Many of these functions have an optional argument Basis which may take one of the following values
The base space of the reflection group W. If W is not a reflection group, an error is flagged
> W := ComplexReflectionGroup( "M", 4 ); > RootSpace(W); Full Vector space of degree 3 over Cyclotomic Field of order 24 and degree 8
The sequence of simple orders of the reflection group W. If W is not a reflection group, an error is flagged
The simple (co)roots of the reflection group W as the rows of a matrix, i.e. A (resp. B).
> W := ComplexReflectionGroup( "M", 4 ); > RootSpace(W); Full Vector space of degree 3 over Cyclotomic Field of order 24 and degree 8 > SimpleRoots( W ); [1 0 0] [0 1 0] [0 0 1] [1 1/3*(2*z^7 - 2*z^5 + 2*z^3 + 4*z) z^7 + z^5 - z^3] > SimpleOrders( W ); [ 2, 6, 6, 6 ]
The number of positive roots of the real reflection group W. This is also the number of positive coroots. The total number of (co)roots is twice the number of positive (co)roots. This number is finite if, and only if, W is finite.
Basis: MonStgElt Default: "Standard"
The indexed set of (co)roots of the real reflection group W, i.e. {@ alpha_1, dotsalpha_(2N) @} (resp. {@ alpha_1^star, dotsalpha_(2N)^star @}). If W is infinite, an error is flagged.
Basis: MonStgElt Default: "Standard"
The indexed set of positive (co)roots of the real reflection group W, that is, {@ alpha_1, dotsalpha_N @} (resp. {@ alpha_1^star, dotsalpha_N^star @}). If W is infinite, an error is flagged.
Basis: MonStgElt Default: "Standard"
The rth (co)root alpha_r (resp. alpha_r^star) of the real reflection group W. If W is infinite, an error is flagged.
Basis: MonStgElt Default: "Standard"
If v is a (co)root in the finite real reflection group W, 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. If W is infinite, an error is flagged.
> W := ReflectionGroup( "A3" );
> Roots( W );
{@
(1 0 0),
(0 1 0),
(0 0 1),
(1 1 0),
(0 1 1),
(1 1 1),
(-1 0 0),
( 0 -1 0),
( 0 0 -1),
(-1 -1 0),
( 0 -1 -1),
(-1 -1 -1)
@}
> PositiveCoroots( W );
{@
( 2 -1 0),
(-1 2 -1),
( 0 -1 2),
( 1 1 -1),
(-1 1 1),
(1 0 1)
@}
> #Roots(W) eq 2*NumPosRoots(W);
true
> Root( W, 4 );
(1 1 0)
> Root( W, 4 : Basis := "Root" );
(1 1 0)
> RootPosition( W, [1,1,0] );
4> W := ReflectionGroup( "A3" );
> Roots( W );
{@
(1 0 0),
(-1 -1 -3),
(1 2 4),
( 0 -1 -3),
(0 1 1),
(1 1 1),
(-1 0 0),
(1 1 3),
(-1 -2 -4),
(0 1 3),
( 0 -1 -1),
(-1 -1 -1)
@}
> PositiveCoroots( W );
{@
( 2 -1 0),
(-1 2 -1),
(0 1 0),
( 1 1 -1),
(-1 3 -1),
( 1 2 -1)
@}
> #Roots(W) eq 2*NumPosRoots(W);
true
> Root( W, 4 );
( 0 -1 -3)
> Root( W, 4 : Basis := "Root" );
(1 1 0)
> RootPosition( W, [0,-1,-3] );
4
The root alpha acts on the root space via the reflection s_alpha; the coroot alpha^star acts on the coroot space via the coreflection s_alpha^star.
The sequence of reflections in the finite real reflection group W. The rth reflection in the sequence corresponds to the rth (co)root.
The reflection in finite real reflection group W corresponding to the rth (co)root. If r=1, ..., n, this is a generator of W.
The sequence of permutations giving the action of the simple (co)roots of the finite reflection group W 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 finite reflection group W 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 finite reflection group W 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 real reflection group W. These words are given as sequences of integers. In other words, if a = [a_1, ..., a_l] = ReflectionWords(R)[r], then s_(alpha_r) = s_(alpha_(a_1)) ... s_(alpha_(a_l)).
The word in the simple reflections for the rth reflection of the real reflection group W. The word is given as a sequence of integers. In other words, if a = [a_1, ..., a_l] = ReflectionWord(R, r), then s_(alpha_r) = s_(alpha_(a_1)) ... s_(alpha_(a_l)).
> Q := RationalField(); > W := ReflectionGroup( "A3" ); > mx := ReflectionMatrix( W, 4 ); > perm := ReflectionPermutation( W, 4 ); > RootPosition( W, Vector(Q, Eltseq(Root(W,2))) * mx ) eq 2^perm; true > mx := CoreflectionMatrix( W, 4 ); > CorootPosition( W, Coroot(W,2) * mx ) eq 2^perm; true
The (co)weight lattice of the real reflection group W.. The roots and coroots of W must have integral components.
Basis: MonStgElt Default: "Standard"
The fundamental weights of the real reflection group W given as the rows of a matrix. The roots and coroots of W must have integral components.
> W := ReflectionGroup( "E6" ); > WeightLattice( W ); 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( W ); [ 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]