Experimental: Complex reflection groups

docs/oscar_references.bib

@@ -104,6 +104,20 @@ @Article{BCL21
   numpages      = {8}
 }
 
+@Article{BCP97,
+  author        = {Bosma, Wieb and Cannon, John and Playoust, Catherine},
+  title         = {The Magma algebra system. I. The user language},
+  mrnumber      = {1484478},
+  journal       = {J. Symbolic Comput.},
+  fjournal      = {Journal of Symbolic Computation},
+  volume        = {24},
+  number        = {3-4},
+  note          = {Computational algebra and number theory (London, 1993)},
+  pages         = {235--265},
+  year          = {1997},
+  doi           = {10.1006/jsco.1996.0125}
+}
+
 @Article{BDEPS04,
   author        = {Berry, Neil and Dubickas, Artūras and Elkies, Noam D. and Poonen, Bjorn and Smyth, Chris},
   title         = {The conjugate dimension of algebraic numbers},
@@ -334,6 +348,20 @@ @Article{BS09
   doi           = {10.1016/j.aim.2009.06.009}
 }
 
+@Article{Bea00,
+  author        = {Beauville, Arnaud},
+  title         = {Symplectic singularities},
+  mrnumber      = {1738060},
+  journal       = {Invent. Math.},
+  fjournal      = {Inventiones Mathematicae},
+  volume        = {139},
+  number        = {3},
+  pages         = {541--549},
+  year          = {2000},
+  doi           = {10.1007/s002229900043},
+  groups        = {Symplectic singularities}
+}
+
 @Book{Ben93,
   author        = {Benson, David J.},
   title         = {Polynomial invariants of finite groups},
@@ -549,6 +577,19 @@ @Book{Coh00
   doi           = {10.1007/978-1-4419-8489-0}
 }
 
+@Article{Coh80,
+  author        = {Cohen, Arjeh M.},
+  title         = {Finite quaternionic reflection groups},
+  mrnumber      = {579063},
+  journal       = {J. Algebra},
+  fjournal      = {Journal of Algebra},
+  volume        = {64},
+  number        = {2},
+  pages         = {293--324},
+  year          = {1980},
+  doi           = {10.1016/0021-8693(80)90148-9}
+}
+
 @Book{Coh93,
   author        = {Cohen, Henri},
   title         = {A course in computational algebraic number theory},
@@ -781,6 +822,19 @@ @Article{EFS03
   zbmath        = {1963988}
 }
 
+@Article{EG02,
+  author        = {Etingof, Pavel and Ginzburg, Victor},
+  title         = {Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism},
+  mrnumber      = {1881922},
+  journal       = {Invent. Math.},
+  fjournal      = {Inventiones Mathematicae},
+  volume        = {147},
+  number        = {2},
+  pages         = {243--348},
+  year          = {2002},
+  doi           = {10.1007/s002220100171}
+}
+
 @Book{EH16,
   author        = {Eisenbud, David and Harris, Joe},
   title         = {3264 and all that. {A} second course in algebraic geometry},
@@ -1624,6 +1678,15 @@ @Article{LS13
   reportnumber  = {KCL-MTH-12-14}
 }
 
+@Book{LT09,
+  author        = {Lehrer, Gustav I. and Taylor, Donald E.},
+  title         = {Unitary reflection groups},
+  series        = {Australian Mathematical Society Lecture Series},
+  volume        = {20},
+  publisher     = {Cambridge University Press, Cambridge},
+  year          = {2009}
+}
+
 @PhDThesis{Lev05,
   author        = {Viktor Levandovskyy},
   title         = {Non-commutative Computer Algebra for polynomial algebras: Gröbner bases, applications and
@@ -1657,6 +1720,19 @@ @Book{Loo84
   year          = {1984}
 }
 
+@Article{MM10,
+  author        = {Marin, I. and Michel, J.},
+  title         = {Automorphisms of complex reflection groups},
+  mrnumber      = {2746138},
+  journal       = {Represent. Theory},
+  fjournal      = {Representation Theory. An Electronic Journal of the American Mathematical Society},
+  volume        = {14},
+  pages         = {747--788},
+  year          = {2010},
+  doi           = {10.1090/S1088-4165-2010-00380-5},
+  groups        = {Reflection groups}
+}
+
 @Article{MR20,
   author        = {Markwig, Thomas and Ren, Yue},
   title         = {Computing tropical varieties over fields with valuation},
@@ -1718,6 +1794,29 @@ @Book{Mar18
   doi           = {10.1007/978-3-319-90233-3}
 }
 
+@Article{Mer12,
+  author        = {Merca, M},
+  title         = {Fast algorithm for generating ascending compositions},
+  journal       = {J. Math. Model. Algorithms},
+  volume        = {11},
+  number        = {1},
+  pages         = {89--104},
+  year          = {2012},
+  doi           = {10.1007/s10852-011-9168-y}
+}
+
+@Article{Mic15,
+  author        = {Michel, Jean},
+  title         = {The development version of the \texttt{CHEVIE} package of \texttt{GAP}3},
+  mrnumber      = {3343221},
+  journal       = {J. Algebra},
+  fjournal      = {Journal of Algebra},
+  volume        = {435},
+  pages         = {308--336},
+  year          = {2015},
+  doi           = {10.1016/j.jalgebra.2015.03.031}
+}
+
 @Article{Nik79,
   author        = {Nikulin, V. V.},
   title         = {Integer symmetric bilinear forms and some of their geometric applications},
@@ -1952,6 +2051,18 @@ @Article{SS12
   doi           = {10.1016/j.jcta.2011.12.005}
 }
 
+@Article{ST54,
+  author        = {Shephard, G. C. and Todd, J. A.},
+  title         = {Finite unitary reflection groups},
+  mrnumber      = {59914},
+  journal       = {Canad. J. Math.},
+  fjournal      = {Canadian Journal of Mathematics. Journal Canadien de Mathématiques},
+  volume        = {6},
+  pages         = {274--304},
+  year          = {1954},
+  doi           = {10.4153/cjm-1954-028-3}
+}
+
 @Article{SV-D-V87,
   author        = {Sommese, Andrew John and Van de Ven, A.},
   title         = {On the adjunction mapping},

experimental/ComplexReflectionGroups/README.md

@@ -0,0 +1,37 @@
+# Complex Reflection Groups
+
+## Aim 
+The aim is to implement [complex reflection groups](https://en.wikipedia.org/wiki/Complex_reflection_group) along with data that is known or can be computed.
+
+By [Ulrich Thiel](https://ulthiel.com/math), 2023.
+
+## Example
+
+```
+julia> W = ComplexReflectionGroupType([ (4,) , (10,5,3)] )
+Complex reflection group type G4 x G(10,5,3)
+
+julia> order(W)
+28800
+```
+
+## Status
+
+- [X] Structure ```ComplexReflectionGroupType``` for type (equivalence up to conjugacy) of complex reflection group following Shephard–Todd notation.
+
+- [X] Basic data for complex reflection group types: ```order```, ```is_imprimitive```, ```num_reflections```, ```is_well_generated```, ```is_pseudo_real```, ```is_real```, ```is_rational```, ```coxeter_number```, ```is_spetsial```, ```num_reflections```, ```num_hyperplanes```, ```num_reflection_classes``` 
+
+- [X] Explicit models via ```complex_reflection_group``` (Magma, Lehrer–Taylor, and CHEVIE)
+
+- [ ] "Reflection library" (list of reflections, conjugacy classes of reflections, hyperplane orbits etc.) 
+
+- [ ] Conjugacy classes and character tables (main problem will be to have compatibility with CHEVIE; it may be easiest/best to [compute](https://webusers.imj-prg.fr/~jean.michel/gap3/htm/chap087.htm) the labeling)
+
+- [ ] Explicit models of the irreducible representations (same problem as above + problem of how to store the data)
+
+- [ ] Because it somehow fits in here as well: symplectic reflection groups
+
+- [ ] With the same explanation as above: Drinfeld–Hecke algebras (which includes symplectic reflection algebras and rational Cherednik algebras)
+
+ - [ ] Identification of the type of a complex reflection group (challenging)
+

experimental/ComplexReflectionGroups/docs/doc.main

@@ -0,0 +1,6 @@
+[
+   "Complex Reflection Groups" => [
+      "introduction.md",
+      "complex_reflection_groups.md",
+   ],
+]