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CurrentModule = Oscar
DocTestSetup = Oscar.doctestsetup()

Weyl groups

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

!!! warning Weyl groups in OSCAR afford both left and right actions on roots and weights. Note however, that the left action is the default (to align with the literature), and all more complex functionality is defined with respect to the left action, e.g. conjugate_dominant_weight_with_elem(::WeightLatticeElem).

Table of contents

Pages = ["weyl_groups.md"]
Depth = 2:5

Constructing Weyl groups

weyl_group(::RootSystem)
weyl_group(::ZZMatrix)
weyl_group(::Symbol, ::Int)
weyl_group(::Vector{Tuple{Symbol,Int}})

Basic properties

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

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

is_finite(::WeylGroup)
one(::WeylGroup)
isone(::WeylGroupElem)
gen(::WeylGroup, ::Int)
gens(::WeylGroup)
number_of_generators(::WeylGroup)
order(::Type{T}, ::WeylGroup) where {T}

root_system(::WeylGroup)

Words and length

word(::WeylGroupElem)
length(::WeylGroupElem)
longest_element(::WeylGroup)

Bruhat order

<(::WeylGroupElem, ::WeylGroupElem)

Conversion to other group types

For many computations, it may be suitable to have a WeylGroup as a different kind of group object, to e.g. use functionality that is only available for that other type.

The conversion functions come in pairs: one only creates an isomorphic group object, the other also computes the isomorphism.

fp_group(::WeylGroup)
isomorphism(::Type{FPGroup}, ::WeylGroup)

Reduced expressions

reduced_expressions(::WeylGroupElem)

Action on roots and weights

*(::WeylGroupElem, ::Union{RootSpaceElem,WeightLatticeElem})
*(::Union{RootSpaceElem,WeightLatticeElem}, ::WeylGroupElem)

Orbits

TODO