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In this paper we give a description of the structure of the based ring of the lowest two-sided cell for a weighted Coxeter group with complete graph. It is proved that this ring is generated, in a simple way, by some subrings which are isomorphic to the based rings of the lowest two-sided cells for affine Weyl groups of type A ˜ 1 or A ˜ 2 .

The main algorithmic problems of group theory posed by M. Dehn are the problem of words, the problem of the conjugation of words for finitely presented groups, and the group’s isomorphism problem. Among the works related to the study of the M. Dehn’s problems, the most outstanding ones are the work of P. S. Novikov who proved the undecidability of the problem of words and the conjugacy problem for finitely presented groups as well as the undecidability of the problem of isomorphism of groups. In this regard, the main algorithmic problems and their various generalizations are studied in certain classes of groups. Coxeter groups were introduced by H. S. M. Coxeter: every reflection group is a Coxeter group if its generating elements are reflections with respect to hyperplanes limiting its fundamental polyhedron. H. S. M. Coxeter listed all the reflection groups in three-dimensional Euclidean space and proved that they are all Coxeter groups and every finite Coxeter group is isomorphic to some reflection group in the three-dimensional Euclidean space which elements have a common fixed point. In an algebraic aspect Coxeter groups are studied starting with works by J. Tits who solved the problem of words in certain Coxeter groups. The article describes the known results obtained in solving algorithmic problems in Coxeter groups; the main purpose of the paper is to analyze of the results of solving algorithmic problems in Coxeter groups that were obtained by members of the Tula algebraic school ’Algorithmic problems of theory of the groups and semigroups ’ under the supervision of V. N. Bezverkhnii. It reviews assertions and theorems proved by the authors of the article for the various classes of Coxeter groups: Coxeter groups of large and extra-large types, Coxeter groups with a tree-structure, and Coxeter groups with n-angled structure. The basic approaches and methods of evidence among which the method of diagrams worked out by van Kampen, reopened by R. Lindon and refined by V. N. Bezverkhnii concerning the introduction of R-cancellations, special R-cancellations, special ring cancellations as well as method of graphs, method of types worked out by V. N. Bezverkhnii, method of special set of words designed by V. N. Bezverkhnii on the basis of the generalization of Nielsen method for free construction of groups. Classes of group considered in the article include all Coxeter groups which may be represented as generalized tree structures of Coxeter groups formed from Coxeter groups with tree structure with replacing some vertices of the corresponding tree-graph by Coxeter groups of large or extra-large types as well as Coxeter groups with n-angled structure.

We prove a characterization of irreducible, non-spherical and non-affine Coxeter groups, motivated by applications to metric commensurators of Kac-Moody groups and twinnings at finite distance.

We present new efficient data structures for elements of Coxeter groups of type $A_m$ and their associated Iwahori--Hecke algebras $H(A_m)$. Usually, elements of $H(A_m)$ are represented as simple coefficient list of length $M = (m+1)!$ with respect to the standard basis, indexed by the elements of the Coxeter group. In the new data structure, elements of $H(A_m)$ are represented as nested coefficient lists. While the cost of addition is the same in both data structures, the new data structure leads to a huge improvement in the cost of multiplication in~$H(A_m)$., Comment: 17 pages; errors corrected; example added. To appear in J. Symb. Comp

Besides simplices, n -cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter n -cubes are not classified. In this work, we first show that there are no Coxeter n -cubes in H n for n ≥ 10 . Then, we show that the ideal ones exist only for n = 2 and 3 , and provide a classification. The methods used are of combinatorial and algebraic nature, using properties of a Coxeter graph, its Schlafli matrix, and the Gram matrix of a polyhedron.

In this paper we prove that the parabolic Kazhdan–Lusztig polynomials and the parabolic R -polynomials of the universal Coxeter group can be computed in a combinatorial way, by using special matchings.

We study the concept of the Coxeter energy of graphs and digraphs (quivers) as an analogue of Gutman's adjacency energy, which has applications in theoretical chemistry and is a recently widely investigated graph invariant. Coxeter energy CE ( G ) of a (di)graph G is defined to be the sum of the absolute values of all complex eigenvalues of the Coxeter matrix associated with G. Our main inspiration for the study comes from the Coxeter formalism appearing in group theory, Lie theory, representation theory of algebras, mathematical physics and other contexts. We focus on the Coxeter energy of trees and we prove that the path (resp. the maximal star) has the smallest (resp. the greatest) Coxeter energy among all trees (resp. two large subclasses of trees) with fixed number of vertices. We provide several other related results, as the characterization of trees with second smallest and second greatest Coxeter energy, bounds for Coxeter energy, and general facts on Coxeter spectra of graphs extending known results e.g. for Salem trees and for certain special real Coxeter eigenvalues of trees. Additionally, we discuss few other energy-like quantities for the Coxeter spectra of (di)graphs, including Coxeter energy “normalized” by the trace of the Coxeter matrix and the quantities derived from variants of Coulson integral formula.

It is an open question whether right-angled Coxeter groups have unique group-equivariant visual boundaries. In Croke and Kleiner (Topology 39(3):549–556, 2000. doi: 10.1016/S0040-9383(99)00016-6 ), presented a right-angled Artin group with more than one visual boundary. In this paper we examine the space constructed by Croke and Kleiner and present a right-angled Coxeter group that is quasi-isometric to this space. Then we change the geometric parameters of this space to show that the proposed right-angled Coxeter group has non-unique equivariant visual boundaries, hence answer the open question negatively.

Let G be a finitely generated Coxeter group with presentation G = , where mij — are the elements of the symmetric Coxeter matrix: ∀i, j ∈ 1, n, mii = 1, mij ≥ 2, i 6= j. If mij ≥ 3(mij > 3), i 6= j, then G is a Coxeter group of large (extra-large) type. These groups introduced by K. Appel and P. Schupp. If the group G corresponds to a finite tree-graph Γ such that if the vertices of some edge e of the graph Γ correspond to generators ai , aj , then the edge e corresponds to the ratio of the species (aiaj ) mij = 1, then G is a Coxeter group with a tree-structure. Coxeter groups with a tree-structure introduced by V. N. Bezverkhnii, algorithmic problems in them was considered by V. N. Bezverkhnii and O. V. Inchenko. The group G can be represented as tree product 2-generated of Coxeter groups, amalgamated by cyclic subgroups. Thus from the graph Γ of G will move to the graph Γ in the following way: the vertices of the graph Γ we will put in line Coxeter group on two generators Gij = and Gjk = , to every edge e joining the vertices corresponding to Gij and Gjk is a cyclic subgroup . In this paper we prove the following theorem: normalizer of finitely generated subgroup of Coxeter group with tree-structure G = Gij∗Gjk, Gij = , Gjk = finitely generated and exist algorithm for generating.

For Coxeter groups acting non-cocompactly but with finite covolume on real hyperbolic space $$\mathbb H^n$$ , new methods are presented to distinguish them up to (wide) commensurability. We exploit these ideas and determine the commensurability classes of all hyperbolic Coxeter groups whose fundamental polyhedra are pyramids over a product of two simplices of positive dimensions.

In well-known work, Kazhdan and Lusztig (1979) defined a new set of Hecke algebra basis elements (actually two such sets) associated to elements in any Coxeter group. Often these basis elements are computed by a standard recursive algorithm which, for Coxeter group elements of long length, generally involves computing most basis elements corresponding to Coxeter group elements of smaller length. Thus, many calculations simply compute all basis elements associated to a given length or less, even if the interest is in a specific Kazhdan-Lusztig basis element. Similar remarks apply to "parabolic" versions of these basis elements defined later by Deodhar (1987, 1990), though the lengths involved are the (smaller) lengths of distinguished coset representatives. We give an algorithm which targets any given Kazhdan-Lusztig basis element or parabolic analog and does not precompute any other Kazhdan-Lusztig basis elements. In particular it does not have to store them. This results in a considerable saving in memory usage, enabling new calculations in an important case (for finite and algebraic group 1-cohomology with irreducible coefficients) analyzed by Scott-Xi (2010).

The growth function W(t) of a Coxeter group W relative to a Coxeter generating set is always a rational function. We prove by an explicit construction that there are infinitely many cocompact Coxeter groups W in hyperbolic 4-space with the following property. All the roots of the denominator of W(t) are on the unit circle except exactly two pairs of real roots.

Let ${\mathcal A}$ be a finite real linear hyperplane arrangement in three dimensions. Suppose further that all the regions of ${\mathcal A}$ are isometric. We prove that ${\mathcal A}$ is necessarily a Coxeter arrangement. As it is well known that the regions of a Coxeter arrangement are isometric, this characterizes three-dimensional Coxeter arrangements precisely as those arrangements with isometric regions. It is an open question whether this suffices to characterize Coxeter arrangements in higher dimensions. We also present the three families of affine arrangements in the plane which are not reflection arrangements, but in which all the regions are isometric., Comment: 6 pages, 1 figure

The affine Weyl group ( C ˜ n , S ) can be realized as the fixed point set of the affine Weyl group ( A ˜ m , S ˜ m ) , m ∈ { 2 n − 1 , 2 n , 2 n + 1 } , under a certain group automorphism α m , n . Let l ˜ m be the length function of A ˜ m . The aim of the present paper is to give a combinatorial description for all the left cells of A ˜ m which have non-empty intersection with C ˜ n . Then we use this description to deduce some formulae for the number of left cells of the weighted Coxeter group ( C ˜ n , l ˜ m ) in the set E λ associated to any partition λ of m + 1 .

An element of a Coxeter group $W$ is fully commutative if any two of its reduced decompositions are related by a series of transpositions of adjacent commuting generators. In the present work, we focus on fully commutative involutions, which are characterized in terms of Viennot's heaps. By encoding the latter by Dyck-type lattice walks, we enumerate fully commutative involutions according to their length, for all classical finite and affine Coxeter groups. In the finite cases, we also find explicit expressions for their generating functions with respect to the major index. Finally in affine type $A$, we connect our results to Fan--Green's cell structure of the corresponding Temperley--Lieb algebra., 25 pages

Lusztig proposed 15 conjectures for any weighted Coxeter groups. In this study, we explicitly describe Kazhdan–Lusztig cells, distinguished involutions and the function a for any weighted universal Coxeter group. We also verify these conjectures for this group.

Chapuy and Stump have given a nice generating series for the number of factorizations of a Coxeter element as a product of reflections. Their method is to evaluate case by case a character-theoretic expression. The goal of this note is to give a uniform evaluation of their character-theoretic expression in the case of Weyl groups, by using combinatorial properties of Deligne-Lusztig representations.

We show that every finitely generated Artin–Tits group admits a finite Garside family, by introducing the notion of a low element in a Coxeter group and proving that the family of all low elements in a Coxeter system (W, S) with S finite includes S and is finite and closed under suffix and join with respect to the right weak order.

Inspired by famous results of Drozd (1974) [13], Nazarova–Roĭter (1975) [42], and Bondarenko–Stepochkina (2005) [8], on the importance of finite posets I with the non-negative Tits quadratic form qˆI:ZI→Z in the representation theory, we study the finite non-negative posets I of corank two, that is, such posets I that the symmetric Gram matrix GI:=12[CI+CItr]∈MI(Q) of I is positive semi-definite of corank two, where CI∈MI(Z) is the incidence matrix of I. A structure of such posets is described in the paper. It is shown that every such a poset I can be obtained in a natural way from a corank zero poset J and a pair of roots of J. Following main ideas of the Coxeter spectral analysis of posets, we associate with any connected corank two poset I the incidence quadratic form qI:ZI→Z, the Coxeter matrix CoxI:=−CI⋅CI−tr, the Coxeter polynomial coxI(t):=det⁡(t⋅E−CoxI)∈Z[t], its complex Coxeter spectrum speccI, the Coxeter number cI∈N∪{∞}, the reduced Coxeter number cˇI∈N, the incidence defect homomorphism ∂˜I:ZI→KerqI⊆ZI, and a simply laced Dynkin diagram DynI. In particular, we study the question when the Coxeter spectrum speccI of I, together with DynI, determines the incidence matrix CI of I (hence the poset I) uniquely, up to a Z-congruence. By applying linear algebra techniques and a combinatorial computer-aided analysis of posets, we prove that the question has an affirmative answer for connected corank two posets I, with at most 15 elements. A complete list of such posets is presented and their Coxeter type invariants are computed. It follows that |I|≥6, there is only one such a poset I of cardinality 6, and 8 such posets of cardinality 7, up to the poset duality I↦Iop. The existence of a ΦI-mesh translation quiver Γ(RI∪KerqI,ΦI) on the set RI∪KerqI is discussed, where RI={v∈ZI;qI(v)=1} is the set of roots of qI.

3-dimensional convex uniform polyhedra have been projected onto their corresponding Coxeter planes defined by the simple roots of the Coxeter diagram

Let ( W , S ) be a Coxeter system with a strictly complete Coxeter graph. The present paper concerns the set Red ( z ) of all reduced expressions for any z ∈ W . By associating each bc-expression to a certain symbol, we describe the set Red ( z ) and compute its cardinal | Red ( z ) | in terms of symbols. An explicit formula for | Red ( z ) | is deduced, where the Fibonacci numbers play a crucial role.

CoxIter is a computer program designed to compute invariants of hyperbolic Coxeter groups. Given such a group, the program determines whether it is cocompact or of finite covolume, whether it is arithmetic in the non-cocompact case, and whether it provides the Euler characteristic and the combinatorial structure of the associated fundamental polyhedron. The aim of this paper is to present the theoretical background for the program. The source code is available online as supplementary material with the published article and on the author’s website (http://coxiter.rgug.ch).Supplementary materials are available with this article.

Let Λ be a k-algebra of finite global dimension. We study tubular families in the Auslander–Reiten quiver of the bounded derived category Db(Λ) satisfying certain natural axioms. In particular, we precisely describe their influence on the cyclotomic factors of the Coxeter polynomial χΛ of Λ and discuss several numerical limitations for their possible shapes. Moreover, we show that our results provide an alternative, relatively simple proof of non-trivial classical facts concerning tubular families in module categories, and also extend them slightly.

In (Howlett and Nguyen in J Algebra 361:188–212, 2012), the concept of W-graph ideal in a Coxeter group was introduced, and it was showed that a W-graph can be constructed from a given W-graph ideal (this is the case of equal parameters). In this paper, we generalize the definition of W-graph ideal in the weighted Coxeter groups, and describe how to construct a W-graph from a given W-graph ideal in the case of unequal parameters.

Let (W,S) be a finite rank Coxeter system with W infinite. We prove that the limit weak order on the blocks of infinite reduced words of W is encoded by the topology of the Tits boundary of the Davis complex X of W. We consider many special cases, including W word hyperbolic, and X with isolated flats. We establish that when W is word hyperbolic, the limit weak order is the disjoint union of weak orders of finite Coxeter groups. We also establish, for each boundary point \xi, a natural order-preserving correspondence between infinite reduced words which "point towards" \xi, and elements of the reflection subgroup of W which fixes \xi., Comment: 28 pages, 2 figures. Version 2: additional references in introduction. Version 3: results are unchanged but exposition has been substantially revised following referee's suggestions. To appear in Int. Math. Res. Not

For a finite Coxeter group, a subword complex is a simplicial complex associated with a pair (Q, π), where Q is a word in the alphabet of simple reflections and π is a group element. We discuss the transformations of such a complex that are induced by braid moves of the word Q. We show that under certain conditions, such a transformation is a composition of edge subdivisions and inverse edge subdivisions. In this case, we describe how the H- and γ-polynomials change under the transformation. This case includes all braid moves for groups with simply laced Coxeter diagrams.

We study spectral properties of the Coxeter transformations for the affine Dynkin cycle and find the Jordan form of the Coxeter transformation and the Coxeter numbers.

Let W be a Coxeter group. In this paper, we establish that, up to going to some finite index normal subgroup W_0 of W, any two cyclically reduced expressions of conjugate elements of W_0 only differ by a sequence of braid relations and cyclic shifts. This thus provides a simple description of conjugacy classes in W_0. As a byproduct of our methods, we also obtain a characterisation of straight elements of W, namely of those elements w in W for which $\ell(w^n)=n\ell(w)$ for any integer n. In particular, we generalise previous characterisations of straight elements within the class of so-called cyclically fully commutative (CFC) elements, and we give a shorter and more transparent proof that Coxeter elements are straight., 12 pages, to appear in Journal of Algebra

In this article, we prove that for any indecomposable dominant character χ of a maximal torus T of a simple adjoint group G over ℂ such that there is a Coxeter element w in the Weyl group W for which , the graded algebra is a polynomial ring if and only if dim(H 0(G/B, ℒχ) T ) ≤rank of G. We also prove that the coordinate ring ℂ[𝔥] of the cartan subalgebra 𝔥 of the Lie algebra 𝔤 of G and are isomorphic if and only if is nonempty for some coxeter element w in W, where α0 denotes the highest long root.

We provide a survey of hyperbolic orbifolds of minimal volume, starting with the results of Siegel in two dimensions and with the contributions of Gehring, Martin and others in three dimensions. For higher dimensions, we summarise some of the most important results, due to Belolipetsky, Emery and Hild, by discussing related features such as hyperbolic Coxeter groups, arithmeticity and consequences of Prasad’s volume, as well as canonical cusps, crystallography and packing densities. We also present some new results about volume minimisers in dimensions six and eight related to Bugaenko’s cocompact arithmetic Coxeter groups.

In the papers of I.K. Zhuk, then more completely of E. Moln?r, I. Prok, J. Szirmai all simplicial 3-tilings have been classified, where a symmetry group acts transitively on the simplex tiles. The involved spaces depends on some rotational order parameters. When a vertex of a such simplex lies out of the absolute, e.g. in hyperbolic space H3, then truncation with its polar plane gives a truncated simplex or simply, trunc-simplex. Looking for symmetries of these tilings by simplex or trunc-simplex domains, with their side face pairings, it is possible to find all their group extensions, especially Coxeter?s reflection groups, if they exist. So here, connections between isometry groups and their supergroups is given by expressing the generators and the corresponding parameters. There are investigated simplices in families F3, F4, F6 and appropriate series of trunc-simplices. In all cases the Coxeter groups are the maximal ones.

Associated with any Coxeter group is a Coxeter monoid, which has the same elements, and the same identity, but a different multiplication. (Some authors call these Coxeter monoids 0-Hecke monoids, because of their relation to the 0-Hecke algebras—the q=0 case of the Hecke algebra of a Coxeter group.) A Coxeter group is defined as a group having a particular presentation, but a pair of isomorphic groups could be obtained via non-isomorphic presentations of this form. We show that when we have both the group and the monoid structure, we can reconstruct the presentation uniquely up to isomorphism and present a characterisation of those finite group and monoid structures that occur as a Coxeter group and its corresponding Coxeter monoid. The Coxeter monoid structure is related to this Bruhat order. More precisely, multiplication in the Coxeter monoid corresponds to element-wise multiplication of principal downsets in the Bruhat order. Using this property and our characterisation of Coxeter groups among structures with a group and monoid operation, we derive a classification of Coxeter groups among all groups admitting a partial order.

We refine Brink's theorem, that the non-reflection part of a reflection centralizer in a Coxeter group W is a free group. We give an explicit set of generators for centralizer, which is finitely generated when W is. And we give a method for computing the Coxeter diagram for its reflection subgroup. In many cases, our method allows one to compute centralizers in one's head., Comment: To appear in Transformation Groups

The purpose of this article is to shed new light on the combinatorial structure of Kazhdan–Lusztig cells in infinite Coxeter groups W. Our main focus is the set D of distinguished involutions in W, which was introduced by Lusztig in one of his first papers on cells in affine Weyl groups. We conjecture that the set D has a simple recursive structure and can be enumerated algorithmically starting from the distinguished involutions of finite Coxeter groups. Moreover, to each element of D we assign an explicitly defined set of equivalence relations on W that altogether conjecturally determine the partition of W into left (right) cells. We are able to prove these conjectures only in a special case, but even from these partial results we can deduce some interesting corollaries.

We give closed combinatorial product formulas for Kazhdan---Lusztig polynomials and their parabolic analogue of type q in the case of boolean elements, introduced in (Marietti in J. Algebra 295:1---26, 2006), in Coxeter groups whose Coxeter graph is a tree. Such formulas involve Catalan numbers and use a combinatorial interpretation of the Coxeter graph of the group. In the case of classical Weyl groups, this combinatorial interpretation can be restated in terms of statistics of (signed) permutations. As an application of the formulas, we compute the intersection homology Poincare polynomials of the Schubert varieties of boolean elements.

Let W be a Coxeter group and WJ a finite parabolic subgroup. We present here a geometric new proof of X. Heʼs result on minimal length elements of an orbit of WJ which acts on W in a certain way.

We present a complete list of the Coxeter polynomials of P-critical bigraphs Δ, for | Δ 0 | ⩽ 8 . Moreover, given P-critical bigraphs Δ, Δ ′ of the Euclidean type D Δ = A ˜ n − 1 and n ⩽ 8 , we establish by a computer calculation the equivalence of the following three statements: (a) cox Δ ( t ) = cox Δ ′ ( t ) , (b) the reduced Coxeter numbers c Δ and c Δ ′ of Δ and Δ ′ coincide, (c) the tubular types tub Δ and tub Δ ′ of Δ and Δ ′ coincide (see Section 2).

We investigate the imaginary cone in hyperbolic Coxeter systems in order to show that any Coxeter system contains universal reflection subgroups of arbitrarily large rank. Furthermore, in the hyperbolic case, the positive spans of the simple roots of the universal reflection subgroups are shown to approximate the imaginary cone (using an appropriate topology on the set of roots), answering a question due to Dyer [9] in the special case of hyperbolic Coxeter systems. Finally, we discuss growth in Coxeter systems and utilize the previous results to extend the results of [16] regarding exponential growth in parabolic quotients in Coxeter groups.

Hyperbolic Coxeter polytopes are defined precisely by combinatorial type. Polytopes in hyperbolic n-space with n + p faces that have the combinatorial type of a pyramid over a product of simplices were classified by Tumarkin for small p. In this article we generalise Tumarkin’s methods and find the remaining hyperbolic Coxeter pyramids.

The depth statistic was defined by Petersen and Tenner for an element of an arbitrary Coxeter group in terms of factorizations of the element into a product of reflections. It can also be defined as the minimal cost, given certain prescribed edge weights, for a path in the Bruhat graph from the identity to an element. We present algorithms for calculating the depth of an element of a classical Coxeter group that yield simple formulas for this statistic. We use our algorithms to characterize elements having depth equal to length. These are the short-braid-avoiding elements. We also give a characterization of the elements for which the reflection length coincides with both depth and length. These are the boolean elements.

Consider an irreducible finite Coxeter system. We show that for any nonnegative integer n the sum of the nth powers of the Coxeter exponents can be written uniformly as a polynomial in four parameters: h (the Coxeter number), r (the rank), and two further parameters., Comment: 14 pages

We prove that if b is a co-contraction of , then the right- angled Coxeter group C(b embeds into C(). Further, we provide a graph without an induced long cycle while C() does not contain a hyperbolic surface group.

If x and y are roots in the root system with respect to the standard (Tits) geometric realization of a Coxeter group W, we say that x dominates y if for all w ∈ W , wy is a negative root whenever wx is a negative root. We call a positive root elementary if it does not dominate any positive root other than itself. The set of all elementary roots is denoted by E . It has been proved by B. Brink and R.B. Howlett [B. Brink, R.B. Howlett, A finiteness property and an automatic structure of Coxeter groups, Math. Ann. 296 (1993) 179–190] that E is finite if (and only if) W is a finite-rank Coxeter group. Amongst other things, this finiteness property enabled Brink and Howlett to establish the automaticity of all finite-rank Coxeter groups. Later Brink has also given a complete description of the set E for arbitrary finite-rank Coxeter groups [B. Brink, The set of dominance-minimal roots, J. Algebra 206 (1998) 371–412]. However the set of non-elementary positive roots has received little attention in the literature. In this paper we answer a collection of questions concerning the dominance behavior between such non-elementary positive roots. In particular, we show that for any finite-rank Coxeter group and for any non-negative integer n, the set of roots each dominating precisely n other positive roots is finite. We give upper and lower bounds for the sizes of all such sets as well as an inductive algorithm for their computation.

Let W be a finite Coxeter group, P a parabolic subgroup of W, and N W (P) the normalizer of P in W. We prove that every element in N W (P) is strongly real in N W (P), and that every irreducible complex character of N W (P) has Frobenius-Schur indicator 1.

We characterize certain properties of the derived series of Coxeter groups by properties of the corresponding Coxeter graphs. In particular, we give necessary and sufficient conditions for a Coxeter group to be quasiperfect.

We present a new axiomatization of what is known as the “root system” of a Coxeter group that does not involve vector spaces or Coxeter groups. We use this to combinatorially characterize the structure of intervals in weak Bruhat order in much the same way that finite distributive lattices are characterized as the lattices of order ideals of partially ordered sets. In fact, the result for distributive lattices follows as a special case.

This note proves Cellini’s conjecture that, in a Coxeter system (W,S) with reflections T, the T-increasing paths in W are self-avoiding. Here, a T-increasing path is a sequence v,t1v,…,tn⋯t1v in W with ti∈T and t1≺⋯≺tn in a reflection order ⪯ of T.