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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.

The aim of the present paper is to introduce a generalized quantum cluster character, which assigns to each object V of a finitary Abelian category C over a finite field FF_q and any sequence ii of simple objects in C the element X_{V,ii} of the corresponding algebra P_{C,ii} of q-polynomials. We prove that if C was hereditary, then the assignments V-> X_{V,ii} define algebra homomorphisms from the (dual) Hall-Ringel algebra of C to the P_{C,ii}, which generalize the well-known Feigin homomorphisms from the upper half of a quantum group to q-polynomial algebras. If C is the representation category of an acyclic valued quiver (Q,d) and ii=(ii_0,ii_0), where ii_0 is a repetition-free source-adapted sequence, then we prove that the ii-character X_{V,ii} equals the quantum cluster character X_V introduced earlier by the second author in [29] and [30]. Using this identification, we deduce a quantum cluster structure on the quantum unipotent cell corresponding to the square of a Coxeter element. As a corollary, we prove a conjecture from the joint paper [5] of the first author with A. Zelevinsky for such quantum unipotent cells. As a byproduct, we construct the quantum twist and prove that it preserves the triangular basis introduced by A. Zelevinsky and the first author in [6]., AMS LaTeX, 46 pages, a reference added, to appear in Selecta Mathematica

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.

In this thesis, our goal is to study various aspects of root systems of simple Lie algebras. In the first part, we study the coordinates of the eigenvectors of the Cartan matrices. We start by generalizing the work of physicists who showed that the particle masses of the affine Toda field theory are equal to the coordinates of the Perron -- Frobenius eigenvector of the Cartan matrix. Then, we adopt another approach. Namely, using the ideas coming from the singularity theory, we compute the coordinates of the eigenvectors of some root systems. In the second part, inspired by Givental's ideas, we introduce q-deformations of Cartan matrices and we study their spectrum and their eigenvectors. Then, we propose a q-deformation of Toda's equations et compute 1-solitons solutions, using the Hirota's method and Hollowood's work. Finally, our interest is focused on a set of transformations which induce an action of the braid group on the set of ordered root basis. In particular, we study an orbit for this action, the set of distinguished basis and some associated matrices.; Dans cette thèse, nous nous intéressons à plusieurs aspects des systèmes de racines des algèbres de Lie simples. Dans un premier temps, nous étudions les coordonnées des vecteurs propres des matrices de Cartan. Nous commençons par généraliser les travaux de physiciens qui ont montré que les masses des particules dans la théorie des champs de Toda affine sont égales aux coordonnées du vecteur propre de Perron -- Frobenius de la matrice de Cartan. Puis nous adoptons une approche différente, puisque nous utilisons des résultats de la théorie des singularités pour calculer les coordonnées des vecteurs propres de certains systèmes de racines. Dans un deuxième temps, en s'inspirant des idées de Givental, nous introduisons les matrices de Cartan q-déformées et étudions leur spectre et leurs vecteurs propres. Puis, nous proposons une q-déformation des équations de Toda et construisons des 1-solitons solutions en adaptant la méthode de Hirota, d'après les travaux de Hollowood. Enfin, notre intérêt se porte sur un ensemble de transformations agissant sur l'ensemble des bases ordonnées de racines comme le groupe de tresses. En particulier, nous étudions les bases distinguées, qui forment l'une des orbites de cette action, et des matrices que nous leur associons.

In this thesis, our goal is to study various aspects of root systems of simple Lie algebras. In the first part, we study the coordinates of the eigenvectors of the Cartan matrices. We start by generalizing the work of physicists who showed that the particle masses of the affine Toda field theory are equal to the coordinates of the Perron -- Frobenius eigenvector of the Cartan matrix. Then, we adopt another approach. Namely, using the ideas coming from the singularity theory, we compute the coordinates of the eigenvectors of some root systems. In the second part, inspired by Givental's ideas, we introduce q-deformations of Cartan matrices and we study their spectrum and their eigenvectors. Then, we propose a q-deformation of Toda's equations et compute 1-solitons solutions, using the Hirota's method and Hollowood's work. Finally, our interest is focused on a set of transformations which induce an action of the braid group on the set of ordered root basis. In particular, we study an orbit for this action, the set of distinguished basis and some associated matrices.; Dans cette thèse, nous nous intéressons à plusieurs aspects des systèmes de racines des algèbres de Lie simples. Dans un premier temps, nous étudions les coordonnées des vecteurs propres des matrices de Cartan. Nous commençons par généraliser les travaux de physiciens qui ont montré que les masses des particules dans la théorie des champs de Toda affine sont égales aux coordonnées du vecteur propre de Perron -- Frobenius de la matrice de Cartan. Puis nous adoptons une approche différente, puisque nous utilisons des résultats de la théorie des singularités pour calculer les coordonnées des vecteurs propres de certains systèmes de racines. Dans un deuxième temps, en s'inspirant des idées de Givental, nous introduisons les matrices de Cartan q-déformées et étudions leur spectre et leurs vecteurs propres. Puis, nous proposons une q-déformation des équations de Toda et construisons des 1-solitons solutions en adaptant la méthode de Hirota, d'après les travaux de Hollowood. Enfin, notre intérêt se porte sur un ensemble de transformations agissant sur l'ensemble des bases ordonnées de racines comme le groupe de tresses. En particulier, nous étudions les bases distinguées, qui forment l'une des orbites de cette action, et des matrices que nous leur associons.

In the coset model $(D_N^{(1)} \oplus D_N^{(1)},D_N^{(1)})$ at levels $(k_1,k_2)$, the higher spin $4$ current that contains the quartic WZW currents contracted with completely symmetric $SO(2N)$ invariant $d$ tensor of rank $4$ is obtained. The three-point functions with two scalars are obtained for any finite $N$ and $k_2$ with $k_1=1$. They are determined also in the large $N$ 't Hooft limit. When one of the levels is the dual Coxeter number of $SO(2N)$, $k_1=2N-2$, the higher spin $\frac{7}{2}$ current, which contains the septic adjoint fermions contracted with the above $d$ tensor and the triple product of structure constants, is obtained from the operator product expansion (OPE) between the spin $\frac{3}{2}$ current living in the ${\cal N}=1$ superconformal algebra and the above higher spin $4$ current. The OPEs between the higher spin $\frac{7}{2}, 4$ currents are described. For $k_1=k_2=2N-2$ where both levels are equal to the dual Coxeter number of $SO(2N)$, the higher spin $3$ current of $U(1)$ charge $\frac{4}{3}$, which contains the six product of spin $\frac{1}{2}$ (two) adjoint fermions contracted with the product of $d$ tensor and two structure constants, is obtained. The corresponding ${\cal N}=2$ higher spin multiplet is determined by calculating the remaining higher spin $\frac{7}{2}, \frac{7}{2}, 4$ currents with the help of two spin $\frac{3}{2}$ currents in the ${\cal N}=2$ superconformal algebra. The other ${\cal N}=2$ higher spin multiplet, whose $U(1)$ charge is opposite to the one of above ${\cal N}=2$ higher spin multiplet, is obtained. The OPE between these two ${\cal N}=2$ higher spin mutiplets is also discussed., Comment: 96 pages

According to a theorem of Bjorner [5], there exists a stratified space whose strata are labeled by the elements of [u, v] for every interval [u, v] in the Bruhat order of a Coxeter group W, and each closed stratum (respectively the boundary of each stratum) has the homology of a ball (respectively of a sphere). In [6], Fomin and Shapiro suggest a natural geometric realization of these stratified spaces for a Weyl group W of a semi-simple Lie group G, and then prove its validity in the case of the symmetric group. The stratified spaces arise as links in the Bruhat decomposition of the totally non-negative part of the unipotent radical of G. In this article, we verify the topological regularity property of the strata formed as a result of Bruhat partial ordering on the elements of theWeyl group (of rank 4) of a semi-simple simply connected algebraic group G which is SL(4,ℝ) in our case here. The Weyl group here is the Coxeter group S 4.

The purpose of this paper is to present a condition for the power of a Coxeter element of $\mathfrak{S}_n$ to become the longest element. To be precise, given a product $C$ of $n-1$ distinct adjacent transpositions of $\mathfrak{S}_n$ in any order, we describe a condition for $C$ such that the $(n/2)$-th power $C^{n/2}$ of $C$ becomes the longest element, in terms of the Amida diagrams.

The affine Coxeter group can be realized as the fixed point set of the affine Weyl group , m ∈ {2n, 2n + 1}, under a certain group automorphism α m, n with . Let be the length function of . We study the properties of the left cells of the weighted Coxeter group and give an explicit description for all the left cells L of with a(L) ≤6.

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

Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex., Comment: This version incorporates the referee's comments. It contains the complete appendix (which will be abbreviated in the journal version). To appear in Transactions of the AMS

One possible way to obtain the quasicrystallographic structure is the projection of the higher-dimensional lattice into two- or three-dimensional subspaces. Here a general technique applicable to any higher-dimensional lattice is introduced. The Coxeter number and the integers of the Coxeter exponents of a Coxeter–Weyl group play a crucial role in determining the plane onto which the lattice is to be projected. The quasicrystal structures display the dihedral symmetry of order twice that of the Coxeter number. The eigenvectors and the corresponding eigenvalues of the Cartan matrix are used to determine the set of orthonormal vectors inn-dimensional Euclidean space which lead to suitable choices for the projection subspaces. The maximal dihedral subgroup of the Coxeter–Weyl group is identified to determine the symmetry of the quasicrystal structure. Examples are given for 12-fold symmetric quasicrystal structures obtained by projecting the higher-dimensional lattices determined by the affine Coxeter–Weyl groupsWa(F4),Wa(B6) andWa(E6). These groups share the same Coxeter numberh= 12 with different Coxeter exponents. The dihedral subgroupD12of the Coxeter groups can be obtained by defining two generatorsR1andR2as the products of generators of the Coxeter–Weyl groups. The reflection generatorsR1andR2operate in the Coxeter planes where the Coxeter elementR1R2of the Coxeter–Weyl group represents the rotation of order 12. The canonical (strip, equivalently, cut-and-project technique) projections of the lattices determine the nature of the quasicrystallographic structures with 12-fold symmetry as well as the crystallographic structures with fourfold and sixfold symmetry. It is noted that the quasicrystal structures obtained from the latticesWa(F4) andWa(B6) are compatible with some experimental results.

We consider the complement $$W\setminus W^J$$W\WJ of any quotient $$W^J$$WJ of a Coxeter system $$(W,S)$$(W,S) and we investigate its algebraic, combinatorial and geometric properties, emphasizing its connection with parabolic Kazhdan---Lusztig theory. In particular, we define two families of polynomials which are the analogues, for the poset $$W\setminus W^J$$W\WJ, of the parabolic Kazhdan---Lusztig and $$R$$R-polynomials. These polynomials, indexed by elements of $$W\setminus W^J$$W\WJ, have interesting connections with the ordinary Kazhdan---Lusztig and $$R$$R-polynomials.

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.

It is well known that a line in R 2 is not a set of injectivity for the spherical means for odd functions about that line. We prove that any line passing through the origin is a set of injectivity for the twisted spherical means (TSM) for functions f ∈ L 2 ( C ) , whose each of spectral projection e 1 4 | z | 2 f × φ k is a polynomial. Then, we prove that any Coxeter system of even number of lines is a set of injectivity for the TSM for L p ( C ) , 1 ⩽ p ⩽ 2 . Further, we deduce that certain Coxeter system of even number of planes is a set of injectivity for the TSM for L p ( C n ) , 1 ⩽ p ⩽ 2 . We observe that a set S R 2 n − 1 × C is a set of injectivity for the TSM for a certain class of functions on C n + 1 .

A new definition of the elliptic algebra $U_{q,p}(\widehat {\mathfrak {g}})$ associated with an untwisted affine Lie algebra $\widehat {\mathfrak {g}}$ is given as a topological algebra over the ring of formal power series in p. We also introduce a quantum dynamical analogue of Lepowsky-Wilson’s Z-algebras. The Z-algebra governs the irreducibility of the infinite dimensional $U_{q,p}({\widehat {\mathfrak {g}}})$ -modules. Some level-1 examples indicate a direct connection of the irreducible $U_{q,p}(\widehat {\mathfrak {g}})$ -modules to those of the W-algebras associated with the coset $\widehat {\mathfrak {g}}\oplus \widehat {\mathfrak {g}}\supset (\widehat {\mathfrak {g}})_{{diag}}$ with level (r − g − 1, 1) (g:the dual Coxeter number), which includes Fateev-Lukyanov’s W B l -algebra.

A complete system of primitive pairwise orthogonal idempotents for the Coxeter groups of type B B and, more generally, for the complex reflection groups G ( m , 1 , n ) G(m,1,n) is constructed by a sequence of evaluations of a rational function in several variables with values in the group ring. The evaluations correspond to the eigenvalues of the two arrays of Jucys–Murphy elements.

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.

For any irreducible real reflection group $W$ with Coxeter number $h$, Armstrong, Reiner, and the author introduced a pair of $W \times \ZZ_h$-modules which deserve to be called {\sf $W$-parking spaces} which generalize the type A notion of parking functions and conjectured a relationship between them. In this paper we give a Fuss analog of their constructions. For a Fuss parameter $k \geq 1$, we define a pair of $W \times \ZZ_{kh}$-modules which deserve to be called {\sf $k$-$W$-parking spaces} and conjecture a relationship between them. We prove the weakest version of our conjectures for each of the infinite families ABCDI of finite reflection groups, together with proofs of stronger versions in special cases. Whenever our weakest conjecture holds for $W$, we have the following corollaries. First, there is a simple formula for the character of either $k$-$W$-parking space. Second, we recover a cyclic sieving result due to Krattenthaler and M\"uller which gives the cycle structure of a generalized rotation action on $k$-$W$-noncrossing partitions. Finally, when $W$ is crystallographic, the restriction of either $k$-$W$-parking space to $W$ isomorphic to the action of $W$ on the finite torus $Q / (kh+1)Q$, where $Q$ is the root lattice., Comment: 46 pages, 10 figures

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.

In this article, we study the \'etale cohomology of the compactification of Deligne-Lusztig varieties associated to a Coxeter element. We prove a result for the integral coefficients in the case of general linear group $GL_d$, and we conjecture that the similar result holds for general reductive groups., Comment: in French

Tetrahedron is the only 3-simplex convex polyhedron having four faces, and its shape has a wide application in science and technology. In this article, using graph theory and combinatorics, a study on a special type of tetrahedron called coxeter Andreev's tetrahedron has been facilitated and it has been found that there are exactly one, four and thirty coxeter Andreev's tetrahedrons having respectively two edges of order 6 n

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.

Let $G$ be a reductive algebraic group over an algebraically closed field $k$ of characteristic $p>0$, and assume $p$ is good for $G$. Let $P$ be a parabolic subgroup with unipotent radical $U$. For $r \ge 1$, denote by $\mathbb{G}_{a(r)}$ the $r$-th Frobenius kernel of $\mathbb{G}_a$. We prove that if the nilpotence class of $U$ is less than $p$, then any embedding of $\mathbb{G}_{a(r)}$ in $U$ lies inside a one-parameter subgroup of $U$, and there is a canonical way in which to choose such a subgroup. Applying this result, we prove that if $p$ is at least as big as the Coxeter number of $G$, then the cohomological variety of $G_{(r)}$ is homeomorphic to the variety of $r$-tuples of commuting elements in $\mathcal{N}_1(\mathfrak{g})$, the $[p]$-nilpotent cone of $Lie(G)$., 15 pages

Let Ω be a finite set and let G be a permutation group acting on it. A subset H of G is called t-intersecting if any two elements in H agree on at least t points. Let SDn and SBn be the classical Coxeter group of type Dn and type Bn, respectively. We show that the maximum-sized (2t)-intersecting families in SDn and SBn are precisely cosets of stabilizers of t points in [n] ≔ {1, 2, …, n}, provided n is sufficiently large depending on t.

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.

We present combinatorial algorithms computing reduced mesh root systems and mesh geometries of roots for the Dynkin diagram Δ = A n by means of symbolic computer algebra computations and numeric algorithmic computations in Maple and C++. The problem is reduced to the computation of W Δ -orbits in the set Mor Δ of all matrix morsifications A ∈ M n ( Z ) for Δ , their Coxeter polynomials cox A ( t ) , and the Coxeter numbers c A ≥ 2 , where W Δ ⊆ M n ( Z ) is the Weyl group of Δ . In case 2 ≤ n is small, a complete classification of W Δ -orbits, is given.

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.

This paper explores the orbit structure and homomesy (constant averages over orbits) properties of certain actions of toggle groups on the collection of independent sets of a path graph. In particular we prove a generalization of a homomesy conjecture of Propp that for the action of a "Coxeter element" of vertex toggles, the difference of indicator functions of symmetrically-located vertices is 0-mesic. Then we use our analysis to show facts about orbit sizes that are easy to conjecture but nontrivial to prove. Besides its intrinsic interest, this particular combinatorial dynamical system is valuable in providing an interesting example of (a) homomesy in a context where large orbit sizes make a cyclic sieving phenomenon unlikely to exist, (b) the use of Coxeter theory to greatly generalize the set of actions for which results hold, and (c) the usefulness of Striker's notion of generalized toggle groups., 27 pages, 9 figures

Let $G$ be an exceptional simple algebraic group over an algebraically closed field $k$ and suppose that the characteristic $p$ of $k$ is a good prime for $G$. In this paper we classify the maximal Lie subalgebras $\mathfrak{m}$ of the Lie algebra $\mathfrak{g}={\rm Lie}(G)$. Specifically, we show that one of the following holds: $\mathfrak{m}={\rm Lie}(M)$ for some maximal connected subgroup $M$ of $G$, or $\mathfrak{m}$ is a maximal Witt subalgebra of $\mathfrak{g}$, or $\mathfrak{m}$ is a maximal $\it{\mbox{exotic semidirect product}}$. The conjugacy classes of maximal connected subgroups of G are known thanks to the work of Seitz, Testerman and Liebeck--Seitz. All maximal Witt subalgebras of $\mathfrak{g}$ are $G$-conjugate and they occur when $G$ is not of type ${\rm E}_6$ and $p-1$ coincides with the Coxeter number of $G$. We show that there are two conjugacy classes of maximal exotic semidirect products in $\mathfrak{g}$, one in characteristic $5$ and one in characteristic $7$, and both occur when $G$ is a group of type ${\rm E}_7$., Comment: This version is accepted for publication in Journal of the American Mathematical Society; 40 pages

This is the second in a series of papers developing a theory of total positivity for loop groups. In this paper, we study infinite products of Chevalley generators. We show that the combinatorics of infinite reduced words underlies the theory, and develop the formalism of infinite sequences of braid moves, called a braid limit. We relate this to a partial order, called the limit weak order, on infinite reduced words.

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.

By computer algebra technique and computer computations, we solve the mesh morsification problems 1.10 and present a classification of irreducible mesh roots systems, for some of the simply-laced Dynkin diagrams $\rmDelta \in \{\mathbb{A}_n,\mathbb{D}_n, \mathbb{E}_6, \mathbb{E}_7, \mathbb{E}_8\}$. The methods we use show an importance of computer algebra tools in solving difficult modern algebra problems of enough high complexity that had no solution by means of standard theoretical tools only. Inspired by results of Sato [Linear Algebra Appl. 4062005, 99-108] and a mesh quiver description of indecomposable representations of finite-dimensional algebras and their derived categories explained in [London Math. Soc. Lecture Notes Series, Vol. 119, 1988] and [Fund. Inform. 1092011, 425-462] see also 5.11, given a Dynkin diagram Δ, with n vertices and the Euler quadratic form q$_\rmDelta : \mathbb{Z}^n \rightarrow \mathbb{Z}$, we study the set Mor$_\rmDelta \subseteq \mathbb{M}_{n} \mathbb{Z}$ of all morsifications of q$_\rmDelta$ [37], i.e., the non-singular matrices A $in \mathbb{M}_{n}\mathbb{Z}$ such that its Coxeter matrix Cox$_A$ := -A · A$^{-tr}$ lies in Gln, \mathbb{Z} and q$_{\rmDelta}$ v = v · A · v$^{tr}$, for all v $\in \mathbb{Z}n$. The matrix Weyl group \mathbb{W}$_\rmDelta$ 2.13 acts on Mor$_\rmDelta$ and the determinant detA $\in$ \mathbb{Z}, the order cA $\ge2$ of CoxA i.e. the Coxeter number, and the Coxeter polynomial cox$_A$ t := dett ·E-Cox$_A$ $\in$ \mathbb{Z}[t] are $\mathbb{W}_\rmDelta$-invariant. Moreover, the finite set $R_{q\rmDelta} = \{v \in \mthbb{Z}^n; q_\rmDelta v = 1\}$ of roots of q$_\rmDelta$ is Cox$_A$-invariant. The following problems are studied in the paper: a determine the $\mathbb{W}_\rmDelta$-orbits \cal{Orb}A of Mor$_\rmDelta$ and the set $\cal{CPol}_\rmDelta = \{cox_{A}t; A \in Mor_\rmDelta\}$, b construct a finite minimal Cox$_A$-mesh quiver in $\mathbb{Z}^n$ containing all Cox$_A$-orbits of the finite set $R_{q\rmDelta}$ of roots of q$_\rmDelta$;. We prove that \cal{CPol}$_\rmDelta$ is a finite set and we construct algorithms allowing us to solve the problems for the morsifications $A = [a_{ij}] \in Mor_\rmDelta$, with $|a_{ij}| \le 2$. In this case, by computer algebra technique and computer computations, we prove that, for $n \le 8$, the number of the $\mathbb{W}_\rmDelta$-orbits \cal{Orb}A is at most 6, $s_\rmDelta := |\cal{CPol}_\rmDelta| \le 9$ and, given A,A' $\in$ Mor$_\rmDelta$ and $n \le 7$, the following three conditions are equivalent: i A' = $B^{tr}$ · A · B, for some B $\in$ Gln, \mathbb{Z}, ii cox$_{A}$t = cox$_{A'}$ t, and iii cA · det A = c$_{A'}$ · det A'. We also show that s$_{\rmDelta}$ equals 6, 5, and 9, if $\rmDelta$ is the diagram $\mathbb{E}_6$, $\mathbb{E}_7$, and $\mathbb{E}_8$, respectively.

Following the spectral Coxeter analysis of matrix morsifications for Dynkin diagrams, the spectral graph theory, a graph coloring technique, and algebraic methods in graph theory, we continue our study of the category 𝒰Bigrn of loop-free edge-bipartite signed graphs Δ, with n ≥ 2 vertices, by means of the Coxeter number cΔ, the Coxeter spectrum speccΔ of Δ, that is, the spectrum of the Coxeter polynomial coxΔt ∈ $\mathbb{Z}$[t] and the $\mathbb{Z}$-bilinear Gram form bΔ : $\mathbb{Z}$n × $\mathbb{Z}$n → $\mathbb{Z}$ of Δ [SIAM J. Discrete Math. 272013]. Our main inspiration for the study comes from the representation theory of posets, groups and algebras, Lie theory, and Diophantine geometry problems. We show that the Coxeter spectral classification of connected edge-bipartite graphs Δ in 𝒰Bigrn reduces to the Coxeter spectral classification of rational matrix morsifications A ∈ $\widehat{M}$orDΔ for a simply-laced Dynkin diagram DΔ associated with Δ. Given Δ in 𝒰Bigrn, we study the isotropy subgroup Gln,$\mathbb{Z}$Δ of Gln, $\mathbb{Z}$ that contains the Weyl group $\mathbb{W}$Δ and acts on the set $\widehat{M}$orΔ of rational matrix morsifications A of Δ in such a way that the map A $\mapsto$ speccA, det A, cΔ is Gln, $\mathbb{Z}$Δ-invariant. It is shown that, for n ≤ 6, speccΔ is the spectrum of one of the Coxeter polynomials listed in Tables 3.11-3.11a we determine them by computer search using symbolic and numeric computation. The question, if two connected positive edge-bipartite graphs Δ,Δ' in 𝒰Bigrn, with speccΔ = speccΔ', are $\mathbb{Z}$-bilinear equivalent, is studied in the paper. The problem if any $\mathbb{Z}$-invertible matrix A ∈ Mn$\mathbb{Z}$ is $\mathbb{Z}$-congruent with its transpose Atr is also discussed.

By Andreev's theorem and Choi's theorem, we proved that the degree of each vertex is three and the number of vertices of orderable compact Coxeter polyhedral is at most 10. Therefore a combinatorial polyhedron is a 3-connected planner graph. From the Plantri program, we found that the number of 3-connected planner graphs with at most 10 vertices of degree 3 is 9. We find that only five planner graphs among these 9 graphs satisfy the properties of orderable compact Coxeter polyhedra. Then we verify the polyhedra which are associated with these 5 planner graphs are orderable. Therefore the number of combinatorial polyhedra of orderable and deformable compact hyperbolic Coxeter polyhedra is five up to symmetry.

By applying computer algebra tools mainly, Maple and C++, given the Dynkin diagram $\Delta = \mathbb{A}_n$, with n ≥ 2 vertices and the Euler quadratic form $q_\Delta : \mathbb{Z}^n \rightarrow \mathbb{Z}$, we study the problem of classifying mesh root systems and the mesh geometries of roots of Δ see Section 1 for details. The problem reduces to the computation of the Weyl orbits in the set $Mor_\Delta \subseteq \mathbb{M}_n\mathbb{Z}$ of all matrix morsifications A of qΔ, i.e., the non-singular matrices $A \in \mathbb{M}_n\mathbb{Z}$ such that i qΔv = v · A · vtr, for all $v \in \mathbb{Z}^n$, and ii the Coxeter matrix CoxA := -A · A-tr lies in $Gln,\mathbb{Z}$. The Weyl group $\mathbb{W}_\Delta \subseteq Gln, \mathbb{Z}$ acts on MorΔ and the determinant det $A \in \mathbb{Z}$, the order cA ≥ 2 of CoxA i.e. the Coxeter number, and the Coxeter polynomial $cox_At := dett \centerdot E \minus Cox_A \in \mathbb{Z}[t]$ are $\mathbb{W}_\Delta$-invariant. The problem of determining the $\mathbb{W}_\Delta$-orbits $\cal{O}rbA$ of MorΔ and the Coxeter polynomials coxAt, with $A \in Mor_\Delta$, is studied in the paper and we get its solution for n ≤ 8, and $A = [a_{ij}] \in Mor_{\mathbb{A}}_n$, with $\vert a_{ij} \vert \le 1$. In this case, we prove that the number of the $\mathbb{W}_\Delta$-orbits $\cal{O}rbA$ and the number of the Coxeter polynomials coxAt equals two or three, and the following three conditions are equivalent: i $\cal{O}rbA = \mathbb{O}rbA\prime$, ii coxAt = coxA't, iii cA · det A = cA' · det A'. We also construct: a three pairwise different $\mathbb{W}_\Delta$-orbits in MorΔ, with pairwise different Coxeter polynomials, if $\Delta = \mathbb{A}_{2m \minus 1}$ and m ≥ 3; and b two pairwise different $\mathbb{W}_\Delta$-orbits in MorΔ, with pairwise different Coxeter polynomials, if $\Delta = \mathbb{A}_{2m}$ and m ≥ 1.