Weight lattices

weight_lattice(R::RootSystem) -> WeightLattice

Return the weight lattice of R, i.e. the lattice spanned by the fundamental weights.

This is the parent of all weights of R.

Examples

julia> weight_lattice(root_system([2 -1; -1 2]))
Weight lattice
  of root system of rank 2
    of type A2

julia> weight_lattice(root_system(matrix(ZZ, 2, 2, [2, -1, -1, 2]); detect_type=false))
Weight lattice
  of root system of rank 2
    of unknown type

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

rank(P::WeightLattice) -> Int

Return the rank of the weight lattice P.

is_finite(P::WeightLattice) -> Bool

Check if the weight lattice P is finite, i.e. if it has rank 0.

zero(P::WeightLattice) -> WeightLatticeElem

Return the neutral additive element in the weight lattice P.

gen(P::WeightLattice, i::Int) -> WeightLatticeElem

Return the i-th generator of the weight lattice P, i.e. the i-th fundamental weight of the root system of P.

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

See also: fundamental_weight(::RootSystem, ::Int).

gens(P::WeightLattice) -> Vector{WeightLatticeElem}

Return the generators of the weight lattice P, i.e. the fundamental weights of the root system of P.

See also: gen(::WeightLattice, ::Int), fundamental_weights(::RootSystem).

root_system(P::WeightLattice) -> RootSystem

Return the underlying root system of P.

WeightLatticeElem(P::WeightLattice, vec::Vector{<:IntegerUnion}) -> WeightLatticeElem

Construct a weight lattice element in P with the given coefficients w.r.t. the fundamental weights of corresponding root system.

WeightLatticeElem(R::RootSystem, vec::Vector{<:IntegerUnion}) -> WeightLatticeElem

Construct a weight lattice element in the root system R with the given coefficients w.r.t. the fundamental weights of R.

WeightLatticeElem(P::WeightLattice, vec::ZZMatrix) -> WeightLatticeElem

Construct a weight lattice element in P with the given coefficients w.r.t. the fundamental weights of corresponding root system.

vec must be a row vector of the same length as the rank of P.

WeightLatticeElem(R::RootSystem, vec::ZZMatrix) -> WeightLatticeElem

Construct a weight lattice element in the root system R with the given coefficient vector w.r.t. the fundamental weights of R.

vec must be a row vector of the same length as the rank of R.

WeightLatticeElem(r::RootSpaceElem) -> WeightLatticeElem

Construct a weight lattice element from the root space element r.

coeff(w::WeightLatticeElem, i::Int) -> ZZRingElem

Return the coefficient of the i-th fundamental weight in w.

This can be also accessed via w[i].

coefficients(w::WeightLatticeElem) -> ZZMatrix

Return the coefficients of the weight lattice element w w.r.t. the fundamental weights as a row vector.

iszero(w::WeightLatticeElem) -> Bool

Return whether w is zero.

is_dominant(w::WeightLatticeElem) -> Bool

Check if w is a dominant weight, i.e. if all coefficients are non-negative.

is_fundamental_weight(w::WeightLatticeElem) -> Bool

Check if w is a fundamental weight, i.e. exactly one coefficient is equal to 1 and all others are zero.

See also: is_fundamental_weight_with_index(::WeightLatticeElem).

is_fundamental_weight_with_index(w::WeightLatticeElem) -> Bool, Int

Check if w is a fundamental weight and return this together with the index of the fundamental weight in fundamental_weights(::RootSystem).

If w is not a fundamental weight, the second return value is arbitrary.

See also: is_fundamental_weight(::WeightLatticeElem).

reflect(w::WeightLatticeElem, s::Int) -> WeightLatticeElem

Return the reflection of w in the hyperplane orthogonal to the s-th simple root.

See also: reflect!(::WeightLatticeElem, ::Int).

reflect!(w::WeightLatticeElem, s::Int) -> WeightLatticeElem

Reflect w in the hyperplane orthogonal to the s-th simple root, and return it.

This is a mutating version of reflect(::WeightLatticeElem, ::Int).

conjugate_dominant_weight(w::WeightLatticeElem) -> WeightLatticeElem

Return the unique dominant weight conjugate to w.

See also: conjugate_dominant_weight_with_elem(::WeightLatticeElem).

conjugate_dominant_weight_with_elem(w::WeightLatticeElem) -> Tuple{WeightLatticeElem, WeylGroupElem}

Returns the unique dominant weight dom conjugate to w and a Weyl group element x such that x * w == dom.

If one wants a group element that takes w todom using a right action, one can use inv(x).