Vinbergs Algorithm (#4152)

* Add Vinberg's algorithm
---------

Co-authored-by: Elias Merkel <[email protected]>
Co-authored-by: Max Horn <[email protected]>
Co-authored-by: Lars Göttgens <[email protected]>

docs/doc.main

@@ -83,7 +83,8 @@
         "Hecke/manual/quad_forms/genusherm.md",
         "Hecke/manual/quad_forms/integer_lattices.md",
         "Hecke/manual/quad_forms/Zgenera.md",
-        "Hecke/manual/quad_forms/discriminant_group.md"
+        "Hecke/manual/quad_forms/discriminant_group.md",
+        "NumberTheory/vinberg.md",
      ],
   ],
 

docs/oscar_references.bib

@@ -2305,6 +2305,16 @@ @Article{Tra04
   doi           = {10.1016/j.cagd.2004.07.001}
 }
 
+@Article{Tur18,
+  author        = {Turkalj, I.},
+  title         = {Reflective Lorentzian lattices of signature (5, 1)},
+  journal       = {Journal of Algebra},
+  volume        = {513},
+  pages         = {516--544},
+  year          = {2018},
+  doi           = {10.1016/j.algebra.2018.06.013}
+}
+
 @Misc{VE22,
   author        = {Vaughan-Lee, Michael and Eick, Bettina},
   title         = {SglPPow, Database of groups of prime-power order for some prime-powers, Version 2.3},
@@ -2314,6 +2324,17 @@ @Misc{VE22
   url           = {https://gap-packages.github.io/sglppow/}
 }
 
+@InCollection{Vin75,
+  author        = {Vinberg, E. B.},
+  title         = {Some arithmetical discrete groups in Lobacevsky spaces},
+  booktitle     = {Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973)},
+  series        = {Tata Inst. Fundam. Res. Stud. Math.},
+  volume        = {No. 7},
+  publisher     = {Published for the Tata Institute of Fundamental Research, Bombay by Oxford University Press, Bombay},
+  pages         = {323--348},
+  year          = {1975}
+}
+
 @MastersThesis{Vol23,
   author        = {Volz, Sebastian},
   title         = {Design and implementation of efficient algorithms for operations on partitions of sets},

docs/src/NumberTheory/vinberg.md

@@ -0,0 +1,53 @@
+```@meta
+CurrentModule = Oscar
+DocTestSetup = Oscar.doctestsetup()
+```
+
+# Vinberg's algorithm
+
+ A *Lorentzian lattice* $L$ is an integral $\Z$-lattice of signature $(s_+, s_-)$ with $s_+=1$ and $s_->0$. 
+ A *root* of $r \in L$ is a primitive vector s.t. reflection in the hyperplane $r^\perp$ maps $L$ to itself.
+ Let $W(L)\leq O(L)$ be the Weyl group, that is the group generated by reflections in the root hyperplanes of $L$. 
+ We say that $L$ is *reflective* if $W(L)$ is of finite index in $O(L)$. 
+ For $x,y \in L$ we write $x.y:=\Phi(x,y)$ and $x^2:=\Phi(x,x)$ for the symmetric bilinear form in $L$.
+
+ See for example [Tur18](@cite) for the theory of Arithmetic Reflection Groups and Reflective Lorentzian Lattices.
+
+## Description 
+ Vinberg's algorithm constructs a fundamental polyhedron $P$ for a Lorentzian lattice $L$ by computing its *fundamental roots* $r$, i.e. the roots $r$ which are perpendicular to the faces of $P$ and which have inner product at least 0 with the elements of $P$.
+ Choose $v_0$ in $L$ primitive with $v_0^2 > 0$ as a point that $P$ should contain.
+
+ Let $Q$ be the Gram matrix of $L$ with respect to some basis. A vector $r$ is a fundamental root, if
+ - the vector $r$ is primitive,
+ - reflection by $r$ preserves the lattice, i.e. $\frac{2}{r^2} r Q$ is an integer matrix,
+ - the pair $(r, v_0)$ is positive oriented, i.e. $r. v_0 > 0$,
+ - the product $r. \~{r} \geq \ 0$ for all roots $\~{r}$ already found.
+ This implies that $r^2$ divides $2 i$ for $i$ being the level of $Q$, i.e. the last invariant of the Smith normal form of $Q$. 
+
+ $P$ can be constructed by solving $r.v_0 = n$ and $r^2 = k$ by increasing order of the value $\frac{n^2}{k}$ and $r$ satisfying the above conditions.
+
+ If $v_0$ lies on a root hyperplane, then $P$ is not uniquely determined.
+ In that case we need a direction vector $v_1$ which satisfies $\~{v}. v_1 \neq 0$ 
+ for all possible roots $\~{v}$ with $v_0. \~{v} = 0$  
+
+ With $v_0$ and $v_1$ fixed $P$ is uniquely determined for any choice of root lengths and maximal distance $v_0.r$.
+ We choose the first roots $r$ by increasing order of the value $\frac{\~{r}. v_1}{r^2}$ for all possible roots $\~{v}$ with $v_0. \~{v} = 0$.
+ For any other root length we continue as stated above.
+
+ For proofs of the statements above and further explanations see [Vin75](@cite).
+
+ ## Function
+
+```@docs
+vinberg_algorithm
+```
+
+## Contact
+
+Please direct questions about this part of OSCAR to the following people:
+* [Simon Brandhorst](https://www.math.uni-sb.de/ag/brandhorst/index.php?lang=en).
+* [Stevell Muller](https://www.math.uni-sb.de/ag/brandhorst/index.php?option=com_content&view=article&id=30:muller&catid=10&lang=de&Itemid=104),
+
+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).