Coxeter System Research Articles

Parabolic recursions for Kazhdan–Lusztig polynomials and the hypercube decomposition

We employ general parabolic recursion methods to demonstrate the recently devised hypercube formula for Kazhdan-Lusztig polynomials of Sn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_n$$\\end{document}, and establish its generalization to the full setting of a finite Coxeter system through algebraic proof. We introduce procedures for positive decompositions of q-derived Kazhdan–Lusztig polynomials within this setting, that utilize classical Hecke algebra positivity phenomena of Dyer-Lehrer and Grojnowski–Haiman. This leads to a distinct algorithmic approach to the subject, based on induction from a parabolic subgroup. We propose suitable weak variants of the combinatorial invariance conjecture and verify their validity for permutation groups.

Shi arrangements and low elements in Coxeter groups

AbstractGiven an arbitrary Coxeter system and a non‐negative integer , the ‐Shi arrangement of is a subarrangement of the Coxeter hyperplane arrangement of . The classical Shi arrangement () was introduced in the case of affine Weyl groups by Shi to study Kazhdan–Lusztig cells for . As two key results, Shi showed that each region of the Shi arrangement contains exactly one element of minimal length in and that the union of their inverses form a convex subset of the Coxeter complex. The set of ‐low elements in were introduced to study the word problem of the corresponding Artin–Tits (braid) group and they turn out to produce automata to study the combinatorics of reduced words in . In this article, we generalize and extend Shi's results to any Coxeter system for any : (1) the set of minimal length elements of the regions in a ‐Shi arrangement is precisely the set of ‐low elements, settling a conjecture of the first and third authors in this case; (2) the union of the inverses of the (0‐)low elements form a convex subset in the Coxeter complex, settling a conjecture by the third author, Nadeau and Williams.

Representation of coxeter group and orthogonal group

The paper is primarily divided into two parts. The main focus of the first part is the construction of a representation of Coxeter groups. This begins with the definition of the Coxeter system and connected components, followed by the introduction of the length function and subsequent theorems. The faithfulness of this representation is then proven, allowing for the identification of isomorphisms that enable the final classification of finite Coxeter groups. This classification is achieved by leveraging the established relationship between irreducible representations of Coxeter groups and positive definite quadratic forms. Given the strong connection between Coxeter groups and orthogonal groups, the primary objective of the second part is to create a specific representation of orthogonal groups. This is accomplished through an examination of the decomposition of harmonic polynomials into subspaces of homogeneous harmonic polynomials, using the action of O(2) on these subspaces. The paper concludes by drawing connections to results in Invariant Theory, demonstrating the applicability of the presented concepts in a more general duality context.

Hecke algebras and edge contractions

We establish an embedding from the Hecke algebra associated with the edge contraction of a Coxeter system along an edge to the Hecke algebra associated with the original Coxeter system.

Coxeter systems with 2–dimensional Davis complexes, growth rates and Perron numbers

Coxeter systems with 2–dimensional Davis complexes, growth rates and Perron numbers

Braid graphs in simply-laced triangle-free Coxeter systems are partial cubes

In this paper, we study the structure of braid graphs in simply-laced Coxeter systems. We prove that every reduced expression has a unique factorization as a product of so-called links, which in turn induces a decomposition of the braid graph into a box product of the braid graphs for each link factor. When the Coxeter graph has no three-cycles, we use the decomposition to prove that braid graphs are partial cubes, i.e., can be isometrically embedded into a hypercube. For a special class of links, called Fibonacci links, we prove that the corresponding braid graphs are Fibonacci cubes.

The galaxy of Coxeter groups

In this paper we introduce the galaxy of Coxeter groups (of finite rank) – an infinite dimensional, locally finite, ranked simplicial complex which captures isomorphisms between finite rank Coxeter systems. In doing so, we would like to suggest a new framework to study the isomorphism problem for Coxeter groups. We prove some structural results about this space, provide a full characterization in small ranks and propose many questions. In addition we survey known tools, results and conjectures.Along the way we show profinite rigidity of triangle Coxeter groups – a result which is possibly of independent interest.

On the continuity of the growth rate on the space of Coxeter systems

Floyd showed that if a sequence of compact hyperbolic Coxeter polygons converges, then so does the sequence of the growth rates of the Coxeter groups associated with the polygons. For the case of the hyperbolic 3-space, Kolpakov discovered the same phenomena for specific convergent sequences of hyperbolic Coxeter polyhedra. In this paper, we show that the growth rate is a continuous function on the space of Coxeter systems. This is an extension of the results due to Floyd and Kolpakov since the convergent sequences of Coxeter polyhedra give rise to that of Coxeter systems in the space of marked groups.

Graphical complexes of groups

Abstract We introduce graphical complexes of groups, which can be thought of as a generalisation of Coxeter systems with 1-dimensional nerves. We show that these complexes are strictly developable, and we equip the resulting Basic Construction with three structures of non-positive curvature: piecewise linear CAT ⁢ ( 0 ) \mathrm{CAT}(0) , C ⁢ ( 6 ) C(6) graphical small cancellation, and a systolic one. We then use these structures to establish various properties of the fundamental groups of these complexes, such as biautomaticity and the Tits Alternative. We isolate an easily checkable condition implying hyperbolicity of the fundamental groups, and we construct some non-hyperbolic examples. We also briefly discuss a parallel theory of C ⁢ ( 4 ) C(4) - T ⁢ ( 4 ) T(4) graphical complexes of groups and outline their basic properties.

On the Induction of p-Cells

AbstractWe study cells with respect to thep-canonical basis of the Hecke algebra of a crystallographic Coxeter system (see Jensen and Williamson (2017) and Jensen (2020)) and their compatibility with standard parabolic subgroups. We show that after induction to the surrounding bigger Coxeter group the cell module of a rightp-cell in a standard parabolic subgroup decomposes as a direct sum of cell modules. Along the way, we state some new positivity properties of thep-canonical basis.

Modular Koszul duality for Soergel bimodules

We generalize the modular Koszul duality of Achar–Riche [4] to the setting of Soergel bimodules associated to any finite Coxeter system. The key new tools are a functorial monodromy action and wall-crossing functors in the mixed modular derived category of [4]. In characteristic 0, this duality together with Soergel's conjecture (proved by Elias–Williamson [14]) imply that our Soergel-theoretic graded category O is Koszul self-dual, generalizing the result of Beilinson–Ginzburg–Soergel [25,8].

Topological quantum field theories from Hecke algebras

We construct one-dimensional non-commutative topological quantum field theories (TQFTs), one for each Hecke algebra corresponding to a finite Coxeter system. These TQFTs associate an invariant to each ciliated surface, which is a Laurent polynomial for punctured surfaces. There is a graphical way to compute the invariant using minimal colored graphs. We give explicit formulas in terms of the Schur elements of the Hecke algebra and prove positivity properties for the invariants when the Coxeter group is of classical type, or one of the exceptional typesH3H_3,E6E_6andE7E_7.

Bruhat intervals, subword complexes and brick polyhedra for finite Coxeter groups

We study the interplay between the discrete geometry of Bruhat poset intervals and subword complexes of finite Coxeter systems. We establish connections between the cones generated by cover labels for Bruhat intervals and of root configurations for subword complexes, culminating in the notion of brick polyhedra for general subword complexes.

Kazhdan-Lusztig polynomials for [formula omitted

In 1979 Kazhdan and Lusztig defined, for an arbitrary Coxeter system (W,S), a family of polynomials indexed by pairs of elements of W. Despite their relevance and elementary definition, the explicit computation of these polynomials is still one of the hardest open problems in algebraic combinatorics. In this paper we explicitly compute Kazhdan-Lusztig polynomials for a Coxeter system of type B˜2.

Extending a word property for twisted Coxeter systems

We prove two extensions of Hansson and Hultman's word property for certain analogues of reduced words associated to twisted involutions in Coxeter groups. Our first extension concerns the superset of such words in which terms with a natural commutativity property may be optionally primed. Our other extension involves variants of these words in which a defining minimal length condition is relaxed. In type A the sets considered are closely related to generating functions for Schur Q-functions and K-theoretic Schur P-functions.

Boolean Complexes of Involutions

Let (W, S) be a Coxeter system. We introduce the boolean complex of involutions of W which is an analogue of the boolean complex of W studied by Ragnarsson and Tenner. By applying discrete Morse theory, we determine the homotopy type of the boolean complex of involutions for a large class of (W, S), including all finite Coxeter groups, finding that the homotopy type is that of a wedge of spheres of dimension vert Svert -1. In addition, we find simple recurrence formulas for the number of spheres in the wedge.

The modular Weyl–Kac character formula

We classify and explicitly construct the irreducible graded representations of anti-spherical Hecke categories which are concentrated one degree. Each of these homogeneous representations is one-dimensional and can be cohomologically constructed via a BGG resolution involving every (infinite dimensional) standard representation of the category. We hence determine the complete first row of the inverse parabolic p-Kazhdan–Lusztig matrix for an arbitrary Coxeter group and an arbitrary parabolic subgroup. This generalises the Weyl–Kac character formula to all Coxeter systems (and their parabolics) and proves that this generalised formula is rigid with respect to base change to fields of arbitrary characteristic.

Subregular J-rings of Coxeter systems via quiver path algebras

We use quivers and their representations to bring new perspectives on the subregular J-ring JC of a Coxeter system (W,S), a subring of Lusztig's J-ring. We prove that JC is isomorphic to a suitable quotient of the path algebra of the double quiver of (W,S). Up to Morita equivalence, such quotients include the group algebras of all free products of finite cyclic groups. We then use quiver representations to study the category mod-AK of finite dimensional right modules of the algebra AK=K⊗ZJC over an algebraically closed field K of characteristic zero. Our results include classifications of Coxeter systems for which mod-AK is semisimple, has finitely many simple modules up to isomorphism, or has a bound on the dimensions of simple modules.

Combinatorial Invariance Conjecture for $\widetilde {A}_2$

Abstract The combinatorial invariance conjecture (due independently to Lusztig and Dyer) predicts that if $[x,y]$ and $[x^{\prime},y^{\prime}]$ are isomorphic Bruhat posets (of possibly different Coxeter systems), then the corresponding Kazhdan–Lusztig polynomials are equal, that is, $P_{x,y}(q)=P_{x^{\prime},y^{\prime}}(q)$. We prove this conjecture for the affine Weyl group of type $\widetilde {A}_2$. This is the first infinite group with non-trivial Kazhdan–Lusztig polynomials where the conjecture is proved.

First-order aspects of Coxeter groups

We lay the foundations of the first-order model theory of Coxeter groups. Firstly, with the exception of the 2-spherical non-affine case (which we leave open), we characterize the superstable Coxeter groups of finite rank, which we show to be essentially the Coxeter groups of affine type. Secondly, we characterize the Coxeter groups of finite rank which are domains, a central assumption in the theory of algebraic geometry over groups, which in many respects (e.g. λ-stability) reduces the model theory of a given Coxeter system to the model theory of its associated irreducible components. In the second part of the paper we move to specific definability questions in right-angled Coxeter groups (RACGs) and 2-spherical Coxeter groups. In this respect, firstly, we prove that RACGs of finite rank do not have proper elementary subgroups which are Coxeter groups, and prove further that reflection independent ones do not have proper elementary subgroups at all. Secondly, we prove that if the monoid Sim(W,S) of S-self-similarities of W is finitely generated, then W is a prime model of its theory. Thirdly, we prove that in reflection independent RACGs of finite rank the Coxeter elements are type-determined. We then move to 2-spherical Coxeter groups, proving that if (W,S) is irreducible, 2-spherical even and not affine, then W is a prime model of its theory, and that if WΓ and WΘ are as in the previous sentence, then WΓ is elementary equivalent to WΘ if and only if Γ≅Θ, thus solving the elementary equivalence problem for most of the 2-spherical Coxeter groups. In the last part of the paper we focus on model theoretic applications of the notion of reflection length from Coxeter group theory, proving in particular that affine Coxeter groups are not connected.