

Returns true if and only if the crystallographic real reflection groups \(W_1\) and \(W_2\) have Cartan equivalent Cartan matrices.

Example: Coxeter Isomorphism and Cartan Equivalence (ex-ce63f7)

> W1 := ReflectionGroup("B3");
> W2 := ReflectionGroup("C3");
> IsCoxeterIsomorphic(W1, W2);
true [ 1, 2, 3 ]
> IsCartanEquivalent(W1, W2);
false

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CartanName(W): GrpMat -> List

The Cartan name of the finite or affine real reflection group \(W\) (Section Finite and Affine Coxeter Groups).

CoxeterDiagram(W): GrpMat

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.

DynkinDiagram(W): GrpMat

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.

Example: Cartan Name and Coxeter Diagram (ex-0c1a24)

> G := CompleteGraph(3);
> W := ReflectionGroup(G);
> CartanName(W);
A~2
> CoxeterDiagram(W);

A~2    1 - 2
       |   |
       - 3 -

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RootSystem(W): GrpMat -> RootDtm

The root system of the finite real reflection group \(W\) (Chapter Root Systems). If \(W\) is infinite, an error is flagged.

RootDatum(W): GrpMat -> RootDtm

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.

CoxeterMatrix(W): GrpMat -> AlgMatElt

The Coxeter matrix of the real reflection group \(W\) (Section Coxeter Matrices).

CoxeterGraph(W): GrpMat -> GrphUnd

The Coxeter graph of the real reflection group \(W\) (Section Coxeter Graphs).

CartanMatrix(W): GrpMat -> AlgMatElt

The Cartan matrix of the real reflection group \(W\) (Section Cartan Matrices).

DynkinDigraph(W): GrpMat -> GrphDir

The Dynkin digraph of the real reflection group \(W\) (Section Dynkin Digraphs).

Rank(W): GrpMat -> RngIntElt

NumberOfGenerators(W): GrpMat -> RngIntElt

The rank of the reflection group \(W\).

Example: Rank and Dimension (ex-6d4455)

> R := StandardRootSystem("A", 4);
> W := ReflectionGroup(R);
> Rank(W);
4
> Dimension(W);
5

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FundamentalGroup(W): GrpMat -> GrpAb

The fundamental group of the real reflection group \(W\) (Subsection Isogeny of Split Reduced Root Data). The roots and coroots of \(W\) must have integral components.

IsogenyGroup(W): GrpMat -> GrpAb, Map

The isogeny group of 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.

CoisogenyGroup(W): GrpMat -> GrpAb, Map

The fundamental group of 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.

BasicDegrees(W): GrpMat -> RngIntElt

The degrees of the basic invariant polynomials of the reflection group \(W\). These are computed using the table in [Car72b, page 155] if the group is real, and using the algorithm of [LT09] in other cases. If \(W\) is infinite, an error is flagged.

BasicCodegrees(W): GrpMat -> RngIntElt

The basic codegrees of the reflection group \(W\). These are computed using the algorithm of [LT09]. If \(W\) is infinite, an error is flagged.

Example: Basic Degrees (ex-9828d5)

The product of the basic degrees is the order of the Coxeter group; the sum of the basic degrees is the sum of the rank and the number of positive roots.

> W := ReflectionGroup("E6");
> degs := BasicDegrees(W);
> degs;
[ 2, 5, 6, 8, 9, 12 ]
> &*degs eq #W;
true
> &+degs eq NumPosRoots(W) + Rank(W);
true

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LongestElement(W): GrpMat -> SeqEnum

The unique longest element in the finite real reflection group \(W\).

CoxeterElement(W): GrpMat -> SeqEnum

The Coxeter element in the reflection group \(W\), ie. the product of the generators.

CoxeterNumber(W): GrpMat -> SeqEnum

The order of the Coxeter element in the real reflection group \(W\).

Example: Longest and Coxeter Elements (ex-ae7bca)

Operations on groups.

> 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]

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LeftDescentSet(W, w): GrpMat, GrpMatElt -> SetEnum

The set of indices \(r\) of simple roots of the finite real reflection group \(W\) such that the length of the product \(s_rw\) is less than that of the element \(w\).

RightDescentSet(W, w): GrpMat, GrpMatElt -> SetEnum

The set of indices \(r\) of simple roots of the finite real reflection group \(W\) such that the length of the product \(ws_r\) is less than that of the element \(w\).

Example: Left and Right Descent Sets (ex-609f66)

> W := ReflectionGroup("A5");
> x := W.1*W.2*W.4*W.5;
> LeftDescentSet(W, x);
{ 1, 4 }
> RightDescentSet(W, x);
{ 2, 5 }

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