Root system documentation (#4297)

Co-authored-by: Felix Röhrich <[email protected]>

experimental/BasisLieHighestWeight/src/MainAlgorithm.jl

@@ -203,7 +203,7 @@ function compute_monomials(
   # if highest_weight is not a fundamental weight, partition into smaller summands is possible. This is the basecase of 
   # the recursion.
   dim = dim_of_simple_module(L, highest_weight)
-  if is_zero(highest_weight) || is_fundamental(highest_weight)
+  if is_zero(highest_weight) || is_fundamental_weight(highest_weight)
     push!(no_minkowski, highest_weight)
     monomials = add_by_hand(
       L, birational_seq, ZZx, highest_weight, monomial_ordering, Set{ZZMPolyRingElem}()
@@ -473,22 +473,6 @@ function operators_lusztig(L::LieAlgebra, reduced_expression::Vector{Int})
   return operators
 end
 
-# TODO: upstream to LieAlgebras/RootSystem.jl
-function is_fundamental(highest_weight::WeightLatticeElem)
-  hasone = false
-  for i in coefficients(highest_weight)
-    if iszero(i)
-      continue
-    elseif isone(i)
-      hasone && return false
-      hasone = true
-    else
-      return false
-    end
-  end
-  return hasone
-end
-
 function sub_weights(w::WeightLatticeElem)
   # returns list of weights v != 0, highest_weight with 0 <= v <= w elementwise
   @req is_dominant(w) "The input must be a dominant weight"

experimental/BasisLieHighestWeight/test/MainAlgorithm-test.jl

@@ -31,12 +31,6 @@ function check_dimension(
   @test gap_dim == dim(basis) == length(monomials(basis)) # check if dimension is correct
 end
 
-@testset "is_fundamental" begin
-  R = root_system(:B, 3)
-  @test BasisLieHighestWeight.is_fundamental(WeightLatticeElem(R, [ZZ(0), ZZ(1), ZZ(0)]))
-  @test !BasisLieHighestWeight.is_fundamental(WeightLatticeElem(R, [ZZ(0), ZZ(1), ZZ(1)]))
-end
-
 @testset "sub_weights(_proper)" begin
   sub_weights = BasisLieHighestWeight.sub_weights
   sub_weights_proper = BasisLieHighestWeight.sub_weights_proper

experimental/LieAlgebras/docs/doc.main

@@ -7,5 +7,7 @@
       "modules.md",
       "module_homs.md",
       "cartan_matrix.md",
+      "root_systems.md",
+      "weyl_groups.md",
    ],
 ]

experimental/LieAlgebras/docs/src/cartan_matrix.md

@@ -5,12 +5,46 @@ DocTestSetup = Oscar.doctestsetup()
 
 # Cartan Matrices
 
+Cartan matrices can be constructed from a Cartan type, and are represented as a square `ZZMatrix`.
+
+Many functions taking a Cartan matrix as input (like [`root_system`](@ref) and [`weyl_group`](@ref)) will also accept a Cartan type as input. Both cases are semantically equivalent, but the latter may be more efficient.
+
+!!! note
+    The convention for Cartan matrices in OSCAR is $(a_{ij}) = (\langle \alpha_i^\vee, \alpha_j \rangle)$ for simple roots $\alpha_i$.
+
+## Table of contents
+
+```@contents
+Pages = ["cartan_matrix.md"]
+Depth = 2:5
+```
+
+## Constructors
+
 ```@docs
 cartan_matrix(::Symbol, ::Int)
-cartan_matrix(::Tuple{Symbol,Int}...)
-is_cartan_matrix(::ZZMatrix; generalized::Bool)
-cartan_symmetrizer(::ZZMatrix; check::Bool)
-cartan_bilinear_form(::ZZMatrix; check::Bool)
-cartan_type(::ZZMatrix; check::Bool)
-cartan_type_with_ordering(::ZZMatrix; check::Bool)
+cartan_matrix(::Vector{Tuple{Symbol,Int}})
+```
+
+
+## Properties
+
+```@docs
+is_cartan_matrix(::ZZMatrix)
+cartan_symmetrizer(::ZZMatrix)
+cartan_bilinear_form(::ZZMatrix)
+```
+
+
+## Cartan types
+
+The following function is used to verify the validity of a Cartan type, and is thus used in input sanitization.
+```@docs
+is_cartan_type(::Symbol, ::Int)
+```
+
+Given a Cartan matrix, the following functions can be used to determine its Cartan type.
+```@docs
+cartan_type(::ZZMatrix)
+cartan_type_with_ordering(::ZZMatrix)
 ```

experimental/LieAlgebras/docs/src/introduction.md

@@ -16,7 +16,7 @@ This part of OSCAR is in an experimental state; please see [Adding new projects
 
 Please direct questions about this part of OSCAR to the following people:
 * [Lars Göttgens](https://lgoe.li/)
-* [Laura Voggesberger](https://www.ruhr-uni-bochum.de/ffm/Lehrstuehle/Lehrstuhl-VI/voggesberger.html)
+* [Felix Röhrich](https://www.art.rwth-aachen.de/cms/~xlgua)
 
 You can ask questions in the [OSCAR Slack](https://www.oscar-system.org/community/#slack).
 

experimental/LieAlgebras/docs/src/root_systems.md

@@ -0,0 +1,223 @@
+```@meta
+CurrentModule = Oscar
+DocTestSetup = Oscar.doctestsetup()
+```
+
+# Root systems
+
+Root systems in this module are meant to be abstract root systems, i.e. they are represented by a set of roots (vectors in an euclidean space).
+
+The relevant types around root systems are:
+- `RootSystem` for the root system itself,
+- `RootSpaceElem` for elements in the root space, i.e. roots and linear combinations thereof,
+- `DualRootSpaceElem` for elements in the dual root space, i.e. coroots and linear combinations thereof,
+- `WeightLatticeElem` for elements in the weight lattice, i.e. weights and linear combinations thereof.
+
+!!! warning
+    Most functionality around root systems is currently only intended to be used with root systems of finite type.
+    For root systems of affine type, some documentation may be ill-phrased or incorrect, and some functions may not work as intended.
+
+## Table of contents
+
+```@contents
+Pages = ["root_systems.md"]
+Depth = 2:5
+```
+
+## Constructing root systems
+
+```@docs
+root_system(::ZZMatrix)
+root_system(::Symbol, ::Int64)
+root_system(::Vector{Tuple{Symbol, Int64}})
+```
+
+
+## Properties of root systems
+
+```@docs
+is_simple(::RootSystem)
+rank(::RootSystem)
+```
+
+
+### Cartan matrix and Weyl group
+```@docs
+cartan_matrix(::RootSystem)
+```
+```@docs; canonical=false
+weyl_group(::RootSystem)
+```
+
+
+### Root system type
+```@docs
+has_root_system_type(::RootSystem)
+root_system_type(::RootSystem)
+root_system_type_with_ordering(::RootSystem)
+```
+
+
+### Root getters
+```@docs
+number_of_roots(::RootSystem)
+number_of_positive_roots(::RootSystem)
+number_of_simple_roots(::RootSystem)
+```
+
+The following functions return roots, see [Root space elements](@ref) for more information.
+```@docs
+root(::RootSystem, ::Int64)
+roots(::RootSystem)
+simple_root(::RootSystem, ::Int64)
+simple_roots(::RootSystem)
+positive_root(::RootSystem, ::Int64)
+positive_roots(::RootSystem)
+negative_root(::RootSystem, ::Int64)
+negative_roots(::RootSystem)
+```
+
+
+### Coroot getters
+The following functions return coroots, see [Dual root space elements](@ref) for more information.
+```@docs
+coroot(::RootSystem, ::Int64)
+coroots(::RootSystem)
+simple_coroot(::RootSystem, ::Int64)
+simple_coroots(::RootSystem)
+positive_coroot(::RootSystem, ::Int64)
+positive_coroots(::RootSystem)
+negative_coroot(::RootSystem, ::Int64)
+negative_coroots(::RootSystem)
+```
+
+
+### Weight getters
+The following functions return weights, see [Weight lattice elements](@ref) for more information.
+```@docs
+fundamental_weight(::RootSystem, ::Int64)
+fundamental_weights(::RootSystem)
+weyl_vector(::RootSystem)
+```
+
+
+## Root space elements
+
+```@docs
+RootSpaceElem(::RootSystem, ::Vector{<:RationalUnion})
+RootSpaceElem(::RootSystem, ::QQMatrix)
+RootSpaceElem(::WeightLatticeElem)
+zero(::Type{RootSpaceElem}, ::RootSystem)
+```
+
+```@docs
+root_system(::RootSpaceElem)
+```
+
+Basic arithmetic operations like `zero`, `+`, `-`, `*` (with rational scalars), and `==` are supported.
+
+```@docs
+coeff(::RootSpaceElem, ::Int)
+coefficients(::RootSpaceElem)
+```
+
+```@docs
+height(::RootSpaceElem)
+iszero(::RootSpaceElem)
+```
+
+### Root testing
+```@docs
+is_root(::RootSpaceElem)
+is_root_with_index(::RootSpaceElem)
+is_simple_root(::RootSpaceElem)
+is_simple_root_with_index(::RootSpaceElem)
+is_positive_root(::RootSpaceElem)
+is_positive_root_with_index(::RootSpaceElem)
+is_negative_root(::RootSpaceElem)
+is_negative_root_with_index(::RootSpaceElem)
+```
+
+### Reflections
+```@docs
+reflect(::RootSpaceElem, ::Int)
+reflect!(::RootSpaceElem, ::Int)
+```
+
+
+## Dual root space elements
+
+```@docs
+DualRootSpaceElem(::RootSystem, ::Vector{<:RationalUnion})
+DualRootSpaceElem(::RootSystem, ::QQMatrix)
+zero(::Type{DualRootSpaceElem}, ::RootSystem)
+```
+
+```@docs
+root_system(::DualRootSpaceElem)
+```
+
+Basic arithmetic operations like `zero`, `+`, `-`, `*` (with rational scalars), and `==` are supported.
+
+```@docs
+coeff(::DualRootSpaceElem, ::Int)
+coefficients(::DualRootSpaceElem)
+```
+
+```@docs
+height(::DualRootSpaceElem)
+iszero(::DualRootSpaceElem)
+```
+
+### Coroot testing
+```@docs
+is_coroot(::DualRootSpaceElem)
+is_coroot_with_index(::DualRootSpaceElem)
+is_simple_coroot(::DualRootSpaceElem)
+is_simple_coroot_with_index(::DualRootSpaceElem)
+is_positive_coroot(::DualRootSpaceElem)
+is_positive_coroot_with_index(::DualRootSpaceElem)
+is_negative_coroot(::DualRootSpaceElem)
+is_negative_coroot_with_index(::DualRootSpaceElem)
+```
+
+
+## Weight lattice elements
+
+```@docs
+WeightLatticeElem(::RootSystem, ::Vector{<:IntegerUnion})
+WeightLatticeElem(::RootSystem, ::ZZMatrix)
+WeightLatticeElem(::RootSpaceElem)
+zero(::Type{WeightLatticeElem}, ::RootSystem)
+```
+
+```@docs
+root_system(::WeightLatticeElem)
+```
+
+Basic arithmetic operations like `zero`, `+`, `-`, `*` (with integer scalars), and `==` are supported.
+
+```@docs
+coeff(::WeightLatticeElem, ::Int)
+coefficients(::WeightLatticeElem)
+```
+
+```@docs
+iszero(::WeightLatticeElem)
+is_dominant(::WeightLatticeElem)
+is_fundamental_weight(::WeightLatticeElem)
+is_fundamental_weight_with_index(::WeightLatticeElem)
+```
+
+### Reflections
+```@docs
+reflect(::WeightLatticeElem, ::Int)
+reflect!(::WeightLatticeElem, ::Int)
+```
+
+### Conjugate dominant weight
+```@docs
+conjugate_dominant_weight(::WeightLatticeElem)
+conjugate_dominant_weight_with_left_elem(::WeightLatticeElem)
+conjugate_dominant_weight_with_right_elem(::WeightLatticeElem)
+```