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 datum R, i.e. X⊗Q (respectively, Y⊗Q).
The latticecontaining the (co)roots of the root datum R, i.e. X (respectively, Y). An inclusion map into the (co)root space of R is returned as the second value.
The latticespanned by the (co)roots of the root datum R. An inclusion map into the (co)root space of R is returned as the second value.
> R := StandardRootDatum("A",2);
> V := RootSpace(R);
> FullRootLattice(R);
Standard Lattice of rank 3 and degree 3
Mapping from: Standard Lattice of rank 3 and degree 3 to ModTupFld: V
> RootLattice(R);
Lattice of rank 2 and degree 3
Basis:
( 1 -1 0)
( 0 1 -1)
Mapping from: Lattice of rank 2 and degree 3 to ModTupFld: V
Return true if, and only if, V is the (co)root space of some root datum.
Return true if, and only if, V is an element of the (co)root space of some root datum.
If V is the (co)root space of some root datum, this returns the datum.
> R := RootDatum("a3");
> V := RootSpace(R);
> v := V.1;
> IsRootSpace(V);
true
> RootDatum(V);
R: Adjoint root datum of dimension 3 of type A3
> IsInRootSpace(v);
true
For the given root datum R, return the lattice X0 and the vector space X0⊗Q, respectively (see Section Extended Root Data).
For the given root datum R, return the vector space bar(X) = (X⊗Q)/(X0⊗Q) containing the relative roots (see Section Extended Root Data). The projection from X⊗Q onto bar(X) is returned as second return value.
The simple (co)rootsof the root datum R as the rows of a matrix, i.e. A (respectively, B).
> R := RootDatum("G2");
> RootSpace(R);
Full Vector space of degree 2 over Rational Field
> FullRootLattice(R);
Standard Lattice of rank 2 and degree 2
Mapping from: Standard Lattice of rank 2 and degree 2 to Full Vector space of
degree 2 over Rational Field
> RootLattice(R);
Standard Lattice of rank 2 and degree 2
Mapping from: Standard Lattice of rank 2 and degree 2 to 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]
> Rank(R) eq Dimension(R);
true
The number of positive roots of the root datum 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 datum R, i.e. {@ α1, ... α2N @} (respectively, {@ α1star, ... α2Nstar @}).
Basis: MonStgElt Default: "Standard"
The indexed set of positive (co)rootsof the root datum R, i.e. {@ α1, ... αN @} (respectively, {@ α1star, ... αNstar @}).
Basis: MonStgElt Default: "Standard"
The rth (co)root αr (respectively, αrstar) of the root datum R.
Basis: MonStgElt Default: "Standard"
If v is a (co)root in the root datum R, return its index; otherwise return 0. These functions will try to coerce v into the appropriate lattice; v should be written with respect to the basis specified by the parameter Basis.
InBasis: MonStgElt Default: "Standard"
OutBasis: MonStgElt Default: "Standard"
Coroots: BoolElt Default: false
Changes the basis of the vector v contained in the space spanned by the (co)roots of the root datum R. The vector v is considered as an element of the root space by default. If the parameter Coroots is set to true, v is considered as an element of the coroot space. The optional arguments InBasis and OutBasis may take the same values as the parameter Basis as described at the beginning of the current section.
> R := RootDatum("A3" : Isogeny := 2);
> Roots(R);
{@
(1 0 0),
(0 1 0),
(1 0 2),
(1 1 0),
(1 1 2),
(2 1 2),
(-1 0 0),
(0 -1 0),
(-1 0 -2),
(-1 -1 0),
(-1 -1 -2),
(-2 -1 -2)
@}
> PositiveCoroots(R);
{@
(2 -1 -1),
(-1 2 0),
(0 -1 1),
(1 1 -1),
(-1 1 1),
(1 0 0)
@}
> #Roots(R) eq 2*NumPosRoots(R);
true
> Coroot(R, 4);
(1 1 -1)
> Coroot(R, 4 : Basis := "Root");
(1 1 0)
> CorootPosition(R, [1,1,-1]);
4
> CorootPosition(R, [1,1,0] : Basis := "Root");
4
> BasisChange(R, [1,0,0] : InBasis:="Root");
(1 0 0)
> BasisChange(R, [1,0,0] : InBasis:="Root", Coroots);
( 2 -1 -1)
Returns true if and only if the vector v is contained in the (co)root space of the root datum R.
Basis: MonStgElt Default: "Standard"
The unique (co)root of greatest heightin the irreducible root datum R.
Basis: MonStgElt Default: "Standard"
The unique long (co)root of greatest heightin the irreducible root datum R.
Basis: MonStgElt Default: "Standard"
The unique short (co)root of greatest heightin the irreducible root datum R.
> R := RootDatum("G2");
> HighestRoot(R);
(3 2)
> HighestLongRoot(R);
(3 2)
> HighestShortRoot(R);
(2 1)
The indexed set of all (resp. positive, negative, simple) relative roots of the root datum R. Note that the relative roots are returned in the order induced by the standard ordering on the (nonrelative) roots of R.
The relative root datum of the root datum R.
The preimage of a relative root δ is a disjoint union of Γ-orbits on the set of all roots of the root datum R. This intrinsic returns a sequence of representatives of these orbits.
> DynkinDiagram(RootDatum("E6"));
E6 1 - 3 - 4 - 5 - 6
|
2
>
> R := RootDatum("E6" : Twist:=2 ); R;
R: Twisted adjoint root datum of type 2E6,4
> OrbitsOnSimples(R);
[
GSet{ 2 },
GSet{ 4 },
GSet{ 1, 6 },
GSet{ 3, 5 }
]
> DistinguishedOrbitsOnSimples(R) eq OrbitsOnSimples(R);
true
> AnisotropicSubdatum(R);
Twisted toral root datum of dimension 6
[]
> RR := RelativeRootDatum(R);RR;
RR: Adjoint root datum of type F4
> _,pi := RelativeRootSpace(R);
> DynkinDiagram(RR);
F4 1 - 2 =>= 4 - 3
> [ Position(Roots(RR), pi(Root(R,i)) ) : i in [1,6, 3,5, 4, 2]];
[ 3, 3, 4, 4, 2, 1 ]
now one with distinguished orbits {2} and {4}:
> R := RootDatum("E6" : Twist := <{{2},{4}},2> ); R;
R: Twisted adjoint root datum of type 2E6,2
> OrbitsOnSimples(R);
[
GSet{ 2 },
GSet{ 4 },
GSet{ 1, 6 },
GSet{ 3, 5 }
]
> DistinguishedOrbitsOnSimples(R);
[
GSet{ 2 },
GSet{ 4 }
]
> AnisotropicSubdatum(R);
Twisted root datum of type 2(A2 A2)4,0
[ 1, 3, 5, 6, 7, 11, 37, 39, 41, 42, 43, 47 ]
> RR := RelativeRootDatum(R);RR;
RR: Adjoint root datum of type G2
> DynkinDiagram(RR);
G2 2 =<= 1
3
> _,pi := RelativeRootSpace(R);
> [ Position(Roots(RR), pi(Root(R,i)) ) : i in [2,4]];
[ 1, 2 ]
and now the one with distinguished orbits {2} and {1, 6},
which has a non-reduced relative root datum:
> R := RootDatum("E6" : Twist := <{{2},{1,6}},2> ); R;
R: Twisted adjoint root datum of dimension 6 of type 2E6,2
> OrbitsOnSimples(R);
[
GSet{ 2 },
GSet{ 4 },
GSet{ 1, 6 },
GSet{ 3, 5 }
]
> DistinguishedOrbitsOnSimples(R);
[
GSet{ 2 },
GSet{ 1, 6 }
]
> AnisotropicSubdatum(R);
Twisted root datum of type 2A3,0
[ 3, 4, 5, 9, 10, 15, 39, 40, 41, 45, 46, 51 ]
> RR := RelativeRootDatum(R);RR;
RR: Adjoint root datum of type BC2
> DynkinDiagram(RR);
BC2 1 =>= 2
> _,pi := RelativeRootSpace(R);
> [ Position(Roots(RR), pi(Root(R,i)) ) : i in [2, 1,6]];
[ 1, 2, 2 ]
Finally, the twisted root Datum of type ()6D4, 1:
> T := RootDatum( "D4" : Twist:=<{{2}},6> );
> T;
T: Twisted adjoint root datum of dimension 4 of type 6D4,1
> RelativeRootDatum(T);
Adjoint root datum of type BC1
> GammaOrbitsRepresentatives(T,1);
[ 11, 5 ]
> GammaOrbitsRepresentatives(T,2);
[ 12 ]
Basis: MonStgElt Default: "Standard"
The matrix of an inner producton the (co)root space of the root datum R which is invariant under the action of the (co)roots. The inner product is normalised so that the short roots in each irreducible 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 datum 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 datum 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 datum 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 datum 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 datum 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 datum 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 datum R. These words are given as sequences of integers. In other words, if a = [a1, ..., al] = ReflectionWords(R)[r], then sαr = s_(αa1) ... s_(αal).
The word in the simple reflections for the rth reflection of the root datum R. The word is given as a sequence of integers. In other words, if a = [a1, ..., al] = ReflectionWord(R, r), then sαr = s_(αa1) ... s_(αal).
> R := RootDatum("A3" : Isogeny := 2);
> 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 root datum R, or 0 if the sum is not a root. In other words, if t = hbox(Sum(R,r,s)) ≠0 then αt=αr + αs. The condition αr≠∓αs must be satisfied.
Returns true if, and only if, the rth (co)root of the root datum R is a positive root.
Returns true if, and only if, the rth (co)root of the root datum R is a negative root.
The index of the negative of the rth (co)root of the root datum R. In other words, if s = hbox(Negative(R,r)) then αs= - αr.
Indices in the root datum 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. The condition αr≠∓αs must be satisfied.
Indices in the root datum 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. The condition αr≠∓αs must be satisfied.
The largest p such that αs - pαr is a root of the root datum R. The condition αs≠∓αr must be satisfied.
The largest q such that αs + qαr is a root of the root datum R. The condition αs≠∓αr must be satisfied.
> R := RootDatum("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 datum 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 datum R.
The square of the length of the rth (co)root of the root datum R.
Returns true if, and only if, the rth root of the root datum R is long. This only makes sense for irreducible crystallographic root data. Note that for non-reduced root data, 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 datum R is short. This only makes sense for irreducible crystallographic root data.
Returns true if, and only if, the rth root of the root system R is indivisible.
> R := RootDatum("G2");
> RootHeight(R, 5);
4
> F := CoxeterForm(R);
> v := VectorSpace(Rationals(),2) ! 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
The closure in the root datum R of the set S of root indices. That is the indices of every root that can be written as a sum of roots with indices in S.
An additive orderon the positive roots of the root datum R, ie. 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 [Pap94]
Returns true if, and only if, Q gives an additive order on a set of positive roots of R. Q must be a sequence of integers in the range [1..N] with no gaps or repeats.
> R := RootDatum("A5");
> a := AdditiveOrder(R);
> Position(a, 2);
6
> Position(a, 3);
10
> Position(a, Sum(R, 2, 3));
7
The weight lattice Λ of the root datum R.i.e. the λ in QΦ≤Q tensor X such that < λ, αstar >∈Z for every coroot αstar.
The coweight lattice Λstar of the root datum R,i.e. the λstar in QΦstar≤Q tensor Y such that < α, λstar >∈Z for every root α.
Basis: MonStgElt Default: "Standard"
The fundamental weightsλ1, ..., λn of the root datum R given as the rows of a matrix. This is the basis of the weight lattice Λ dual to the simple coroots, i.e. < λi, αjstar >=δij.
Basis: MonStgElt Default: "Standard"
The fundamental coweightsλ1star, ..., λnstar of the root datum R given as the rows of a matrix. This is the basis of the coweight lattice Λstar dual to the simple roots, i.e. < αi, λjstar >=δij.
> R := RootDatum("E6");
> WeightLattice(R);
Lattice of rank 6 and degree 6
Basis:
(4 3 5 6 4 2)
(3 6 6 9 6 3)
(5 6 10 12 8 4)
(6 9 12 18 12 6)
(4 6 8 12 10 5)
(2 3 4 6 5 4)
Basis Denominator: 3
> FundamentalWeights(R);
[ 4/3 1 5/3 2 4/3 2/3]
[ 1 2 2 3 2 1]
[ 5/3 2 10/3 4 8/3 4/3]
[ 2 3 4 6 4 2]
[ 4/3 2 8/3 4 10/3 5/3]
[ 2/3 1 4/3 2 5/3 4/3]
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 dominant weight in the same W-orbit as the weight v, where W is the Weyl group of the root datum R. The second value returned is a Weyl group element taking v to the dominant weight. 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 W-orbit of the weight v as an indexed set, where W is the Weyl group of the root datum R. The first element in the orbit is always dominant. The second value returned is a sequence of Weyl group elements taking the weight to the corresponding element of the orbit. 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.
> R := RootDatum("B3");
> DominantWeight(R, [1,-1,0] : Basis:="Weight");
(1 0 0)
[ 2, 3, 2, 1 ]
> #WeightOrbit(R, [1,-1,0] : Basis:="Weight");
6