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Weyl groups

Weyl groups are represented by objects of type WeylGroup <: Group, and their elements by WeylGroupElem <: GroupElement.

Warning

Weyl groups in OSCAR only afford right actions on roots and weights. Note however, that this may differ from the literature, but is to stay consistent with the conventions in the rest of OSCAR.

Note

See Cartan types for our conventions on Cartan types and ordering of simple roots.

Table of contents

Constructing Weyl groups

weyl_group — Method
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)
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weyl_group — Method
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.

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weyl_group — Method
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
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weyl_group — Method
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
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Basic properties

Basic group arithmetic like *, and inv are defined for WeylGroupElem objects.

Finite Weyl groups support iteration over all group elements (in an arbitrary order).

is_finite — Method
is_finite(W::WeylGroup) -> Bool

Return whether W is finite.

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one — Method
one(W::WeylGroup) -> WeylGroupElem

Return the identity element of W.

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isone — Method
isone(x::WeylGroupElem) -> Bool

Return whether x is the identity element of its parent.

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gen — Method
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].

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gens — Method
gens(W::WeylGroup) -> WeylGroupElem

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

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

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number_of_generators — Method
number_of_generators(W::WeylGroup) -> Int

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

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order — Method
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.

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is_finite_order — Method
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).

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order — Method
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.

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root_system — Method
root_system(W::WeylGroup) -> RootSystem

Return the underlying root system of W.

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Element constructors

Using (W::WeylGroup)(word::Vector{<:Integer}), one can construct group elements from a word in the generators.

gen — Method
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].

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gens — Method
gens(W::WeylGroup) -> WeylGroupElem

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

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

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reflection — Method
reflection(beta::RootSpaceElem) -> WeylGroupElem

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

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Words and length

word — Method
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
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length — Method
length(x::WeylGroupElem) -> Int

Return the length of x.

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longest_element — Method
longest_element(W::WeylGroup) -> WeylGroupElem

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

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Bruhat order

< — Method
<(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.

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Reduced expressions

reduced_expressions — Method
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.

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Action on roots and weights

geometric_representation — Method
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]
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dual_geometric_representation — Method
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]
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Orbits

weyl_orbit — Method
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.

Note

This function currently only supports finite Weyl groups.

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