Merge remote-tracking branch 'origin/adv/algebraic-shifting' into adv…

…/algebraic-shifting

experimental/AlgebraicShifting/README.md

@@ -2,12 +2,11 @@
 
 ## Aims
 
-This is an example for a file structure to set up a new package 
-in the experimental section. All files you find here are part of the 
-minimum requirements. See also the official Oscar documentation.
+This package grew out of a project of Antony della Vecchia, Michael Joswig and Fabian Lenzen on "Partial Algebraic Shifting".
+Part of this project is the development of new methods for algebraic shifting; these are implemented here.
 
 ## Status
 
-We plan to also provide a function to automatically copy this template 
-for you to start your own package. 
+Full and partial shifting in the exterior algebra fully functional.
+
 

experimental/AlgebraicShifting/docs/src/exterior_shifting.md

@@ -10,7 +10,7 @@ DocTestSetup = Oscar.doctestsetup()
 uniform_hypergraph
 ```
 
-## Helpful Matrix constructions
+## Matrix Constructions
 
 ```@docs
 generic_unipotent_matrix

experimental/AlgebraicShifting/docs/src/introduction.md

@@ -6,7 +6,11 @@ DocTestSetup = Oscar.doctestsetup()
 
 # Introduction
 
-This project aims to provide functionality for Algebraic Shifting, for some background on the subject we refer to [KAL02](@cite).
+Algebraic shifting is a widely applicable method for converting a uniform hypergraph $S$ into another hypergraph, $\Delta(S)$, which is somehow simpler but still retains key properties of $S$.
+Often this is applied to simplicial complexes, where each layer of $(k-1)$-faces forms a $k$-uniform hypergraph.
+
+Here we focus on full and partial algebraic shifting in the exterior algebra.
+For some background on the subject we refer to [KAL02](@cite).
 
 
 ## Status
@@ -17,6 +21,7 @@ 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:
 * [Antony Della Vecchia](https://antonydellavecchia.github.io)
+* [Michael Joswig](https://page.math.tu-berlin.de/~joswig/)
 
 
 You can ask questions in the [OSCAR Slack](https://www.oscar-system.org/community/#slack).

experimental/AlgebraicShifting/src/PartialShift.jl

@@ -32,7 +32,7 @@ end
 
 Constructs a unipotent matrix with entries in a polynomial ring `R`.
 One can also provide a field `F` and an integer `n`,
-then the entries of the unipotent matrix will lie in a multi variate
+then the entries of the unipotent matrix will lie in a multivariate
 polynomial ring over `F` with `n^2` variables.
 
 # Examples
@@ -58,17 +58,17 @@ end
 @doc raw"""
      rothe_matrix(F::Field, w::WeylGroupElem; K::Union{SimplicialComplex, Nothing} = nothing)
 
-For a base field `F` and a weyl group element `w` return the matrix with entries in the
+For a base field `F` and a Weyl group element `w` return the matrix with entries in the
 multivariate polynomial ring `R` with `n^2` many indeterminants where `n - 1` is the rank of the
-root system of the weyl group.
-We know that since `general_linear_group(n^2, R)` has a Bruhat decomposition, any element lies in some double coset $BwB$.
+root system of the Weyl group.
+As `general_linear_group(n^2, R)` has a Bruhat decomposition, any element lies in a unique double coset $BwB$, where $B$ is the Borel group of upper triangular matrices.
 The Rothe matrix is a normal form for the matrix on the left of a representative for the double coset corresponding to `w`.
-(this might need to be explained further and reference the preprint)
-We use the name Rothe matrix because of its resemblance with a Rothe diagram. (add ref?)
+This will be explained further once the corresponding preprint is on the arXiv.
+We use the name Rothe matrix because of its resemblance with a Rothe diagram. (add ref? Knuth?)
 
 # Examples
 ```jldoctest
-  julia> W = weyl_group(:A, 4)
+julia> W = weyl_group(:A, 4)
 Weyl group for root system defined by Cartan matrix [2 -1 0 0; -1 2 -1 0; 0 -1 2 -1; 0 0 -1 2]
 
 julia> s = gens(W)
@@ -105,12 +105,11 @@ end
      compound_matrix(w::WeylGroupElem, k::Int)
      compound_matrix(m::MatElem, K::Vector{Vector{Int}})
 
-Given a matrix `m` return the matrix where each entry is a `k` minor of `m`.
+Given a matrix `m`, return the matrix where each entry is a `k`$\times$`k`-minor of `m`.
 The entries of the compound matrix are ordered with respect to the lexicographic order on sets.
-When passed a `PermGroupElem` or `WeylGroupElem`, return the copound matrix for their
-permutation matrix representation.
+When passed a `PermGroupElem` or `WeylGroupElem`, return the compound matrix for their permutation matrix representation.
 
-Alternatively, passing a `UniformHypergraph` `K` will return the compound matrix with entries the `face_size(K)` minors, and restrict the rows to the rows corresponding to `K`
+Alternatively, passing a `UniformHypergraph` `K` will return the compound matrix with entries the `face_size(K)` minors, and restrict the rows to the rows corresponding to `K`.
 
 # Examples
 ```jldoctest
@@ -219,7 +218,7 @@ end
 
 Computes the (partial) exterior shift of a simplical complex or uniform hypergraph `K` with respect to the Weyl group element `w` and the field `F`.
 If the field is not given then `QQ` is used during the computation.
-If `w` is not given then `longest_element(weyl_group(:A, n_vertices(K) - 1))` is used
+If `w` is not given then `longest_element(Weyl_group(:A, n_vertices(K) - 1))` is used
 
 # Examples
 ```jldoctest
@@ -248,7 +247,7 @@ true
 julia> betti_numbers(L) == betti_numbers(K)
 true
 
-julia> W = weyl_group(:A, n_vertices(K) - 1)
+julia> W = Weyl_group(:A, n_vertices(K) - 1)
 Weyl group for root system defined by Cartan matrix [2 -1 0 0 0; -1 2 -1 0 0; 0 -1 2 -1 0; 0 0 -1 2 -1; 0 0 0 -1 2]
 
 julia> s = gens(W)