The functions in this section give access to the underlying root system (or datum) of a permutation Coxeter group. These functions do not apply to finitely presented Coxeter groups
Roots are stored as an indexed set
{@ α1, ..., αN, αN + 1, ..., α2N @},
where α1, ..., αN are the positive roots in an order compatible with height; and αN + 1, ..., α2N are the corresponding negative roots (i.e. αi + N= - αi). The simple roots are α1, ..., αn where n is the rank.
Many of these functions have an optional argument Basis which may take one of the following values
The (co)root spaceof the Coxeter group W. This can be a vector space over a field of characteristic zero (Chapter VECTOR SPACES), or an integer lattice in the crystallographic case (Chapter LATTICES). The (co)reflection group of W acts on the (co)root space.
The simple (co)rootsof the Coxeter group W as the rows of a matrix.
> W := CoxeterGroup("G2");
> RootSpace(W);
Full Vector space of degree 2 over Rational Field
> CorootSpace(W);
Full Vector space of degree 2 over Rational Field
> SimpleRoots(W);
[1 0]
[0 1]
> SimpleCoroots(W);
[ 2 -3]
[-1 2]
> CartanMatrix(W);
[ 2 -1]
[-3 2]
The number of positive roots of the Coxeter group W.
Basis: MonStgElt Default: "Standard"
An indexed set containing the (co)rootsof the Coxeter group W.
Basis: MonStgElt Default: "Standard"
An indexed set containing the positive (co)rootsof the Coxeter group W.
Basis: MonStgElt Default: "Standard"
The rth (co)root of the Coxeter group W.
Basis: MonStgElt Default: "Standard"
If v is a (co)root of the Coxeter group W, this returns its position; otherwise it returns 0. These functions will try to coerce v, which can be a vector or a sequence representing a vector, into the appropriate vector space; v should be written with respect to the basis specified by the parameter Basis.
> A := Matrix(2,3, [1,-1,0, -1,1,-1]);
> B := Matrix(2,3, [1,-1,1, 0,1,-1]);
> W := CoxeterGroup(A, B);
> Roots(W);
{@
(1 -1 0),
(-1 1 -1),
(0 0 -1),
(1 -1 -1),
(2 -2 -1),
(1 -1 -2),
(-1 1 0),
(1 -1 1),
(0 0 1),
(-1 1 1),
(-2 2 1),
(-1 1 2)
@}
> PositiveCoroots(W);
{@
(1 -1 1),
(0 1 -1),
(1 2 -2),
(2 1 -1),
(1 0 0),
(1 1 -1)
@}
> #Roots(W) eq 2*NumPosRoots(W);
true
> Root(W, 4);
(1 -1 -1)
> Root(W, 4 : Basis := "Root");
(2 1)
> RootPosition(W, [1,-1,-1]);
4
> RootPosition(W, [2,1] : Basis := "Root");
4
Basis: MonStgElt Default: "Standard"
The unique (long) root of greatest heightof the irreducible Coxeter group W.
Basis: MonStgElt Default: "Standard"
The unique short root of greatest heightof the irreducible Coxeter group W.
> W := RootDatum("G2");
> HighestRoot(W);
(3 2)
> HighestLongRoot(W);
(3 2)
> HighestShortRoot(W);
(2 1)
Basis: MonStgElt Default: "Standard"
The matrix of an inner producton the (co)root space of the finite Coxeter group W which is invariant under the action of W. This inner product is uniquely determined up to a constant on each irreducible component of W. The inner product is normalised so that the short roots in each crystallographic component have length one.
An additive orderon the positive roots of the finite Coxeter group W, i.e. a sequence containing the numbers 1, ..., N in some order such that αr + αs=αt implies t is between r and s. This is computed using the techniques of Papi [Pap94].
An additive order for the set of positive roots of the finite Coxeter group W sent negative by w - 1. The element w can be supplied either as an element of the permutation group W or as a sequence representing a word in its generators.
The index of the sum of the rth and sth roots in the Coxeter group W, or 0 if the sum is not a root. In other words, if t = hbox(Sum(W,r,s)) ≠0 then αt=αr + αs. The condition αr≠∓αs must be satisfied. If W is noncrystallographic, an error is flagged.
Returns true if, and only if, the rth (co)root of the Coxeter group W is a positive root.
Returns true if, and only if, the rth (co)root of the Coxeter group W is a negative root.
The index of the negative of the rth (co)root of the Coxeter group W. In other words, if s = hbox(Negative(W,r)) then αs= - αr.
Root indices in the Coxeter group W of the left string through αs in the direction of αr, i.e. the indices of αs - αr, αs - 2αr, ..., αs - pαr. In other words, this returns the sequence [r1, ..., rp] where αri=αs - iαr and αs - (p + 1)αr is not a root. The condition αr≠∓αs must be satisfied. If W is noncrystallographic, an error is flagged.
Root indices of the Coxeter group W of the left string through αs in the direction of αr, i.e. the indices of αs + αr, αs + 2αr, ..., αs + qαr. In other words, this returns the sequence [r1, ..., rq] where αri=αs + iαr and αs + (q + 1)αr is not a root. The condition αr≠∓αs must be satisfied. If W is noncrystallographic, an error is flagged.
The largest p such that αs - pαr is a root of the Coxeter group W. The condition αr≠∓αs must be satisfied. If W is noncrystallographic, an error is flagged.
The largest q such that αs + qαr is a root of the Coxeter group W. The condition αr≠∓αs must be satisfied. If W is noncrystallographic, an error is flagged.
> W := RootDatum("G2");
> Sum(W, 1, Negative(W,5));
10
> IsPositive(W, 10);
false
> Negative(W, 10);
4
> P := PositiveRoots(W);
> P[1] - P[5] eq -P[4];
true
The height of the rth (co)root of the Coxeter group W, i.e. the sum of the coefficients of αr (respectively, αrstar) with respect to the simple (co)roots.
The sequence of squares of the lengthsof the (co)roots of the Coxeter group W.
The square of the length of the rth (co)root of the Coxeter group W.
Returns true if, and only if, the rth root of the Coxeter group W is long, i.e. the rth coroot is short. An error is flagged unless W is irreducible and crystallographic.
Returns true if, and only if, the rth root of the Coxeter group W is short, i.e. the rth coroot is long. An error is flagged unless W is irreducible and crystallographic.
> W := RootDatum("G2");
> RootHeight(W, 5);
4
> F := CoxeterForm(W);
> v := VectorSpace(Rationals(),2) ! Root(W, 5);
> (v*F, v) eq RootNorm(W, 5);
true
> IsLongRoot(W, 5);
true
> LeftString(W, 1, 5);
[ 4, 3, 2 ]
> roots := Roots(W);
> for i in [1..3] do
> RootPosition(W, roots[5]-i*roots[1]);
> end for;
4
3
2
The (co)weight latticeof the Coxeter group W. The roots and coroots of W must have integral components.
Basis: MonStgElt Default: "Standard"
The fundamental (co)weightsof the Coxeter group W. The roots and coroots of W must have integral components.
Basis: MonStgElt Default: "Standard"
Returns true if, and only if, v is a dominant weight for the root datum R, ie, a nonnegative integral linear combination of the fundamental weights.
Basis: MonStgElt Default: "Standard"
The unique element in the W-orbit of the weight v which lies in the fundamental Weyl chamber, and the word in the generators which sends v to this element. The Coxeter group W must have a root datum. The weight v can be given either as a vector or as a sequence representing the vector and is coerced into the weight lattice first.
Basis: MonStgElt Default: "Standard"
The orbit of the weight v under the action of W. The Coxeter group W must have a root datum. The weight v can be given either as a vector or as a sequence representing the vector and is coerced into the weight lattice first.
> W := CoxeterGroup("B3");
> DominantWeight(W, [1,-1,0] : Basis:="Weight");
(1 0 0)
[ 2, 3, 2, 1 ]
> #WeightOrbit(W, [1,-1,0] : Basis:="Weight");
6