Tits Cone Research Articles

Apollonian packings and Kac-Moody root systems

We study Apollonian circle packings using the properties of a certain rank 4 indefinite Kac-Moody root system Φ \Phi . We introduce the generating function Z ( s ) Z(\mathbf {s}) of a packing, an exponential series in four variables with an Apollonian symmetry group, which is a symmetric function for Φ \Phi . By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of Φ \Phi , with automorphic Weyl denominators, we express Z ( s ) Z(\mathbf {s}) in terms of Jacobi theta functions and the Siegel modular form Δ 5 \Delta _5 . We also show that the domain of convergence of Z ( s ) Z(\mathbf {s}) is the Tits cone of Φ \Phi , and discover that this domain inherits the intricate geometric structure of Apollonian packings.

On the convergence of the Kac-Moody correction factor

The Kac-Moody correction factor, first studied by Macdonald in the affine case, corrects the failure of an identity found by Macdonald in finite-dimensional root systems in 1972. Subsequntly this factor appeared in several formulas in the affine or Kac-Moody analogue of p-adic spherical theory for reductive groups. In this article we view the inverse of this correction factor as a function, prove the convergency and holomorphy of this function on the complexified Tits cone.

On the Tits cone of a Weyl groupoid

We translate the axioms of a Weyl groupoid with (not necessarily finite) root system in terms of arrangements. The result is a correspondence between Weyl groupoids permitting a root system and Tits arrangements satisfying an integrality condition which we call the crystallographic property.

Imaginary cone and reflection subgroups of Coxeter groups

The imaginary cone of a Kac-Moody Lie algebra is the convex hull of zero and the positive imaginary roots. This paper studies the imaginary cone for a class of root systems of general Coxeter groups W. It is shown that the imaginary cone of a reflection subgroup of W is contained in that of W, and that for irreducible infinite W of finite rank, the closed imaginary cone is the only non-zero, closed, pointed W-stable cone contained in the pointed cone spanned by the simple roots. For W of finite rank, various natural notions of faces of the imaginary cone are shown to coincide, the face lattice is explicitly described in terms of the lattice of facial reflection subgroups and it is shown that the Tits cone and imaginary cone are related by a duality closely analogous to the standard duality for polyhedral cones, even though neither of them is a closed cone in general. Some of these results have application, to be given in sequels to this paper, to dominance order of Coxeter groups, associated automata, and construction of modules for generic Iwahori-Hecke algebras.

On the double-affine Bruhat order: the $\varepsilon =1$ conjecture and classification of covers in ADE type

For any Kac–Moody group G, we prove that the Bruhat order on the semidirect product of the Weyl group and the Tits cone for G is strictly compatible with a ℤ-valued length function. We conjecture in general and prove for G of affine ADE type that the Bruhat order is graded by this length function. We also formulate and discuss conjectures relating the length function to intersections of “double-affine Schubert varieties”.

Invariant convex subcones of the Tits cone of a linear Coxeter group

We investigate the faces and the face lattices of arbitrary Coxeter group invariant convex subcones of the Tits cone of a linear Coxeter system as introduced by E.B. Vinberg. Particular examples are given by certain Weyl group invariant polyhedral cones, which underlie the theory of normal reductive linear algebraic monoids as developed by M.S. Putcha and L.E. Renner. We determine the faces and the face lattice of the Tits cone and the imaginary cone, generalizing some of the results obtained for linear Coxeter systems with symmetric root bases by M. Dyer, and for linear Coxeter systems with free root bases by E. Looijenga, P. Slodowy, and the author.

Even More Infinite Ball Packings from Lorentzian Root Systems

Boyd (1974) proposed a class of infinite ball packings that are generated by inversions. Later, Maxwell (1983) interpreted Boyd's construction in terms of root systems in Lorentz spaces. In particular, he showed that the space-like weight vectors correspond to a ball packing if and only if the associated Coxeter graph is of "level 2"'. In Maxwell's work, the simple roots form a basis of the representations space of the Coxeter group. In several recent studies, the more general based root system is considered, where the simple roots are only required to be positively independent. In this paper, we propose a geometric version of "level'' for root systems to replace Maxwell's graph theoretical "level''. Then we show that Maxwell's results naturally extend to the more general root systems with positively independent simple roots. In particular, the space-like extreme rays of the Tits cone correspond to a ball packing if and only if the root system is of level $2$. We also present a partial classification of level-$2$ root systems, namely the Coxeter $d$-polytopes of level-$2$ with $d+2$ facets.

NONORTHOGONAL GEOMETRIC REALIZATIONS OF COXETER GROUPS

AbstractWe define in an axiomatic fashion a Coxeter datum for an arbitrary Coxeter group $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}W$. This Coxeter datum will specify a pair of reflection representations of $W$ in two vector spaces linked only by a bilinear pairing without any integrality or nondegeneracy requirements. These representations are not required to be embeddings of $W$ in the orthogonal group of any vector space, and they give rise to a pair of inter-related root systems generalizing the classical root systems of Coxeter groups. We obtain comparison results between these nonorthogonal root systems and the classical root systems. Further, we study the equivalent of the Tits cone in these nonorthogonal representations.

K(π, 1) and word problems for infinite type Artin–Tits groups, and applications to virtual braid groups

Let Γ be a Coxeter graph, let (W, S) be its associated Coxeter system, and let (A, Σ) be its associated Artin–Tits system. We regard W as a reflection group acting on a real vector space V. Let I be the Tits cone, and let E Γ be the complement in I + iV of the reflecting hyperplanes. Recall that Salvetti, Charney and Davis have constructed a simplicial complex Ω(Γ) having the same homotopy type as E Γ. We observe that, if $${T \subset S}$$ , then Ω(Γ T ) naturally embeds into Ω (Γ). We prove that this embedding admits a retraction $${\pi_T: \Omega(\Gamma) \to \Omega (\Gamma_T)}$$ , and we deduce several topological and combinatorial results on parabolic subgroups of A. From a family $${\mathcal{S}}$$ of subsets of S having certain properties, we construct a cube complex Φ, we show that Φ has the same homotopy type as the universal cover of E Γ, and we prove that Φ is CAT(0) if and only if $${\mathcal{S}}$$ is a flag complex. We say that $${X \subset S}$$ is free of infinity if Γ X has no edge labeled by ∞. We show that, if $${E_{\Gamma_X}}$$ is aspherical and A X has a solution to the word problem for all $${X \subset S}$$ free of infinity, then E Γ is aspherical and A has a solution to the word problem. We apply these results to the virtual braid group VB n . In particular, we give a solution to the word problem in VB n , and we prove that the virtual cohomological dimension of VB n is n−1.

Looping of the numbers game and the alcoved hypercube

We study the so-called looping case of Mozesʼs game of numbers, which concerns the (finite) orbits in the reflection representation of affine Weyl groups situated on the boundary of the Tits cone. We give a simple proof that all configurations in the orbit are obtainable from each other by playing the numbers game, and give a strategy for going from one configuration to another. This strategy gives rise to a partition of the finite Weyl group into finitely many graded posets, one for each extending vertex of the associated extended Dynkin diagram. These posets are self-dual and mutually isomorphic, and their Hasse diagrams are dual to the triangulation of the unit hypercube by reflecting hyperplanes. Unlike the weak and Bruhat orders, the top degree is cubic in the number of vertices of the graph. We explicitly compute the rank generating function of the poset.

Reflection subgroups of finite and affine Weyl groups

We discuss the classification of reflection subgroups of finite and affine Weyl groups from the point of view of their root systems. A short case free proof is given of the well known classification of the isomorphism classes of reflection subgroups using completed Dynkin diagrams, for which there seems to be no convenient source in the literature. This is used as a basis for treating the affine case, where finer classifications of reflection subgroups are given, and combinatorial aspects of root systems are shown to appear. Various parameter sets for certain types of subsets of roots are interpreted in terms of alcove geometry and the Tits cone, and combinatorial identities are derived.

Root Systems for Asymmetric Geometric Representations of Coxeter Groups

Results are obtained concerning root systems for asymmetric geometric representations of Coxeter groups. These representations were independently introduced by Vinberg and Eriksson, and generalize the standard geometric representation of a Coxeter group in such a way as to include certain restrictions of all Kac–Moody Weyl groups. In particular, a characterization of when a nontrivial multiple of a root may also be a root is given in the general context. Characterizations of when the number of such multiples of a root is finite and when the number of positive roots sent to negative roots by a group element is finite are also given. These characterizations are stated in terms of combinatorial conditions on a graph closely related to the Coxeter graph for the group. Other finiteness results for the symmetric case which are connected to the Tits cone and to a natural partial order on positive roots are extended to this asymmetric setting.

Masures Affines

We give an abstract definition of affine hovels which generalizes the definition of affine buildings (eventually non simplicial) given by Jacques Tits and includes the hovels built by Stephane Gaussent and the author for some Kac-Moody groups over ultrametric fields. We prove that, in such an affine hovel I, there exist retractions with center a sector germ and that we can add at the infinity of I a pair of twin buildings or two microaffine buildings. For some affine hovels I, we prove that the residue at a point of I has a natural structure of pair of twin buildings and that there exists on I a preorder which induces on each apartment the preorder associated to the Tits cone.

Actions of the face monoid associated to a Kac–Moody group on its building

We described in [C. Mokler, An analogue of a reductive algebraic monoid whose unit group is a Kac–Moody group, Mem. Amer. Math. Soc. 823 (2005), 90 pp.] a monoid G ˆ acting on the integrable highest weight modules of a symmetrizable Kac–Moody algebra. It has similar structural properties as a reductive algebraic monoid with unit group a Kac–Moody group G. Now we find natural extensions of the action of the Kac–Moody group G on its building Ω to actions of the monoid G ˆ on Ω. These extensions are partly motivated by representation theory and the combinatorics of the faces of the Tits cone.

Generalisations of the Tits Representation

We construct a group $K_n$ with properties similar to infinite Coxeter groups. In particular, it has a geometric representation featuring hyperplanes, simplicial chambers and a Tits cone. The generators of $K_n$ are given by $2$-element subsets of $\{0,\ldots,n\}$. We provide some generalities to deal with groups like these. We give some easy combinatorial results on the finite residues of $K_n$, which are equivalent to certain simplicial real central hyperplane arrangements.

The F-valued points of the algebra ofstrongly regular functions of a Kac-Moody group

Let G m , resp. G f , be the minimal, resp. formal, Kac-Moody group, associated to a symmetrizable generalized Cartan matrix, over a field F of characteristic 0. Let F[G m ] be the algebra of strongly regular functions on G m . We denote by G m , resp. G f , certain monoid completions of G m , resp. G f , built by using the faces of the Tits cone. We show that there is an action of G f x G f on the spectrum of F-valued points of F[G m ]. As a G f x Gf-set it can be identified with a certain quotient of the G f x G f -set G f x G f , built by using G m . We prove a Birkhoff decomposition for the F-valued points of F[G m ]. We describe the stratification of the spectrum of F-valued points of F[G m ] in G f x G f -orbits. We show that every orbit can be covered by suitably defined big cells.

Laplace operators of infinite-dimensional Lie algebras and theta functions

Until recently, the generalized Casimir operator constructed by Kac [Kac, V. G. (1974) Funct. Anal. Appl. 8, 68-70] has been the only known element of the center of a completion of the enveloping algebra of a Kac-Moody algebra. It has been conjectured [Deodhar, V. V., Gabber, O. & Kac, V. G. (1982) Adv. Math. 45, 92-116], however, that the image of the Harish-Chandra homomorphism contains all theta functions defined on the interior of the complexified Tits cone and hence separates the orbits of the Weyl group. Developing the ideas of Feigin and Fuchs [Feigin, B. L. & Fuchs, D. B. (1983) Dokl. Akad. Nauk SSSR 269, 1057-1060], I prove this conjecture. Another application of this method is the Chevalley type restriction theorem for simple finite-dimensional Lie superalgebras.

Root Systems and Cartan Matrices

This paper is concerned with two things. The first is a (primarily) geometric axiomatic description for the systems of real roots of Lie algebras arising from (generalized) Cartan matrices. The description is base free and is a natural extension of the well-known axiomatic description of finite root systems. The primary component of our description is an open convex cone which, following Looijenga [3], we call the Tits cone. In fact it was Looijenga's paper that led to this axiomatic formulation. Unlike his construction, the dimension of the Tits cone is not tightly connected to the dimension of the Cartan matrix which it eventually yields. This leads us to the second part of the paper which concerns the construction of Cartan matrices of low row rank. We can show that if we have an l × l Cartan matrix of row rank n, then we can model an axiomatic description of it with a cone of dimension n + 1.