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Coxeter arrangements
 Proceedings of Symposia in Pure Mathematics 40
, 1983
"... Let V be an ℓdimensional Euclidean space. Let G ⊂ O(V) be a finite irreducible orthogonal reflection group. Let A be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For H ∈ A choose αH ∈ V ∗ such that H = ker(αH). For each nonnegative integer m, define the ..."
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Cited by 87 (10 self)
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). Their degrees are all equal to mh/2 (when m is even) or are equal to ((m − 1)h/2) + mi(1 ≤ i ≤ ℓ) (when m is odd). Here m1 ≤ · · · ≤ mℓ are the exponents of G and h = mℓ + 1 is the Coxeter number. The construction heavily uses the primitive derivation D which plays a central role in the theory of flat
Deformations of Coxeter Hyperplane Arrangements
 J. Combin. Theory Ser. A
, 1997
"... We investigate several hyperplane arrangements that can be viewed as deformations of Coxeter arrangements. In particular, we prove a conjecture of Linial and Stanley that the number of regions of the arrangement x i \Gamma x j = 1; 1 i ! j n; is equal to the number of alternating trees. Remarkab ..."
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Cited by 48 (5 self)
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We investigate several hyperplane arrangements that can be viewed as deformations of Coxeter arrangements. In particular, we prove a conjecture of Linial and Stanley that the number of regions of the arrangement x i \Gamma x j = 1; 1 i ! j n; is equal to the number of alternating trees
Coxeter groups, Salem numbers and the Hilbert metric
, 2001
"... this paper we prove a similar result for loops in the fundamental polyhedron of a Coxeter group W , and use it to study the spectral radius (w), w 2 W for the geometric action of W . In particular we prove: Theorem 1.1 Let (W; S) be a Coxeter system and let w 2 W . Then either (w) = 1 or (w) Le ..."
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Cited by 32 (6 self)
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) Lehmer 1:1762808. Here Lehmer denotes Lehmer's number, a root of the polynomial 1 + x \Gamma x 3 \Gamma x 4 \Gamma x 5 \Gamma x 6 \Gamma x 7 + x 9 + x 10 (1.1) and the smallest known Salem number. Billiards. Recall that a Coxeter system (W; S) is a group W with a finite generating
Conjugacy relation on Coxeter elements
 Advances in Mathematics 161
"... Abstract. Let (W,S,Γ) be an irreducible finitely presented Coxeter system. The present paper is mainly concerned with conjugacy relation on Coxeter elements in the case where Γ containing just one circle, in particular when Γ is itself a circle. In the cases where Γ is either a three multiple circle ..."
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Cited by 11 (0 self)
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Abstract. Let (W,S,Γ) be an irreducible finitely presented Coxeter system. The present paper is mainly concerned with conjugacy relation on Coxeter elements in the case where Γ containing just one circle, in particular when Γ is itself a circle. In the cases where Γ is either a three multiple
Sylvester Waves in the Coxeter Groups
, 2008
"... A new recursive procedure of the calculation of partition numbers function W(s,d m) is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of W(s,d m) and their periods τ(G) for the irreducible Coxeter groups and a list for the first ten symmetric ..."
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A new recursive procedure of the calculation of partition numbers function W(s,d m) is suggested. We find its zeroes and prove a lemma on the function parity properties. The explicit formulas of W(s,d m) and their periods τ(G) for the irreducible Coxeter groups and a list for the first ten
Eriksson’s numbers game and finite Coxeter groups
 European J. Combin
"... The numbers game is a oneplayer game played on a finite simple graph with certain “amplitudes” assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at the nodes using the amplitudes in a certain way. This game an ..."
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Cited by 6 (6 self)
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and its interactions with Coxeter/Weyl group theory and Lie theory have been studied by many authors. In particular, Eriksson connects certain geometric representations of Coxeter groups with games on graphs with certain real number amplitudes. Games played on such graphs are “Egames. ” Here we
SUBLATTICES OF CERTAIN COXETER LATTICES
"... Abstract. In this paper, we describe the sublattices of some lattices, extending previous results of [Ber]. Our description makes intensive use of graphs. Résumé. Dans cet article, nous décrivons les sousréseaux de certains réseaux de Coxeter, prolongeant les résultats de [Ber]. Notre description u ..."
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Abstract. In this paper, we describe the sublattices of some lattices, extending previous results of [Ber]. Our description makes intensive use of graphs. Résumé. Dans cet article, nous décrivons les sousréseaux de certains réseaux de Coxeter, prolongeant les résultats de [Ber]. Notre description
A GENERALIZATION OF EULER NUMBERS TO FINITE COXETER GROUPS
, 2013
"... It is known that Euler numbers, defined as the Taylor coefficients of the tangent and secant functions, count alternating permutations in the symmetric group. Springer defined a generalization of these numbers for each finite Coxeter group by considering the largest descent class, and computed the ..."
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Cited by 1 (1 self)
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It is known that Euler numbers, defined as the Taylor coefficients of the tangent and secant functions, count alternating permutations in the symmetric group. Springer defined a generalization of these numbers for each finite Coxeter group by considering the largest descent class, and computed
On the Number of Reduced Decompositions of Elements of.Coxeter Groups
"... Let r ( w) denote the number of reduced decompositions of the element w of a Coxeter group W Using the theory of symmetric functions, a formula is found for r ( w) when W is the symmetric group S". For the element Wo E S " of longest length and certain other WE S", the formula for r ( ..."
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Let r ( w) denote the number of reduced decompositions of the element w of a Coxeter group W Using the theory of symmetric functions, a formula is found for r ( w) when W is the symmetric group S". For the element Wo E S " of longest length and certain other WE S", the formula for r
Rigidity Of Coxeter Groups And Artin Groups
, 2000
"... A Coxeter group is rigid if it cannot be defined by two nonisomorphic diagrams. There have been a number of recent results showing that various classes of Coxeter groups are rigid, and a particularly interesting example of a nonrigid Coxeter group has been given in [17]. We show that this examp ..."
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Cited by 10 (1 self)
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A Coxeter group is rigid if it cannot be defined by two nonisomorphic diagrams. There have been a number of recent results showing that various classes of Coxeter groups are rigid, and a particularly interesting example of a nonrigid Coxeter group has been given in [17]. We show
Results 1  10
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285