June 19, 2018

The Coxeter Legacy: Reflections and Projections
May 12-16, 2004


Alexandre Borovik, UMIST
Coxeter matroids
The talk will describe a variety of ways in which the concept of a Coxeter group (in its most ubiquitous manifestation, the symmetric group) emerges, again and again, in matroid theory and in combinatorics in general:

* Gale's maximality principle which leads to the Bruhatorder on the symmetric group;
* Jordan--Holder permutation which measures distance between two maximal chains in a semimodular lattice and which happens to be closely related to Tits' axioms for buildings;
* matroid polytopes and associated reflection groups;
* Gaussian elimination procedure, BN-pairs and their Weyl groups.

These observations suggest a very natural generalisation of matroids; the new objects are called Coxeter matroids and are related to other Coxeter groups in the same way as(classical) matroids are related to the symmetric group.
The language of Coxeter groups and Coxeter matroids is very flexible and versatile and can be applied to a variety of problems in geometry and topology. We shall briefly outline the possible directions of further research. Joint work with Israel Gelfand and Neil White.

Michel Broué, Université Paris VII
Coxeter like presentations for complex reflection groups
Coxeter systems provide very powerful tools to understand finite real reflection groups (the "Coxeter groups"), their geometry, and their braid groups. Complex reflection groups share many properties with real reflection groups, but until recently, no analogue (but empirical facts for some cases) of Coxeter presentations was known for these groups. We shall present a survey of recent results which provide presentations "à la Coxeter" for complex reflection groups and for their associated braid groups, leaning on combinatorial, topological and geometrical methods.

John Conway, Princeton University
Fabulous Groups
Coxeter's theory of reflection groups is the only case known to me in which an interesting class of presentations characterizes an interesting class of groups.

Many interesting groups have presentations as quotients of hyperbolic Coxeter groups, and the notion of "fabulous group" gives us a good idea of which non-Coxeter relations to adjoin. As examples, I'll discuss the smallest and largest of the 20th-century simple groups, namely Janko's first group and the Monster group.

Michele Emmer, Universita di Roma "La Sapienza"
The Visual Mind: Math, Art, Cinema
I first met Donald Coxeter in 1976. I had a project to make films on art and mathematics and I asked him to hel p me. We met the first time in Toronto. One of the films was on solids in space, another on symmetry, the third on geometries and impossible worlds; the last two related to the Dutch graphic artist M.C. Escher. Then we organized together the conference on Escher in Rome in 1986. But he could not make it to the second one which took place in 1998. He wrote a paper in the proceedings. I will try to present "visually" the ideas and suggestions of Coxeter I have used in my films and how we cooperated in making them. I will also present some of his "visual" ideas that were of great importance for me.

Branko Grünbaum, University of Washington
Perspectives on configurations and polyhedra
Both topics were of enduring interest to Coxeter.
Configurations of points and lines were intensively studied for several decades more than a century ago, only to be ignored during most of the twentieth century. Coxeter was one of the few contributors that obtained significant results in the third quarter of that century. The history and present status of the theory of configurations will be presented, together with an outlook on future developments.
A brief report will be given on the theory of polyhedra, in particular those with considerable degree of symmetry. The main purpose here will be to point out the close parallels of the perspectives on configurations and on polyhedra; both have received a significant infusion of new ideas in recent years.

J.E. Humphreys, University of Massachusetts, Amherst
The unifying role of Coxeter groups in representation theory
After recalling how a large class of Coxeter groups arise as "Weyl groups" in semisimple Lie theory, one classical and one less-classical branch of representation theory will be discussed: (1) highest weight representations in characteristic 0, where the Weyl group is finite;
(2) representations of semisimple groups and their Lie algebras in prime characteristic, where the same finite Weyl group appears but affine Weyl groups also play a deep role (for reasons not yet fully understood). In both cases, Kazhdan-Lusztig theory for Coxeter groups provides the essential framework. In prime characteristic, the geometry of cells in the affine Weyl group as well as the geometry of the flag variety and the nilpotent variety are conjecturally involved in many of the unsolved problems.

Ruth Kellerhals, University of Fribourg
Aspects of hyperbolic space forms
We shall present some structural results for hyperbolic quotient spaces by discrete group actions. Several concepts and constructions strongly influenced by H.S.M. Coxeter will be used to derive extremality properties for some of their characteristic
invariants: Hyperbolic isometries and Clifford matrices, Coxeter groups and polytopes, sphere packings and the simplicial density function, polylogarithms and Lobachevsky functions.

Askold Khovanskii, The University of Toronto
Combinatorics of affine sections of convex polyhedra
A bounded polyhedron is called simple if it is the intersection of half-spaces in general position. We estimate the number and the proportion of $k$-dimensional faces of a simple $n$-dimensional polyhedron that intersect a generic affine subspace of dimension $l$. Our estimates depend only on the number of faces of the simple polyhedron in each dimension and on the dimension $l$ of the section. In the case when the affine subspace is a hyperplane such estimates were found in [1]. Using them one can prove [1--2] the following

{\bf Theorem.}{\sl In Lobachevsky space of dimension $> 995$, there are no discrete groups generated by reflections with fundamental polyhedra of finite volume.}

In fact, the case of an $l$-dimensional affine section is completely analogous to the case of a hyperplane section. The result could be considered as a generalization of the famous Upper Bound Theorem.

[1] A.G. Khovanskii. Hyperplane sections of polyhedra, toric varieties and discrete groups in Lobachevsky space, ``Functional Analysis and its applications", V. 20, N 1, 50-61, 1986; translation in Funct. Anal. Appl. V. 20 (1986), no. 1, 41--50 (1986). \medskip

[2] M.N. Prokhorov. Non-existence of
crystallographic groups of reflections in Lobachevsky spaces of large dimension, ``Izv. Akad. Nauk SSSR'', Ser. Mat., V. 50, no. 2, 320-332 (1986).

Bertram Kostant, MIT
The McKay correspondence, the Coxeter Element and representation theory
The McKay correspondence sets up a bijection, $\Gamma\mapsto \g_{\Gamma}$, of the set of all finite subgroups $\Gamma$ of $SU(2)$ and a set of complex simple Lie algebras of type $A,D$ and $E$, representing all isomorphism classes. In addition if $N(\g_{\Gamma})$ is the set of nodes of extended Coxeter-Dynkin diagram of $\g_{\Gamma}$, the correspondence defines a bijection $$\tau: \widehat{\Gamma}\to N(\g_{\Gamma})\qquad $$ where $\widehat{\Gamma}$ is the unitary dual of $\Gamma$. Now the embedding $\Gamma\s SU(2)$ defines a natural action of $\Gamma$ on the symmetric algebra $S(\Bbb C^2)$. For each $\pi\in \widehat{\Gamma}$ one can define the Poincar\'e series, $P_{\pi}(t)$, for the occurrence of $\pi$ in $S(\Bbb C^2)$. If $\pi$ is the identity representation then $P_{\pi}(t)$ is famous and well known in the theory of Kleinian singularities. If $\pi$ is not the identity representation then $\tau(\pi)$ can be identified with a simple root and we have determined $P_{\pi}(t)$ in terms of certain properties of that root orbit of the Coxeter element which contains $\tau(\pi)$.

Greg Kuperberg, University of California at Davis
Numerical cubature from geometry and coding theory
The numerical cubature problem is the generalization to higher dimensions of integration methods such as Simpson's rule. Given a measure mu on R^n, a t-cubature formula is a finite set C such that integral of any polynomial P of degree t with respect to mu equals a weighted sum over values on C. The main interest is in cubature formulas with few points, with positive weights, and without points outside of the domain of mu. Gaussian quadrature satisfies all three conditions in one dimension, but the problem is already open-ended in two dimensions and increasingly non-trivial in higher dimensions.

I will discuss new methods for the cubature problem coming from error-correcting codes, symplectic moment maps, and lattice packings of discretized convex bodies. The methods yield many new explicit, efficient, positive, interior, cubature formulas for the most standard choices of mu. In one context, they also lead to an interesting local lower bound on the number of points needed for cubature.

Peter McMullen, University College London
Regular polytopes of full rank
An abstract polytope $\CP$ is a poset whose structure mimics that of the regular convex polytopes. It has rank $n$ if its flags (or maximal chains) contain $n+2$ elements (including the unique minimal and maximal elements), and it is regular if its automorphism group $\Ga$ is transitive on its flags. A (faithful) realization of $\CP$ represents it in some euclidean space (with symmetry group isomorphic to $\Ga$). If the abstract regular polytope $\CP$ is finite, then the dimension of a faithful realization of $\CP$ is no smaller than its rank. Similarly, that of a discrete faithful realization of a regular (infinite) apeirotope is at least one fewer than the rank. Realizations which attain the minimum are said to be of full rank. In this talk, the classification of the regular polytopes and apeirotopes of full rank in all dimensions is outlined.

Robert Moody, University of Alberta
Non-crystallographic Coxeter Groups and Quasicrystals
In Lie theory only the crystallographic Coxeter groups occur. But the discoveries of the Penrose tilings and new crystal-like materials based on long-range icosahedral order have brought fresh interest in the non-crystallographic Coxeter groups, especially H2, H3, and H4. In this talk will discuss the phenomena of long-range aperiodic order, the non-crystallographic root systems, and their connections with icosahedral phases of quasi-crystalline materials.

Since my interaction with Donald Coxeter was to profoundly affect my mathematical career, it will be impossible not to include a few personal reminiscences at this special occasion.

Bernhard Mühlherr, Université Libre de Bruxelles
On the isomorphism problem for Coxeter groups
Coxeter groups are groups which admit a particulary nice presentation using a set of involutory generators. This presentation is given by a symmetric
matrix whose entries are natural numbers or a symbol which represents infinity.

Since the 198O's the automorphism groups of Coxeter groups had been investigated. It turned out that in order to determine these groups it is more natural to solve the isomorphism problem for Coxeter groups.

In my talk I will give an overview about several contributions to the isomorphism problem. The main result of my talk is the solution of the isomorphism problem for 2-spherical Coxeter groups. This means that there are no infinities in the matrix. I describe the two main steps of its proof and indicate to which extent the results generalize to arbitrary matrices.

The first step is to characterize reflections in abstract Coxeter groups. This is joint work with W. Franzsen, F. Haglund and R. Howlett. The second step is to show reflection rigidity. This is joint work with P.-E. Caprace.

Jürgen Richter-Gebert, Technische Universität München
"Geometry Revisited" revisited
In 1967 H.S.M. Coxeter and S.L. Greitzer published the book "Geometry Revisited", which is an excellent and very understandable monography about severel topics in elementary geometry. In a sense the book is written in the same spirit as Felix Kleins Trilogy "Elementary mathematics from the Advanced Point of View". Seemingly simple effects and theorems are presented and viewed from a higher level of mathematical understanding.

Now, almost 50 years after the appearance of the first edition of this book many aspects are interesting from a new perspective. Methods of algebraization, proving techniques and visualization aspects that are used in the book can be used as paradigms to give rise to automatic methods to treat geometry on computers.

In the presentation we will pick a few of these aspects and show their relevance for contemporary computer geometry. We will also show, how the use of computers creates new and fascinating aspects and problems in the context of elementary geometry. Among the topics of the talk will be 'algebraic proving techniques', 'inversive geometry', 'transformations' and 'monodromy aspects'.

Mark Ronan, University of Illinois at Chicago
Buildings and their Classification
The theory of buildings was created by Jacques Tits in the late 1950's and early 1960's. In 1974 he published a definitive book on the subject in which he classified all buildings of spherical type and dimension at least 2. A building is of spherical type when its apartments arise from finite Coxeter groups, in which case they can be represented as tilings of a sphere.

When the apartments of a building are tilings of Euclidean space, the building is called affine. Tits classified affine buildings in 1981 using the fact that an affine building admits a spherical building at infinity. The question of classifying hyperbolic buildings is the main topic of this talk.

All types of buildings have important applications to group theory. Spherical buildings arise from algebraic groups, affine buildings arise from groups over a field having a discrete valuation, and other types of buildings arise from Kac-Moody groups. In the talk I shall explain why a Kac-Moody group, in contrast to an algebraic group, yields two buildings rather than one. These buildings are naturally twinned with one another, forming a twin building that looks remarkably like a spherical building. Twin buildings appear to offer a new way of approaching classification questions for hyperbolic buildings.

Doris Schattschneider, Moravian College
Coxeter and the Artists: two-way inspiration
H.S.M. Coxeter's delight in geometric form was the catalyst for fruitful two-way interactions with artists, sculptors, and model-makers. Their artistic expressions, often guided only by intuition, sparked his geometric curiosity and led to analysis of their works. His articles, books, and personal correspondence in turn encouraged and inspired them to create breathtaking expressions of the power and beauty of geometry.

Egon Schulte, Northeastern University
Chiral Polyhedra in Ordinary Space
Symmetric polyhedra have been investigated since antiquity. With the passage of time, the concept of a polyhedron has undergone a number of changes which have brought to light new classes of highly-symmetric polyhedra. Donald Coxeter's famous "Regular Polytopes" and his various other writings treat the Platonic solids, the Kepler-Poinsot polyhedra and the Petrie-Coxeter in great detail, and cover what might be called the classical theory.

A lot has happened in this area in the past 30 years. In particular, around 1980, the class of regular polyhedra in Euclidean 3-space was considerably extended by Branko Grunbaum and Andreas Dress, and an alternative approach to the full classification was later described by Peter McMullen and the speaker (and is contained in their recent book "Abstract Regular Polytopes").

This talk presents the complete enumeration of chiral polyhedra in Euclidean 3-space. Chiral, or irreflexibly regular, polyhedra are nearly regular polyhedra; their geometric symmetry groups have two orbits on the flags (regular polyhedra have just one orbit), such that adjacent flags are in distinct orbits. There are several infinite families of chiral polyhedra, each with finite skew, or infinite helical, faces, and with finite skew vertex-figures. Their geometry and combinatorics are rather complicated.

Marjorie Senechal, Smith College
Donald and the Golden Rhombohedra
Donald Coxeter found puzzles, problems, and pleasure in reflecting on, projecting, and constructing simple geometric shapes. The two golden rhombohedra were among his favorite toys; in recent years he often wrote and spoke about them, especially as building blocks for golden zonohedra. In 1976, Robert Ammann proposed a new role for the golden rhombohedra: three-dimensional Penrose-tile-like aperiodic tiles. Donald found this possibility intriguing, as their correspondence shows. I will track the golden rhombohedra from Regular Polytopes to aperiodic tilings, where many puzzles and problems remain.

Neil J. A. Sloane, AT&T Shannon Labs
Nonclassical Codes and Their Associated Lattices
The problem of finding the densest packings of spheres in n-dimensional space is one that often appears in Donald Coxeter's papers, even as far
back as 1930. It is well known that many of these packings are closely related to classical error-correcting codes. In this talk I will discuss some nonclassical codes, in particular deletion-correcting and transposition-correcting codes, and show how they are related to some not-so-well known lattices. Many unsolved problems will be mentioned.

Ravi Vakil, Stanford University
A geometric Littlewood-Richardson rule
I will describe an explicit geometric Littlewood-Richardson rule, interpreted in terms of the intersection theory of the Grassmannian. There are no restrictions on the base field, and all multiplicities arising are 1; this is important for applications. This rule should be seen as a generalization of Pieri's rule, by way of explicit homotopies. It has a straightforward bijection to other Littlewood-Richardson rules, such as tableaux and Knutson and Tao's puzzles.

This gives the first geometric proof and interpretation of the Littlewood-Richardson rule. It has a host of geometric consequences, which I may describe, time permitting.

Asia Ivic Weiss, York University
Polytopes, honeycombs, groups and graphs
A symmetrical honeycomb (or polytope) which is not symmetrical by reflection, may be called twisted, to use Donald Coxeter's terminology. When the symmetry group of such a polytope has precisely two orbits on the flags, the polytope is said to be chiral. We briefly review the theory of (abstract) chiral polytopes. Each polytope of rank 4 which can be assigned Schl\"afli symbol $\{3, q, 3\}$ yields a bipartite 3-valent graph whose vertices are the faces of the polytope of ranks 1 and 2. Two vertices of such a graph are adjacent whenever the corresponding faces are incident. The finite quotients of such polytopes are a source of many interesting graphs. We investigate the implications which the symmetry group of the graph has for the structure of the polytope. This is joint work with Barry Monson.

Jörg Wills, University of Siegen
Polyhedral Manifolds; Symmetry and Minimality
In the first part we consider polyhedral embeddings of regular maps in Euclidean 3-space; with maximal geometric symmetry group. Remarkable examples are Klein's quartic and Dyck's quartic of genus 3; further Coxeter's regular maps of genus 6 and 73.
In the second part we consider equivelar polyhedra; i.e. with local regularity like the Archimedean solids; but of positive genus. Some of them have remarkable extremality properties, e.g. their genus is larger than their vertex- or face-number. Some have 5 nonconvex vertices only; independently of the genus. We refer to remarkable polyhedra by Grünbaum, Shephard, McMullen, Schulte, Barnette et al. and, of course, by H.S.M. Coxeter. We further present some open problems.

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