Constructions of Regular Polytopes


A. Key Topics:
                       
* The regular polytopes correspond naturally to string C-groups (each with its specified list of generators).
    Hence such groups will be our primary focus.

* reflection groups (real and unitary); Coxeter groups

* orthogonal geometries, particularly over finite fields

* crystallographic groups and their modular images --> lots of finite polytopes

* the intersection condition in linear groups --> various criteria

* crystallographic groups extended through the Golden Mean;  the natural constructions
   of the 11-cell and 57-cell

                                          And, if time permits, a briefer look at

* non-constructive constructions; residually finite groups; Malcev's Theorem; applications
  to regular polytopes

* semiregular polytopes; amalgamation of compatible string C-groups.
                       
                                    

B. Some Notes for the Course
            1. The notes are on file. Please check for updates: here


And here are some assigned problems (as of Nov.4): here


2. Here are summaries of the lectures:

Lecture 1


Lecture 2

Group Diagrams

The 3-cube and generators for its group B_3

Lecture 3


Proof that the standard representation is faithful


Lecture 4


Invariant lattices for the cube group B_3


Lecture 5 actually finishes off Lecture 4


To go with that, here is a brief look at orthogonal geometries and their groups


Lecture 6


C. Main References.

1. Some papers *  with Egon Schulte:  Reflection groups an polytopes over finite fields - I,II,II,
                                                                     Advances in Applied Mathematics:  33(2004) 290-217; 38(2007) 327-356; 41 (2008) 76-94.
                                                                     Modular reduction in abstract polytopes, Canadian Mathematical Bulletin: 52(2009) 435-450.
                                                                     Semiregular Polytopes and Amalgamated C-groups, under review ( a prelimary
                                                                                                                     version is  here)
                              * with Daniel Pellicer and Gordon Williams: The Tomotope, under review (ditto here)

2. P.  McMullen and E. Schulte, Abstract Regular Polytopes, Cambridge University Press,  2002.

3. E. Artin, Geometric Algebra, Interscience, 1957.

4. H. S. M. Coxeter, Regular Complex Polytopes,  Cambridge University Press, 1991.

5. J. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, 1990.

D. Other Stuff.


1.  An extended Gap session for learning the language.
                               Text files for copying into a Gap seesion.

2.  On C-groups

3.  On  presentations and coset enumeration

4.  Some notes on rings

5. Some notes on amalgamation of groups (for example, free products).