ref.bib

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@Book{Humphreys90,
  title      = {Reflection Groups and Coxeter Groups},
  publisher  = {Cambridge University Press},
  author     = {Humphreys, James E.},
  series     = {Cambridge Studies in Advanced Mathematics},
  collection = {Cambridge Studies in Advanced Mathematics},
  date       = {1990},
  doi        = {10.1017/CBO9780511623646},
  place      = {Cambridge},
}

@Article{Maxwell82,
  author       = {George Maxwell},
  title        = {Sphere packings and hyperbolic reflection groups},
  volume       = {79},
  number       = {1},
  pages        = {78--97},
  issn         = {0021-8693},
  date         = {1982},
  doi          = {10.1016/0021-8693(82)90318-0},
  journaltitle = {Journal of Algebra},
  url          = {https://www.sciencedirect.com/science/article/pii/0021869382903180},
}

@Article{Maxwell89,
  author       = {George Maxwell},
  title        = {Wythoff's construction for Coxeter groups},
  volume       = {123},
  number       = {2},
  pages        = {351--377},
  issn         = {0021-8693},
  date         = {1989},
  doi          = {10.1016/0021-8693(89)90051-3},
  journaltitle = {Journal of Algebra},
  url          = {https://www.sciencedirect.com/science/article/pii/0021869389900513},
}

@Article{HaoChen,
  author       = {Hao Chen and Jean-Philippe Labb{\'{e}}},
  title        = {Lorentzian Coxeter systems and Boyd-Maxwell ball packings},
  abstract     = {In the recent study of infinite root systems, fractal patterns of ball packings were observed while visualizing roots in affine space. In this paper, we show that the observed fractals are exactly the ball packings described by Boyd and Maxwell. This correspondence is a corollary of a more fundamental result: Given a geometric representation of a Coxeter group in a Lorentz space, the set of limit directions of weights equals the set of limit roots. Additionally, we use Coxeter complexes to describe tangency graphs of the corresponding Boyd--Maxwell ball packings. Finally, we enumerate all the Coxeter systems that generate Boyd-Maxwell ball packings.},
  date         = {2013-10-31},
  doi          = {10.1007/s10711-014-0004-1},
  eprint       = {1310.8608v6},
  eprintclass  = {math.GR},
  eprinttype   = {arXiv},
  file         = {online:http\:/arxiv.org/pdf/1310.8608v6:PDF},
  journaltitle = {Geometriae Dedicata: Volume 174, Issue 1 (2015), Page 43-73},
  keywords     = {math.GR, math.CO, math.MG, Primary 52C17, 20F55, Secondary 05C30},
}

@Book{Humphreys1973,
  title     = {{Introduction to Lie Algebras and Representation Theory}},
  publisher = {Springer},
  author    = {Humphreys, James E.},
  isbn      = {0387900535},
  date      = {1973},
  pages     = {173},
}

@Book{Kac1990,
  title     = {Infinite-Dimensional Lie Algebras},
  publisher = {Cambridge University Press},
  author    = {Kac, Victor G.},
  edition   = {3},
  date      = {1990},
  place     = {Cambridge},
}

@Book{curtis-reiner,
  title     = {Representation theory of finite groups and associative algebras},
  publisher = {Interscience Publishers, a division of John Wiley \& Sons},
  author    = {Curtis, C.\thinspace{}W. and Reiner, I.},
  note      = {Pure and Applied Mathematics, Vol. XI},
  date      = {1962},
  location  = {New York},
  pages     = {xiv+685},
}

@Book{Lam01,
  title     = {A first course in noncommutative rings},
  publisher = {Springer-Verlag, New York},
  author    = {Lam, T. Y.},
  volume    = {131},
  series    = {Graduate Texts in Mathematics},
  edition   = {Second edition},
  isbn      = {0-387-95183-0},
  date      = {2001},
  pages     = {xx+385},
}

@Book{herstein-rings,
  title     = {Noncommutative rings},
  publisher = {Mathematical Association of America, Washington, DC},
  author    = {Herstein, I. N.},
  volume    = {15},
  series    = {Carus Mathematical Monographs},
  isbn      = {0-88385-015-X},
  note      = {Reprint of the 1968 original, With an afterword by Lance W. Small},
  date      = {1994},
  pages     = {xii+202},
}

@Article{Mills1983,
  author       = {W.H Mills and David P Robbins and Howard Rumsey},
  title        = {Alternating sign matrices and descending plane partitions},
  volume       = {34},
  number       = {3},
  pages        = {340--359},
  issn         = {0097-3165},
  date         = {1983},
  journaltitle = {Journal of Combinatorial Theory, Series A},
}

@Book{Bressoud1999,
  title      = {Proofs and Confirmations: The Story of the Alternating-Sign Matrix Conjecture},
  publisher  = {Cambridge University Press},
  author     = {Bressoud, David M.},
  series     = {Spectrum},
  collection = {Spectrum},
  date       = {1999},
  place      = {Cambridge},
}

@Book{thebook,
  title     = {Proofs from THE BOOK},
  publisher = {Springer Publishing Company, Incorporated},
  author    = {Aigner, Martin and Ziegler, Gnter M.},
  edition   = {6\textsuperscript{th}},
  isbn      = {3662572648},
  abstract  = {This revised and enlarged sixth edition of Proofs from THE BOOKfeatures an entirely new chapter on Van der Waerdens permanent conjecture, as well as additional, highly original and delightful proofs in other chapters. From the citation on the occasion of the 2018 "Steele Prize for Mathematical Exposition" It is almost impossible to write a mathematics book that can be read and enjoyed by peopleof all levels and backgrounds, yet Aigner and Ziegler accomplish this feat of exposition with virtuoso style. [] This book does an invaluable service to mathematics, by illustrating for non-mathematicians what it is that mathematicians mean when they speak about beauty. From the Reviews"... Inside PFTB (Proofs from The Book) is indeed a glimpse of mathematical heaven, where clever insights and beautiful ideas combine in astonishing and glorious ways. There is vast wealth within its pages, one gem after another. ... Aigner and Ziegler... write: "... all we offer is the examples that we have selected, hoping that our readers will share our enthusiasm about brilliant ideas, clever insights and wonderful observations." I do. ... "Notices of the AMS, August 1999"... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures and beautiful drawings ... It is a pleasure to read as well: the style is clear and entertaining, the level is close to elementary, the necessary background is given separately and the proofs are brilliant. ..."LMS Newsletter, January 1999" Martin Aigner and Gnter Ziegler succeeded admirably in putting together a broad collection of theorems and their proofs that would undoubtedly be in the Book of Erds. The theorems are so fundamental, their proofs so elegant and the remaining open questions so intriguing that every mathematician, regardless of speciality, can benefit from reading this book. ... " SIGACT News, December 2011},
  date      = {2018},
}

@Book{aigner07,
  title     = {A Course in Enumeration},
  publisher = {Springer Berlin Heidelberg},
  author    = {Aigner, M.},
  series    = {Graduate Texts in Mathematics},
  isbn      = {9783540390350},
  date      = {2007},
  lccn      = {2007928344},
}

@Book{Williams1991,
  title     = {Probability with Martingales},
  publisher = {Cambridge University Press},
  author    = {Williams, David},
  date      = {1991},
  place     = {Cambridge},
}

@Book{Durrett2019,
  title      = {Probability: Theory and Examples},
  publisher  = {Cambridge University Press},
  author     = {Durrett, Rick},
  series     = {Cambridge Series in Statistical and Probabilistic Mathematics},
  edition    = {5},
  collection = {Cambridge Series in Statistical and Probabilistic Mathematics},
  date       = {2019},
  place      = {Cambridge},
}

@Book{Donoghue2014,
  title     = {Distributions and Fourier Transforms},
  publisher = {Elsevier Science},
  author    = {Donoghue, W.F.},
  series    = {ISSN},
  isbn      = {9780080873442},
  date      = {2014},
  url       = {https://books.google.com/books?id=P30Y7daiGvQC},
}

@Article{Li1980,
  author       = {Shuo-Yen Robert Li},
  title        = {{A Martingale Approach to the Study of Occurrence of Sequence Patterns in Repeated Experiments}},
  volume       = {8},
  number       = {6},
  pages        = {1171--1176},
  date         = {1980},
  doi          = {10.1214/aop/1176994578},
  journaltitle = {The Annals of Probability},
  keywords     = {Leading number, martingale, stopping time, waiting time},
  publisher    = {Institute of Mathematical Statistics},
}

@Book{RiskNeutralValuation,
  title     = {Risk-neutral valuation : pricing and hedging of financial derivatives / N.H. Bingham and R. Kiesel.},
  publisher = {Springer},
  author    = {Bingham, N. H. and Kiesel, R{\"{u}}diger},
  series    = {Springer finance},
  edition   = {Second edition.},
  isbn      = {9781447138563},
  booktitle = {Risk-neutral valuation : pricing and hedging of financial derivatives},
  date      = {2004},
  keywords  = {Investments -- Mathematical models},
  language  = {eng},
  location  = {London},
}

@Book{Macdonald2008,
  title     = {Symmetric functions and hall polynomials},
  publisher = {Oxford science pub.},
  author    = {Macdonald, I. G.},
  edition   = {2\textsuperscript{nd} ed.},
  date      = {2008},
}

@Book{Needham1997,
  title      = {Visual complex analysis},
  publisher  = {The Clarendon Press, Oxford University Press, New York},
  author     = {Needham, Tristan},
  isbn       = {0-19-853447-7},
  date       = {1997},
  mrclass    = {30-01},
  mrnumber   = {1446490},
  mrreviewer = {D. H. Armitage},
  pages      = {xxiv+592},
}

@Book{Jacobson_alg,
  title      = {Basic algebra. {II}},
  publisher  = {W. H. Freeman and Co.},
  author     = {Jacobson, Nathan},
  isbn       = {0-7167-1079-X},
  date       = {1980},
  location   = {San Francisco, Calif.},
  mrclass    = {00A05},
  mrnumber   = {MR571884 (81g:00001)},
  mrreviewer = {M. F. Smiley},
  pages      = {xix+666},
}

@TechReport{Howlett-note,
  author = {Robert B. Howlett},
  title  = {Introduction to Coxeter groups},
  year   = {1996},
  url    = {https://www.maths.usyd.edu.au/u/ResearchReports/Algebra/How/1997-6.html},
}

@Article{CasselmanCoxeterElement,
  author = {Bill Casselman},
  title  = {Coxeter elements in finite Coxeter groups},
  year   = {2017},
  volume = {Essays on Coxeter groups},
  url    = {https://personal.math.ubc.ca/~cass/research/pdf/Element.pdf},
}

@Book{Bea95,
  title     = {The geometry of discrete groups},
  publisher = {Springer-Verlag, New York},
  year      = {1995},
  author    = {Beardon, Alan F.},
  volume    = {91},
  series    = {Graduate Texts in Mathematics},
  isbn      = {0-387-90788-2},
  note      = {Corrected reprint of the 1983 original},
  mrclass   = {22E40 (11F06 20H15 30F35 57N10)},
  mrnumber  = {1393195},
  pages     = {xii+337},
}

@book{indra,
author = {Mumford, David and Series, Caroline and Wright, David J.},
title = {Indra's Pearls: An Atlas of Kleinian Groups},
year = {2002},
isbn = {0521352533},
publisher = {Cambridge University Press},
}

@Book{palka1991,
  title     = {An Introduction to Complex Function Theory},
  publisher = {World Publishing Corporation},
  year      = {1991},
  author    = {Palka, B.P.},
  series    = {An Introduction to Complex Function Theory},
  isbn      = {9780387974279},
  lccn      = {90047375},
}

@Article{dominoshuffling,
  author       = {Noam Elkies and Greg Kuperberg and Michael Larsen and James Propp},
  title        = {Alternating sign matrices and domino tilings},
  abstract     = {We introduce a family of planar regions, called Aztec diamonds, and study the ways in which these regions can be tiled by dominoes. Our main result is a generating function that not only gives the number of domino tilings of the Aztec diamond of order $n$ but also provides information about the orientation of the dominoes (vertical versus horizontal) and the accessibility of one tiling from another by means of local modifications. Several proofs of the formula are given. The problem turns out to have connections with the alternating sign matrices of Mills, Robbins, and Rumsey, as well as the square ice model studied by Lieb.},
  date         = {1991-06-01},
  eprint       = {math/9201305v1},
  eprintclass  = {math.CO},
  eprinttype   = {arXiv},
  file         = {online:http\://arxiv.org/pdf/math/9201305v1:PDF},
  journaltitle = {J. Algebraic Combin. 1 (1992), no. 2, 111--132; J. Algebraic Combin. 1 (1992), no. 3, 219-234},
  keywords     = {math.CO},
}

@Article{spidermove,
  author       = {James Propp},
  title        = {Generalized domino-shuffling},
  abstract     = {The problem of counting tilings of a plane region using specified tiles can often be recast as the problem of counting (perfect) matchings of some subgraph of an Aztec diamond graph A_n, or more generally calculating the sum of the weights of all the matchings, where the weight of a matching is equal to the product of the (pre-assigned) weights of the constituent edges (assumed to be non-negative). This article presents efficient algorithms that work in this context to solve three problems: finding the sum of the weights of the matchings of a weighted Aztec diamond graph A_n; computing the probability that a randomly-chosen matching of A_n will include a particular edge (where the probability of a matching is proportional to its weight); and generating a matching of A_n at random. The first of these algorithms is equivalent to a special case of Mihai Ciucu's cellular complementation algorithm and can be used to solve many of the same problems. The second of the three algorithms is a generalization of not-yet-published work of Alexandru Ionescu, and can be employed to prove an identity governing a three-variable generating function whose coefficients are all the edge-inclusion probabilities; this formula has been used as the basis for asymptotic formulas for these probabilities, but a proof of the generating function identity has not hitherto been published. The third of the three algorithms is a generalization of the domino-shuffling algorithm described by Elkies, Kuperberg, Larsen and Propp; it enables one to generate random ``diabolo-tilings of fortresses'' and thereby to make intriguing inferences about their asymptotic behavior.},
  date         = {2001-11-02},
  eprint       = {math/0111034v2},
  eprintclass  = {math.CO},
  eprinttype   = {arXiv},
  file         = {online:http\://arxiv.org/pdf/math/0111034v2:PDF},
  journaltitle = {Theoret. Comput. Sci. 303, no. 2-3, 267--301, Tilings of the plane (2003).},
  keywords     = {math.CO, 05C70; 05C85},
}

@article{Conway1990,
title = {Tiling with polyominoes and combinatorial group theory},
journal = {Journal of Combinatorial Theory, Series A},
volume = {53},
number = {2},
pages = {183-208},
year = {1990},
issn = {0097-3165},
doi = {https://doi.org/10.1016/0097-3165(90)90057-4},
url = {https://www.sciencedirect.com/science/article/pii/0097316590900574},
author = {J.H Conway and J.C Lagarias},
abstract = {When can a given finite region consisting of cells in a regular lattice (triangular, square, or hexagonal) in R2 be perfectly tiled by tiles drawn from a finite set of tile shapes? This paper gives necessary conditions for the existence of such tilings using boundary invariants, which are combinatorial group-theoretic invariants associated to the boundaries of the tile shapes and the regions to be tiled. Boundary invariants are used to solve problems concerning the tiling of triangular-shaped regions of hexagons in the hexagonal lattice with certain tiles consisting of three hexagons. Boundary invariants give stronger conditions for nonexistence of tilings than those obtainable by weighting or coloring arguments. This is shown by considering whether or not a region has a signed tiling, which is a placement of tiles assigned weights 1 or −1, such that all cells in the region are covered with total weight 1 and all cells outside with total weight 0. Any coloring (or weighting) argument that proves nonexistence of a tiling of a region also proves nonexistence of any signed tiling of the region as well. A partial converse holds: if a simply connected region has no signed tiling by simply connected tiles, then there is a generalized coloring argument proving that no signed tiling exists. There exist regions possessing a signed tiling which can be shown to have no perfect tiling using boundary invariants.}
}

@article{Thurston89,
	author = {W. P. Thurston},
	title = {Groups, tilings and finite state automata},
	year = {1989},
	journal = {AMS Colloquium Lectures},
	lockkey = {Y}
}

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