Operations on Reflection Groups

Operations on Reflection Groups#

See Chapter ChapGrpMat 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: Isomorphism#

> W1 := ReflectionGroup("B3");
> W2 := ReflectionGroup("C3");
> IsCoxeterIsomorphic(W1, W2);
true [ 1, 2, 3 ]
> IsCartanEquivalent(W1, W2);
false
%%a> assert not $1;

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: Name And Diagram#

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

A~2    1 - 2
       |   |
       - 3 -

RootSystem(W): GrpMat → RootDtm#: The root system of the finite real reflection group $W$ (Chapter ChapRootSys). If $W$ is infinite, an error is flagged.

RootDatum(W): GrpMat → RootDtm#: The root datum of the finite real reflection group $W$ (Chapter ChapRootDtm). 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 Dimension#

> R := StandardRootSystem("A", 4);
> W := ReflectionGroup(R);
> Rank(W);
4
%%a> assert $1 eq 4;
> Dimension(W);
5
%%a> assert $1 eq 5;

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 :raw-latex:`\cite[page 155]{Carter-small}` if the group is real, and using the algorithm of [Lehrer and Taylor, 2009] 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 [Lehrer and Taylor, 2009]. If $W$ is infinite, an error is flagged.

Example: Basic Degrees#

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
%%a> assert $1;
> &+degs eq NumPosRoots(W) + Rank(W);
true
%%a> assert $1;

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: Operations#

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]

LeftDescentSet(W, w): GrpMat, GrpMatElt → $\{\}$#: 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 → $\{\}$#: 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: Descent Sets#

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

Operations on Reflection Groups

Contents

Operations on Reflection Groups#