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weight_lattices.md

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CurrentModule = Oscar
DocTestSetup = Oscar.doctestsetup()

Weight lattices

Weight lattices are represented by objects of type WeightLattice <: AdditiveGroup, and their elements by WeightLatticeElem <: AdditiveGroupElement.

They are introduced to have a formal parent object of all weights that correspond to a common given root system.

Table of contents

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

Constructing weight lattices

weight_lattice(::RootSystem)

Properties of weight lattices

rank(::WeightLattice)
is_finite(::WeightLattice)
zero(::WeightLattice)
gen(::WeightLattice, ::Int)
gens(::WeightLattice)

root_system(::WeightLattice)

Weight lattice elements

WeightLatticeElem(::WeightLattice, ::Vector{<:IntegerUnion})
WeightLatticeElem(::RootSystem, ::Vector{<:IntegerUnion})
WeightLatticeElem(::WeightLattice, ::ZZMatrix)
WeightLatticeElem(::RootSystem, ::ZZMatrix)
WeightLatticeElem(::RootSpaceElem)

Basic arithmetic operations like zero, +, -, * (with integer scalars), and == are supported.

coeff(::WeightLatticeElem, ::Int)
coefficients(::WeightLatticeElem)

iszero(::WeightLatticeElem)
is_dominant(::WeightLatticeElem)
is_fundamental_weight(::WeightLatticeElem)
is_fundamental_weight_with_index(::WeightLatticeElem)

Reflections

reflect(::WeightLatticeElem, ::Int)
reflect!(::WeightLatticeElem, ::Int)

Conjugate dominant weight

conjugate_dominant_weight(::WeightLatticeElem)
conjugate_dominant_weight_with_elem(::WeightLatticeElem)