Weyl groups

weyl_group(R::RootSystem) -> WeylGroup

Return the Weyl group of R.

Examples

julia> weyl_group(root_system([2 -1; -1 2]))
Weyl group
  of root system of rank 2
    of type A2

julia> weyl_group(root_system(matrix(ZZ, 2, 2, [2, -1, -1, 2]); detect_type=false))
Weyl group
  of root system of rank 2
    of unknown type

julia> weyl_group(root_system(matrix(ZZ, [2 -1 -2; -1 2 0; -1 0 2])))
Weyl group
  of root system of rank 3
    of type C3 (with non-canonical ordering of simple roots)

weyl_group(cartan_matrix::ZZMatrix; check::Bool=true) -> WeylGroup
weyl_group(cartan_matrix::Matrix{<:IntegerUnion}; check::Bool=true) -> WeylGroup

Construct the Weyl group defined by the given (generalized) Cartan matrix.

If check=true the function will verify that cartan_matrix is indeed a generalized Cartan matrix.

weyl_group(fam::Symbol, rk::Int) -> WeylGroup

Construct the Weyl group of the given type.

The input must be a valid Cartan type, see is_cartan_type(::Symbol, ::Int).

Examples

julia> weyl_group(:A, 2)
Weyl group
  of root system of rank 2
    of type A2

weyl_group(type::Vector{Tuple{Symbol,Int}}) -> WeylGroup
weyl_group(type::Tuple{Symbol,Int}...) -> WeylGroup

Construct the Weyl group of the given type.

Each element of type must be a valid Cartan type, see is_cartan_type(::Symbol, ::Int). The vararg version needs at least one element.

Examples

julia> weyl_group([(:G, 2), (:D, 4)])
Weyl group
  of root system of rank 6
    of type G2 x D4

is_finite(W::WeylGroup) -> Bool

Return whether W is finite.

one(W::WeylGroup) -> WeylGroupElem

Return the identity element of W.

isone(x::WeylGroupElem) -> Bool

Return whether x is the identity element of its parent.

gen(W::WeylGroup, i::Int) -> WeylGroupElem

Return the i-th simple reflection (with respect to the underlying root system) of W.

This is a more efficient version for gens(W)[i].

gens(W::WeylGroup) -> WeylGroupElem

Return the simple reflections (with respect to the underlying root system) of W.

See also: gen(::WeylGroup, ::Int).

number_of_generators(W::WeylGroup) -> Int

Return the number of generators of the W, i.e. the rank of the underlying root system.

order(W::WeylGroup) -> ZZRingELem
order(::Type{T}, W::WeylGroup) where {T} -> T

Return the order of W.

If W is infinite, an InfiniteOrderError exception will be thrown. Use is_finite(::WeylGroup) to check this prior to calling this function if in doubt.

is_finite_order(x::WeylGroupElem) -> Bool

Return whether x is of finite order, i.e. there is a natural number n such that is_one(x^n).

order(x::WeylGroupElem) -> ZZRingELem
order(::Type{T}, x::WeylGroupElem) where {T} -> T

Return the order of x, i.e. the smallest natural number n such that is_one(x^n).

If x is of infinite order, an InfiniteOrderError exception will be thrown. Use is_finite_order(::WeylGroupElem) to check this prior to calling this function if in doubt.

root_system(W::WeylGroup) -> RootSystem

Return the underlying root system of W.

gen(W::WeylGroup, i::Int) -> WeylGroupElem

Return the i-th simple reflection (with respect to the underlying root system) of W.

This is a more efficient version for gens(W)[i].

gens(W::WeylGroup) -> WeylGroupElem

Return the simple reflections (with respect to the underlying root system) of W.

See also: gen(::WeylGroup, ::Int).

reflection(beta::RootSpaceElem) -> WeylGroupElem

Return the Weyl group element corresponding to the reflection at the hyperplane orthogonal to the root beta.

word(x::WeylGroupElem) -> Vector{UInt8}

Return x as a list of indices of simple reflections, in reduced form.

This function is right inverse to calling (W::WeylGroup)(word::Vector{<:Integer}).

Examples

julia> W = weyl_group(:A, 2);

julia> x = longest_element(W)
s1 * s2 * s1

julia> word(x)
3-element Vector{UInt8}:
 0x01
 0x02
 0x01

length(x::WeylGroupElem) -> Int

Return the length of x.

longest_element(W::WeylGroup) -> WeylGroupElem

Return the unique longest element of W. This only exists if W is finite.

<(x::WeylGroupElem, y::WeylGroupElem) -> Bool

Return whether x is smaller than y with respect to the Bruhat order, i.e., whether some (not necessarily connected) subexpression of a reduced decomposition of y, is a reduced decomposition of x.

reduced_expressions(x::WeylGroupElem; up_to_commutation::Bool=false)

Return an iterator over all reduced expressions of x as Vector{UInt8}.

If up_to_commutation is true, the iterator will not return an expression that only differs from a previous one by a swap of two adjacent commuting simple reflections.

The type of the iterator and the order of the elements are considered implementation details and should not be relied upon.

Examples

julia> W = weyl_group(:A, 3);

julia> x = W([1,2,3,1]);

julia> collect(reduced_expressions(x))
3-element Vector{Vector{UInt8}}:
 [0x01, 0x02, 0x01, 0x03]
 [0x01, 0x02, 0x03, 0x01]
 [0x02, 0x01, 0x02, 0x03]

julia> collect(reduced_expressions(x; up_to_commutation=true))
2-element Vector{Vector{UInt8}}:
 [0x01, 0x02, 0x01, 0x03]
 [0x02, 0x01, 0x02, 0x03]

The second expression of the first iterator is not contained in the second iterator because it only differs from the first expression by a swap of two the two commuting simple reflections s1 and s3.

*(r::RootSpaceElem, x::WeylGroupElem) -> RootSpaceElem
*(w::WeightLatticeElem, x::WeylGroupElem) -> WeightLatticeElem

Return the result of acting with x from the right on r or w.

See also: *(::WeylGroupElem, ::Union{RootSpaceElem,WeightLatticeElem}).

geometric_representation(W::WeylGroup) -> ZZMatrixGroup, Map{WeylGroup, ZZMatrixGroup}

Return the geometric representation G of the Weyl group W, together with the isomorphism hom from W to G.

This representation is defined by coefficients(a) * hom(x) == coefficients(a * x) for all x in W and a a simple root of root_system(W). By linear extension, this also holds for all elements a in the root space of root_system(W).

See [Hum90, Sect. 5.3] for more details.

Examples

julia> R = root_system(:B, 3); W = weyl_group(R);

julia> r = positive_root(R, 8)
a_1 + a_2 + 2*a_3

julia> x = W([1, 2, 1, 3])
s1 * s2 * s1 * s3

julia> G, hom = geometric_representation(W)
(Matrix group of degree 3 over ZZ, Map: W -> G)

julia> coefficients(r) * hom(x)
[1   1   0]

julia> coefficients(r * x)
[1   1   0]

dual_geometric_representation(W::WeylGroup) -> ZZMatrixGroup, Map{WeylGroup, ZZMatrixGroup}

Return the dual geometric representation G of the Weyl group W, together with the isomorphism hom from W to G.

This representation is defined by coefficients(w) * hom(x) == coefficients(w * x) for all x in W and w a fundamental weight of root_system(W). By linear extension, this also holds for all elements w in weight_lattice(root_system(W)).

Examples

julia> R = root_system(:B, 3); W = weyl_group(R);

julia> w = WeightLatticeElem(R, [1, 4, -3])
w_1 + 4*w_2 - 3*w_3

julia> x = W([1, 2, 1, 3])
s1 * s2 * s1 * s3

julia> G, hom = dual_geometric_representation(W)
(Matrix group of degree 3 over ZZ, Map: W -> G)

julia> coefficients(w) * hom(x)
[-4   6   -7]

julia> coefficients(w * x)
[-4   6   -7]

weyl_orbit(wt::WeightLatticeElem)

Return an iterator over the Weyl group orbit at the weight wt.

The type of the iterator and the order of the elements are considered implementation details and should not be relied upon.