The 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 vector spacecontaining the (co)roots of the root system R, i.e. X (respectively, Y).
The simple (co)rootsof the root system R as the rows of a matrix, i.e. A (respectively, B).
> R := RootSystem("G2");
> RootSpace(R);
Full Vector space of degree 2 over Rational Field
> CorootSpace(R);
Full Vector space of degree 2 over Rational Field
> SimpleRoots(R);
[1 0]
[0 1]
> SimpleCoroots(R);
[ 2 -3]
[-1 2]
> CartanMatrix(R);
[ 2 -1]
[-3 2]
The number of positive roots of the root system R. This is also the number of positive coroots. The total number of (co)roots is twice the number of positive (co)roots.
Basis: MonStgElt Default: "Standard"
The indexed set of (co)roots of the root system R, i.e. {@ α1, ... α2N @} (respectively, {@ α1star, ... α2Nstar @}).
Basis: MonStgElt Default: "Standard"
The indexed set of positive (co)rootsof the root system R, i.e. {@ α1, ... αN @} (respectively, {@ α1star, ... αNstar @}).
Basis: MonStgElt Default: "Standard"
The rth (co)root αr (respectively, αrstar) of the root system R.
Basis: MonStgElt Default: "Standard"
If v is a (co)root in the root system R, return its index; otherwise return 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]);
> R := RootSystem(A, B);
> Roots(R);
{@
(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(R);
{@
(1 -1 1),
(0 1 -1),
(1 2 -2),
(2 1 -1),
(1 0 0),
(1 1 -1)
@}
> #Roots(R) eq 2*NumPosRoots(R);
true
> Root(R, 4);
(1 -1 -1)
> Root(R, 4 : Basis := "Root");
(2 1)
> RootPosition(R, [1,-1,-1]);
4
> RootPosition(R, [2,1] : Basis := "Root");
4
Basis: MonStgElt Default: "Standard"
The unique (co)root of greatest heightin the irreducible root system R.
Basis: MonStgElt Default: "Standard"
The unique long (co)root of greatest heightin the irreducible root system R.
Basis: MonStgElt Default: "Standard"
The unique short (co)root of greatest heightin the irreducible root system R.
> R := RootSystem("G2");
> HighestRoot(R);
(3 2)
> HighestLongRoot(R);
(3 2)
> HighestShortRoot(R);
(2 1)
Basis: MonStgElt Default: "Standard"
The matrix of an inner producton the (co)root space of the root system R which is invariant under the action of the (co)roots. This inner product is uniquely determined up to a constant on each irreducible component of R. The inner product is normalised so that the short roots in each crystallographic component have length one.
The root α acts on the root space via the reflection sα; the coroot αstar acts on the coroot space via the coreflection sαstar.
Basis: MonStgElt Default: "Standard"
The sequence of matrices giving the action of the simple (co)roots of the root system R on the (co)root space, i.e. the matrices of sα1, ..., sαn (respectively, sα1star, ..., sαnstar).
Basis: MonStgElt Default: "Standard"
The sequence of matrices giving the action of the (co)roots of the root system R on the (co)root space, i.e. the matrices of sα1, ..., s_(α2N) (respectively, sα1star, ..., s_(α2N)star).
Basis: MonStgElt Default: "Standard"
The matrix giving the action of the rth (co)root of the root system R on the (co)root space, i.e. the matrix of sαr (respectively, sαrstar).
The sequence of permutations giving the action of the simple (co)roots of the root system R on the (co)roots. This action is the same for roots and coroots.
The sequence of permutations giving the action of the (co)roots of the root system R on the (co)roots. This action is the same for roots and coroots.
The permutation giving the action of the rth (co)root of the root system R on the (co)roots. This action is the same for roots and coroots.
The sequence of words in the simple reflections for all the reflections of the root system R. These words are given as sequences of integers. In other words, if [a1, ..., al] = ReflectionWords(R)[r], then sαr = s_(αa1) ... s_(αal).
The word in the simple reflections for the rth reflection of the root system R. The word is given as a sequence of integers. In other words, if [a1, ..., al] = ReflectionWord(R, r), then sαr = s_(αa1) ... s_(αal).
> R := RootSystem("B3");
> mx := ReflectionMatrix(R, 4);
> perm := ReflectionPermutation(R, 4);
> wd := ReflectionWord(R, 4);
> RootPosition(R, Root(R,2) * mx) eq 2^perm;
true
> perm eq &*[ ReflectionPermutation(R, r) : r in wd ];
true
>
> mx := CoreflectionMatrix(R, 4);
> CorootPosition(R, Coroot(R,2) * mx) eq 2^perm;
true
The index of the sum of the rth and sth roots in the crystallographic root system R, or 0 if the sum is not a root. In other words, if t=hbox(Sum(R,r,s))ne0 then αt=αr + αs. We require αr≠∓αs.
Returns true if, and only if, the rth (co)root of the root system R is a positive root.
Returns true if, and only if, the rth (co)root of the root system R is a negative root.
The index of the negative of the rth (co)root of the root system R. In other words, if s = hbox(Negative(R,r)) then αs= - αr.
> R := RootSystem("G2");
> Sum(R, 1, Negative(R,5));
10
> IsPositive(R, 10);
false
> Negative(R, 10);
4
> P := PositiveRoots(R);
> P[1] - P[5] eq -P[4];
true
The height of the rth (co)root of the root system R, 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 root system R.
The square of the length of the rth (co)root of the root system R.
Returns true if, and only if, the rth root of the root system R is long. This only makes sense for irreducible crystallographic root systems. Note that for non-reduced root systems, the roots which are not indivisible are actually longer than the long ones.
Returns true if, and only if, the rth root of the root system R is short. This only makes sense for irreducible crystallographic root systems.
Returns true if, and only if, the rth root of the root system R is indivisible, ie, αr/2 is not a root.
Indices in the crystallographic root system R 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. We require that αr≠∓αs.
Indices in the crystallographic root system R 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. We require that αr≠∓αs.
The largest p such that αs - pαr is a root. We require that the root system R be crystallographic and αs≠∓αr.
The largest q such that αs + qαr is a root. We require that the root system R be crystallographic and αs≠∓αr.
> R := RootSystem("G2");
> RootHeight(R, 5);
4
> F := CoxeterForm(R);
> v := Root(R, 5);
> (v*F, v) eq RootNorm(R, 5);
true
> IsLongRoot(R, 5);
true
> LeftString(R, 1, 5);
[ 4, 3, 2 ]
> roots := Roots(R);
> for i in [1..3] do
> RootPosition(R, roots[5]-i*roots[1]);
> end for;
4
3
2
> R := RootSystem("BC2");
> Root(R,2), IsIndivisibleRoot(R,2);
(0 1) true
> Root(R,4), IsIndivisibleRoot(R,4);
(0 2) false
An additive orderon the positive roots of the root system R, ie. a sequence containing the numbers 1, ..., N in some order so that αr + αs=αt implies t is between r and s. This is computed using the techniques of [Pap94].
Returns true if, and only if, the sequence Q gives an additive order on a set of positive roots of the root system R. Q must be a sequence of integers in the range [1..N], where N is the number of positive roots of R, with no gaps or repeats.
> R := RootSystem("A5");
> a := AdditiveOrder(R);
> Position(a, 2);
6
> Position(a, 3);
10
> Position(a, Sum(R, 2, 3));
7