B1. Anders Bjorner and Francesco Brenti, Combinatorics of Coxeter Groups
      Springer, New York, 2005.
      In Chapter 4, the authors develop some aspects of so-called geometric representations of Coxeter groups using as their starting point a possibly asymmetric matrix meeting the requirements of the amplitude matrix of an SC graph, as discussed in Units 1 and 3 of the talks.

B2. N. Bourbaki, Elements of Mathematics: Lie Groups and Lie Algebras, Chapters 4--6
      Springer-Verlag, Berlin - Heidelberg - New York, 2002.
      Originally published as Groupes et Algebres de Lie, Hermann, Paris, 1968. This English language translation is by Andrew Pressley.
      I understand that much of this standard/classical treatment of the foundations of Coxeter groups was authored by Jacques Tits.

B3. Michael W. Davis, The Geometry and Topology of Coxeter Groups
      Princeton University Press, Princeton, New Jersey, 2008.
      A pdf draft version of the entire text can be found here.

B4. James E. Humphreys, Reflection Groups and Coxeter Groups
      Cambridge University Press, Cambridge, 1990.
      My talks on asymmetric geometric representations of Coxeter groups (Unit 3) basically extend much of Chapter 5 of this standard reference.

B5. Victor Kac, Infinite-dimensional Lie Algebras, 3rd edition
      Cambridge University Press, Cambridge, 1990.
      This book is the standard reference on the subject of Kac--Moody algebras and their associated Weyl groups. For comparison, you might look at the Kumar reference.

B6. Shrawan Kumar, Kac--Moody Groups, Their Flag Varieties and Representation Theory
      Birkhauser Boston Inc, Boston, MA, 2002.
      The first chapter exposition on Kac--Moody algebras and their associated Weyl groups connects to our discussion of asymmetric geometric representations of Coxeter groups and associated objects such as the Tits cone (see Unit 3 of the talks).

P1. N. Alon, I. Krasikov, and Y. Peres, "Reflection sequences"
      American Mathematical Monthly  96 (1989), 820-823.
      The authors treat a special case of the numbers game (cyclic graphs).

P2. M. Benoumhani, "A sequence of binomial coefficients related to Lucas and Fibonacci numbers"
      Journal of Integer Sequences  6 (2003), Article 03.2.1, 10 pp. (Electronic, available here.)
      Results from this paper are used in Unit 1.

P3. A. Bjorner, "On a combinatorial game of S. Mozes"
      Preprint, 1988.
      Bjorner was Kimmo Eriksson's advisor and helped inspire Kimmo's interest in the numbers game.

P4. B. Brink and R. Howlett, "A finiteness property and an automatic structure for Coxeter groups"
      Mathematics Annalen  296  (1993), 179-190.
      Brink and Howlett prove a finiteness results about root systems for the standard geometric representations of Coxeter groups. A consequence is that Coxeter groups have an automatic structure. If a group is automatic, one consequence is that the Word Problem can be solved in quadratic time. It is not clear whether their finitness result extends to root systems for asymmetric geometric representations of Coxeter groups as presented in Unit 3 of the talks.

P5. P.-E. Caprace, "Conjugacy of 2-spherical subgroups of Coxeter groups and parallel walls"
      Algebraic and Geometric Topology  6 (2006), 1987-2029. Available here.
      This paper uses connections between the root system/geometric representation viewpoint for Coxeter groups (cf. Unit 3 of the talks) and Coxeter/Davis complexes (as presented earlier in the seminar by Dr. Schroeder) in part to extend the finiteness result of [P4] above.

P6. Bill Casselman, "Computation in Coxeter groups II: Constructing minimal roots"
      Journal of Representation Theory  12 (2008), 260--293. (electronic)
      Preprint available here, 31 pp.

P7. Bill Casselman, Lecture notes on Coxeter groups.
      CRM Winter School on Coxeter Groups, 2002, notes archived here.
      In Part II of these notes, Cassellman proves Tits' Theorem on the Word Problem for Coxeter groups (cf. Unit 2 of the talks).

P8. M. W. Davis and M. D. Shapiro, "Coxeter groups are automatic"
      Ohio State Mathematical Research Institute preprint. Available here, 17 pp.
      Davis and Shapiro put forward a result they call the Parallel Wall Theorem, from which they deduce that Coxeter groups are automatic. However, their proof of the Parallel Wall Theorem incomplete. This issue was resolved by Brink and Howlett in [P4] above: their Theorem 2.8 concerning "minimal roots" is equivalent to the Parallel Wall Theorem.

P9. V. V. Deodhar, "On the root system of a Coxeter group"
      Communications in Algebra  10 (1982), 611--630.

P10. R. G. Donnelly, "Eriksson's numbers game and finite Coxeter groups"
      European Journal of Combinatorics  29 (2008), 1764-1781.
      PDF Preprint, 18 pp.

P11. R. G. Donnelly, "Eriksson's numbers game on certain edge-weighted three-node cyclic graphs"
      PDF Preprint, 5 pp.   Also available on the arXiv as 0708.0880, 5 pp.
      This manuscript provides some supporting details for paper [P10] above.

P12. R. G. Donnelly, "Root systems for asymmetric geometric representations of Coxeter groups"
      Accepted and to appear in Communications in Algebra.
      PDF Preprint, 15 pp.   Also available on the arXiv as 0707.3310, 15 pp.

P13. R. G. Donnelly, "Convergent and divergent numbers games for certain collections of edge-weighted graphs"
      PDF Preprint, 24 pp.   Also available on the arXiv as 0810.5362, 24 pp.
      This manuscript provides some supporting details for paper [P14] below.

P14. R. G. Donnelly and K. Eriksson, "The numbers game and Dynkin diagram classification results"
      PDF Preprint, 20 pp.   Also available on the arXiv as 0810.5371, 20 pp.

P15. K. Eriksson, "Strongly Convergent Games and Coxeter Groups"
      Ph.D. thesis, KTH, Stockholm, 1993.

P16. K. Eriksson, "The numbers game and Coxeter groups"
      Discrete Mathematics  139 (1995), 155--166.

P17. K. Eriksson, "Strong convergence and a game of numbers"
      European Journal of Combinatorics  17 (1996), 379--390.

P18. P. E. Gunnells, "Cells in Coxeter groups"
      Notices of American Mathematical Society  53 (2006), 528--535.
      This paper givs exposition of some uses of Coxeter groups and some current directions of research, available here.

P19. R. B. Howlett, "Introduction to Coxeter groups"
      Preprint, 1997. Available online as Algebra Research Report 1997-06 from the Mathematics and Statistics Department at the University of Sydney, http://www.maths.usyd.edu.au/res/Algebra/How/1997-6.html.

P20. R. B. Howlett, P. J. Rowley, and D. E. Taylor, "On outer automorphism groups of Coxeter groups"
      Manuscripta Mathematica  93 (1997), 499--513.

P21. G. Lusztig, "Some examples of square-integrable representations of semisimple p-adic groups"
      Transactions of the AMS  277 (1983), 623--653.
      In Section 3 of this paper, Lusztig defines some representations of Hecke algebras in a context which allows for an asymmetric matrix as in Unit 3 of the talks.

P22. S. Mozes, "Reflection processes on graphs and Weyl groups"
      Journal of Combinatorial Theory Series A  53 (1990), 128--142.
      The origins of the numbers game are often traced to this paper by Mozes, who was apparently inspired by a problem from the 1986 International Mathematics Olympiad. A version of the game had also been used by Proctor in a Lie theoretic context to compute Weyl group orbits of weights. See [P23] below.

P23. R. A. Proctor, "Bruhat lattices, planepartition generating functions, and minuscule representations"
      European Journal of Combinatorics  5 (1984), 331-350.

P24. R. A. Proctor, "Minuscule elements of Weyl groups, the numbers game, and d-complete posets"
      Journal of Algebra  213 (1999), 272-303.

P25. J. R. Stembridge, "On the fully commutative elements of Coxeter groups"
      Journal of Algebraic Combinatorics  5 (1996), 353--385.

P26. E. B. Vinberg, "Discrete linear groups generated by reflections"
      Math. USSR-Izvestiya  5 (1971), 1083--1119.
      In this paper Vinberg introduces asymmetric geometric representations of a Coxeter group and studies the behavior of a certain convex polyhedral cone under the action of the group.

P27. E. Wegert and C. Reiher, "Relaxation procedures on graphs"
      Discrete Applied Mathematics  157 (2009), 2207--2216.
      This paper gives some good historical and new perspectives on the numbers game.

P28. N. J. Wildberger, "The mutation game, Coxeter graphs, and partially ordered multisets"
      Preprint.
      This preprint gives Norman's take on the numbers game as well as some applications to a poset theoretic approach to Lie representation theory.