Coxeter Research Articles

Symmetric association schemes arising from abstract regular polytopes

This article investigates the question of when every double coset of a string $C$-group $G$ relative to its vertex stabilizer subgroup $H$ is represented by an involution. We show that this is the case for every finite string Coxeter group except in the $\{5,3,3\}$ case of type $H_4$, and for the infinite Coxeter groups of Schlafli type $\{4,4\}$ and $\{3,6\}$. From this it is immediate that, for every string $C$-group of these types, the double coset algebra $\mathbb{C}[G/\!\!/H]$ is commutative and all of its characters are realizable over $\mathbb{R}$. In particular, the abstract regular polytopes with these automorphism groups have a polyhedral realization cone.

Coxeter: "New Encounters with Geometry"

Coxeter: "New Encounters with Geometry"

The groups (2, 𝑚 | 𝑛, 𝑘 | 1, 𝑞): Finiteness and homotopy

Abstract We initiate the study of the groups ( l , m ∣ n , k ∣ p , q ) (l,m\mid n,k\mid p,q) defined by the presentation ⟨ a , b ∣ a l , b m , ( a ⁢ b ) n , ( a p ⁢ b q ) k ⟩ \langle a,b\mid a^{l},b^{m},(ab)^{n},(a^{p}b^{q})^{k}\rangle . When p = 1 p=1 and q = m - 1 q=m-1 , we obtain the group ( l , m ∣ n , k ) (l,m\mid n,k) , first systematically studied by Coxeter in 1939. In this paper, we restrict ourselves to the case l = 2 l=2 and 1 n + 1 k ≤ 1 2 \frac{1}{n}+\frac{1}{k}\leq\frac{1}{2} and give a complete determination as to which of the resulting groups are finite. We also, under certain broadly defined conditions, calculate generating sets for the second homotopy group π 2 ⁢ ( Z ) \pi_{2}(Z) , where 𝑍 is the space formed by attaching 2-cells corresponding to ( a ⁢ b ) n (ab)^{n} and ( a ⁢ b q ) k (ab^{q})^{k} to the wedge sum of the Eilenberg–MacLane spaces 𝑋 and 𝑌, where π 1 ⁢ ( X ) ≅ C 2 \pi_{1}(X)\cong C_{2} and π 1 ⁢ ( Y ) ≅ C m \pi_{1}(Y)\cong C_{m} ; in particular, π 1 ( Z ) ≅ ( 2 , m ∣ n , k ∣ 1 , q ) \pi_{1}(Z)\cong(2,m\mid n,k\mid 1,q) .

A twisted duality for parabolic Kazhdan–Lusztig R-polynomials

We prove a duality result for the parabolic Kazhdan–Lusztig R-polynomials of a finite Coxeter system. This duality is similar to, but different from, the one obtained in [9]. As a consequence of our duality we obtain an identity between the parabolic Kazhdan–Lusztig and inverse Kazhdan–Lusztig polynomials of a finite Coxeter system. We also obtain applications to certain modules defined by Deodhar and derive a result that gives evidence in favor of Marietti's combinatorial invariance conjecture for parabolic Kazhdan–Lusztig polynomials.

加权~Coxeter~群(C3,l6)的胞腔(英)

加权~Coxeter~群(C3,l6)的胞腔(英)

加权Coxeter群(B3, l)的胞腔

仿射Coxeter群（-B3,S）可以被看做仿射Coxeter群（-D4,-S）在满足条件α（-S）=-S的某种群自同构α下的不动点集合,设-l是-D4的长度函数,本文明显地刻画了加权Coxeter群（-B3,-l）的所有左胞腔.同时证明了：加权Coxeter群（-D4,-l）和（-B3,-l）的所有左胞腔都是左连通的,所有双边胞腔都是双边连通的。

Singular Artin monoids of finite coxeter type are automatic

Singular Artin monoids of finite coxeter type are automatic

King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts

King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts

The groups [formula omitted

We determine (except for two outstanding cases) for which values of m, n and p the group G m , n , p defined by the presentation 〈 a , b , c | a 2 , b 2 , ( a b ) 2 , c 2 , ( a c ) m , ( b c ) n , ( a b c ) p 〉 is finite. These groups formed one of three families of groups first studied systematically by Coxeter in 1939.

King of infinite space: Donald Coxeter, the man who saved geometry

King of infinite space: Donald Coxeter, the man who saved geometry

A Reclusive Artist Meets Minds with a World-Famous Geometer: George Odom and H.S.M. (Donald) Coxeter

The author discusses the wide-ranging correspondence of renowned geometer H.S.M. (Donald) Coxeter with George Odom, an artist who has made several geometric discoveries while living in seclusion. Coxeter was responsible for bringing these insights to wider attention

Harold Scott MacDonald Coxeter. 9 February 1907 — 31 March 2003

Donald Coxeter was the twentieth century's greatest classical geometer and the leading authority on polytopes. His Coxeter groups, deepening the investigation of symmetries, have found wide application in the sciences.

Donald in Wonderland: The Many-Faceted Life of H. S. M. Coxeter

Donald in Wonderland: The Many-Faceted Life of H. S. M. Coxeter

Coxeter and Friends

Coxeter and Friends

Donald Coxeter – Master of many dimensions

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Complex reflection groups, braid groups, Hecke algebras

Presentations a la Coxeter are given for all irreducible nite com plex re ection groups They provide presentations for the corresponding generalized braid groups for all but six cases which allow us to generalize some of the known properties of nite Coxeter groups and their associated braid groups such as the computation of the center of the braid group and the construction of deformations of the nite group algebra Hecke algebras We introduce monodromy representations of the braid groups which factorize through the Hecke algebras extending results of Cherednik Opdam Kohno and others Summary Introduction Complex re ection groups and their presentations A Background from complex re ection groups B Presentations Braid groups and their diagrams A Generalities about hyperplane complements B Generalities about the braid groups C The braid diagrams Proofs of the main theorems for the braid groups B de e r A Notation and prerequisites B Computation of B de e r and of its center for d C Computation of B e e r and of its center Hecke algebras A Background from di erential equations and monodromy B A family of monodromy representations of the braid group C Hecke algebras Diagrams and tables Appendix Generators of the monodromy around a divisor Appendix tables to Mathematics Subject Classi cation Primary G We thank Jean Michel Peter Orlik Pierre Vogel for useful conversations and the Isaac Newton Institute for its hospitality while the last version of this manuscript was written up The second named author gratefully acknowledges nancial support by the Fondation Alexander von Humboldt for his stays in Paris Michel Brou e Gunter Malle Raphael Rouquier

On the Efficiency of Coxeter Groups

On the Efficiency of Coxeter Groups

On flatness of the Coxeter graphE8

On flatness of the Coxeter graph<i>E</i>₈

Some remarks on the theorem of M�bius

H.S.M. Coxeter in his book “Introduction to Geometry” quotes a theorem of Mobius. In the paper two counterexamples are given. A corrected version of the theorem is stated and proved.

The Groups (2, 3, p; q); Asphericity and a Conjecture of Coxeter

The Groups (2, 3, p; q); Asphericity and a Conjecture of Coxeter