Adv/algebraic shifting (#4260)

* init algebraic shifting * copied from template * copied from template * concrete function * fix for creating simplicial cmplex from from graph incidence matrix * use sparse matrices * computes correct result for octohedron * fixed function use * started on delta_sym * some progress on delta sym * ext and sym shifting work, return algebras still neds work * updtes to delta_sym * added field for sym shift * fixed apply on facets * delta_sym bug fixed * added fix for delta_sym with finite fields * Revert "added fix for delta_sym with finite fields" This reverts commit 398d1f9. * name changes * optional change_of_basis * fix regression * added some comments * cond fix * adds comments * starting tests * oscar updates * fixes change of basis matrix * fix for exterior shift * update for n_vertices * needed for serialization of results * fixes loading polymake simplicial complex and other types * adds test case * removing unnecessary else * fix test * allow change of basis to be generic + comment about sort * adds some functions for simplicial complexes from polymake * get rid of monomial sort * cleans up symmetric shift a bit * adds gb details to stanley reisner ideal * symmetric shift working over finite characteristic * set description on john solid files * Revert "set description on john solid files" This reverts commit df8279e. * updates for symmetric shift * adds functionality for automorphisms of simplicial complexes * adds docs + missing docs for the other recently added functions * adjustments to the docs * adds n_facets + uses combinatorial symmetries from polymake * acts on_simplicial_complex * fixes for docstrings * warning * fixes require check * add missing vertices * simplicial product * added link_subcomplex * reduce over quotients * barycentric sub div * json -> JSON3.write * Update src/Serialization/main.jl * fixes upgrade scripts * Update src/Serialization/main.jl but, s.obj should always be from JSON3, at least until we change something Co-authored-by: Benjamin Lorenz <[email protected]> * use io instead of string * typo * fix arguments to deserializer_open * remove white space * no need for key when serializing weyl group * root system uses ref * unifiying the exterior shift function * things are starting to work * partial shift graph * tests passing * still need partial graph vertices to work properly * tests passing and correct for partial shift graph * comparing speed with old code * still trying to not count things twice * started docs * more docs * working as it should now * progress on docs * adds is_shifted * docs almost done * added progress meter + finished docs * algebraic shifting: doc additions/fixes * Update docs/oscar_references.bib Co-authored-by: Max Horn <[email protected]> * bibtool formatting * removed tabs * added some docs to simplicial complex functions * Update experimental/LieAlgebras/src/serialization.jl Co-authored-by: Lars Göttgens <[email protected]> * Apply suggestions from code review Co-authored-by: Lars Göttgens <[email protected]> * create isomorphisms between weyl group and perm group * missing line from docs * fixes to bib refs * removes the use of perm for weyl group elements * fixed to docs * speed up from using transpose * using transpose isn't quite right * creates isomorphism only once when finding vertices + constructing graph * fixes some regressions * fix for empty grpah * suppres progress bar in doc strings * fixes for merge * remove permutation matrix function * suppress progress bar in docs * fix _set_to_zero * fixes white space * fixes for parallel and oscar tests * docs for kwargs * typo * shifting docs: ref to arXiv:2410.24044 added * bibtool on references * Apply suggestions from code review Co-authored-by: Lars Göttgens <[email protected]> * fixes for doctests * fix doctest * untabify` * moved symmetric shift * avoid caching polynomial rings * option to pass polynomial ring * fix compound matrix signature * cleanup white space * adds reference for rother diagram * minor adjustments * fixes regression in symmetric shifting and adds test * adds catch for partial shift graph * added ref to preprint in function docs * Update experimental/AlgebraicShifting/src/PartialShiftGraph.jl Co-authored-by: Lars Göttgens <[email protected]> * fixes doctest + only uses @showprogress * fix regression in test * adds uniform hypergraphs as input to partial_shift_graph_vertices * fix test * Update experimental/AlgebraicShifting/src/PartialShift.jl Co-authored-by: Max Horn <[email protected]> --------- Co-authored-by: Benjamin Lorenz <[email protected]> Co-authored-by: Michael Joswig <[email protected]> Co-authored-by: Max Horn <[email protected]> Co-authored-by: Lars Göttgens <[email protected]>
oscar-system · Dec 3, 2024 · 18af71a · 18af71a
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diff --git a/Project.toml b/Project.toml
@@ -16,6 +16,7 @@ Markdown = "d6f4376e-aef5-505a-96c1-9c027394607a"
 Nemo = "2edaba10-b0f1-5616-af89-8c11ac63239a"
 Pkg = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"
 Polymake = "d720cf60-89b5-51f5-aff5-213f193123e7"
+ProgressMeter = "92933f4c-e287-5a05-a399-4b506db050ca"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 RandomExtensions = "fb686558-2515-59ef-acaa-46db3789a887"
 Serialization = "9e88b42a-f829-5b0c-bbe9-9e923198166b"
@@ -37,6 +38,7 @@ Markdown = "1.6"
 Nemo = "0.47.1"
 Pkg = "1.6"
 Polymake = "0.11.20"
+ProgressMeter = "1.10.2"
 Random = "1.6"
 RandomExtensions = "0.4.3"
 Serialization = "1.6"

diff --git a/docs/oscar_references.bib b/docs/oscar_references.bib
@@ -683,6 +683,14 @@ @Book{Cut04
   doi           = {10.1090/gsm/063}
 }
 
+@Article{D-VJL24,
+  author        = {Della Vecchia, Antony and Joswig, Michael and Lenzen, Fabian},
+  title         = {Partial Algebraic Shifting},
+  year          = {2024},
+  eprint        = {2410.24044},
+  archiveprefix = {arXiv}
+}
+
 @InCollection{DE02,
   author        = {Decker, Wolfram and Eisenbud, David},
   title         = {Sheaf algorithms using the exterior algebra},
@@ -1736,6 +1744,24 @@ @Article{Kah10
   doi           = {10.1007/s10463-010-0290-9}
 }
 
+@InProceedings{Kal02,
+  author        = {Kalai, Gil},
+  title         = {Algebraic shifting},
+  editor        = {Hibi, Takayuki},
+  booktitle     = {Computational commutative algebra and combinatorics},
+  series        = {Advanced Studies in Pure Mathematics},
+  number        = {33},
+  publisher     = {Mathematical Society of Japan},
+  pages         = {121--163},
+  year          = {2002},
+  doi           = {10.2969/aspm/03310121},
+  venue         = {Osaka},
+  eventdate     = {1999},
+  shortseries   = {Adv. Stud. Pure Math.},
+  shortpublisher= {Math. Soc. Japan},
+  location      = {Tokyo}
+}
+
 @Article{Kem02,
   author        = {Kemper, Gregor},
   title         = {The calculation of radical ideals in positive characteristic},
@@ -1789,6 +1815,18 @@ @Book{Knu11
   year          = {2011}
 }
 
+@Book{Knu98,
+  author        = {Knuth, Donald Ervin},
+  title         = {Sorting and searching},
+  volume        = {3},
+  publisher     = {Addison-Wesley},
+  edition       = {2},
+  year          = {1998},
+  maintitle     = {The Art of Computer Programming},
+  location      = {Boston},
+  langid        = {english}
+}
+
 @Book{Kol13,
   author        = {Kollár, János},
   title         = {Singularities of the minimal model program},

diff --git a/experimental/AlgebraicShifting/README.md b/experimental/AlgebraicShifting/README.md
@@ -0,0 +1,10 @@
+# Algebraic Shifting
+
+## Aims
+
+This package grew out of an ongoing project of Antony Della Vecchia, Michael Joswig and Fabian Lenzen on [Partial Algebraic Shifting](https://arxiv.org/abs/2410.24044).
+Part of this project is the development of new methods for algebraic shifting; these are implemented here.
+
+## Status
+
+Full and partial shifting in the exterior algebra fully functional.
diff --git a/experimental/AlgebraicShifting/docs/doc.main b/experimental/AlgebraicShifting/docs/doc.main
@@ -0,0 +1,7 @@
+[
+  "Algebraic Shifting" => [
+    "introduction.md",
+    "exterior_shifting.md",
+    "partial_shift_graph.md"
+  ]
+]
diff --git a/experimental/AlgebraicShifting/docs/src/exterior_shifting.md b/experimental/AlgebraicShifting/docs/src/exterior_shifting.md
@@ -0,0 +1,24 @@
+```@meta
+CurrentModule = Oscar
+DocTestSetup = Oscar.doctestsetup()
+```
+
+# Exterior Shifting
+
+## Uniform Hypergraphs
+```@docs
+uniform_hypergraph
+```
+
+## Matrix Constructions
+
+```@docs
+generic_unipotent_matrix
+rothe_matrix
+compound_matrix
+```
+
+## Exterior (Partial) Shifting
+```@docs
+exterior_shift
+```
diff --git a/experimental/AlgebraicShifting/docs/src/introduction.md b/experimental/AlgebraicShifting/docs/src/introduction.md
@@ -0,0 +1,29 @@
+```@meta
+CurrentModule = Oscar
+DocTestSetup = Oscar.doctestsetup()
+```
+
+
+# Introduction
+
+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 general background on the subject we refer to [Kal02](@cite); for details concerning the implementation see [D-VJL24](@cite).
+
+
+## Status
+
+This part of OSCAR is in an experimental state; please see [Adding new projects to experimental](@ref) for what this means.
+
+## Contact
+
+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).
+
+Alternatively, you can [raise an issue on github](https://www.oscar-system.org/community/#how-to-report-issues).
diff --git a/experimental/AlgebraicShifting/docs/src/partial_shift_graph.md b/experimental/AlgebraicShifting/docs/src/partial_shift_graph.md
@@ -0,0 +1,12 @@
+```@meta
+CurrentModule = Oscar
+DocTestSetup = Oscar.doctestsetup()
+```
+
+# Partial Shift Graph
+
+```@docs
+partial_shift_graph_vertices
+partial_shift_graph
+contracted_partial_shift_graph
+```
diff --git a/experimental/AlgebraicShifting/src/AlgebraicShifting.jl b/experimental/AlgebraicShifting/src/AlgebraicShifting.jl
@@ -0,0 +1,29 @@
+include("UniformHypergraph.jl")
+include("PartialShift.jl")
+include("PartialShiftGraph.jl")
+
+export UniformHypergraph
+
+export compound_matrix
+export contracted_partial_shift_graph
+export exterior_shift
+export face_size
+export generic_unipotent_matrix
+export partial_shift_graph
+export partial_shift_graph_vertices
+export rothe_matrix
+export uniform_hypergraph
+
+function independent_columns(A::MatElem)
+  col_indices = Int[]
+  row_index = 1
+  row_bound = nrows(A)
+  for col_index in 1:ncols(A)
+    if !is_zero(A[row_index, col_index])
+      push!(col_indices, col_index)
+      row_index += 1
+    end
+    row_index > row_bound && break
+  end
+  return col_indices
+end