Documentation

Reflections

An element of a Coxeter group is called a reflection if it is conjugate to one of the Coxeter generators.

In a permutation Coxeter group, the root α acts on the root space via the reflection s_α; the coroot α^star acts on the coroot space via the coreflection s_α^star.

IsReflection(w) : GrpFPElt -> BoolElt

IsReflection(w) : GrpPermElt -> BoolElt, ., ., RngInt

Returns true if, and only if, w is a reflection, i.e. w is conjugate to a Coxeter generator. If w is in a permutation Coxeter group, the root, coroot and root index are also returned.

Reflections(W) : GrpFPCox -> [GrpFPCoxElt]

Reflections(W) : GrpPermCox -> [GrpPermElt]

The sequence of reflections in the finite Coxeter group W. If W is a permutation Coxeter group, the rth reflection in the sequence corresponds to the rth (co)root.

Example `GrpCox_Reflections (H105E22)`

> W<a,b> := CoxeterGroup(GrpFPCox, "A2");
> Reflections(W);
[ a, b, a * b * a, a, b, a * b * a ]
> IsReflection(a*b);
false

SimpleReflections(W) : GrpFPCox -> [GrpFPCoxElt]

SimpleReflections(W) : GrpPermCox -> [GrpPermElt]

The sequence of simple reflections in the Coxeter group W, ie, the generators of W.

SimpleReflectionPermutations(W) : GrpPermCox -> [GrpPermElt]

The sequence of simple reflections in the permutation Coxeter group W, ie, the generators of W.

Reflection(W, r) : GrpPermCox, RngIntElt -> GrpPermElt

ReflectionPermutation(W, r) : GrpPermCox, RngIntElt -> GrpPermElt

The reflection in permutation Coxeter group W corresponding to the rth (co)root. If r=1, ..., n, this is a generator of W.

SimpleReflectionMatrices(W) : GrpPermCox -> []

SimpleCoreflectionMatrices(W) : GrpPermCox -> []

    Basis: MonStgElt                    Default: "Standard"

The matrices giving the action of the simple (co)roots on the (co)root space of the permutation Coxeter group W.

ReflectionMatrices(W) : GrpPermCox -> []

CoreflectionMatrices(W) : GrpPermCox -> []

    Basis: MonStgElt                    Default: "Standard"

The matrices giving the action of the (co)roots on the (co)root space of the permutation Coxeter group W.

ReflectionMatrix(W, r) : GrpPermCox, RngIntElt -> []

CoreflectionMatrix(W, r) : GrpPermCox, RngIntElt -> []

    Basis: MonStgElt                    Default: "Standard"

The matrix giving the action of the rth (co)root on the (co)root space of the permutation Coxeter group W.

ReflectionWords(W) : GrpPermCox -> []

The sequence of words in the simple reflections for all the reflections of the Coxeter group W. These words are given as sequences of integers. In other words, if a = [a₁, ..., a_l] = ReflectionWords(W)[r], then s_{α_r} = s_(α_a₁) ... s_(α_{a_l}).

ReflectionWord(W, r) : GrpPermCox, RngIntElt -> []

The word in the simple reflections for the rth reflection of the Coxeter group W. The word is given as a sequence of integers. In other words, if a = [a₁, ..., a_l] = ReflectionWord(W, r), then s_{α_r} = s_(α_a₁) ... s_(α_{a_l}).

Example `GrpCox_Action (H105E23)`

> W := CoxeterGroup("B3");
> IsReflection(W.1*W.2);
false
> mx := ReflectionMatrix(W, 4);
> perm := Reflection(W, 4);
> wd := ReflectionWord(W, 4);
> rt := VectorSpace(Rationals(), 3) ! Root(W,2);
> RootPosition(W, rt * mx) eq 2^perm;
true
> perm eq &*[ Reflection(W, r) : r in wd ];
true
>
> mx := CoreflectionMatrix(W, 4);
> CorootPosition(W, Coroot(W,2) * mx) eq 2^perm;
true

V2.28, 13 July 2023