On symmetry breaking of dual polyhedra of non-crystallographic group H 3

The study of the polyhedra described in this paper is relevant to the icosahedral symmetry in the assembly of various spherical molecules, biomolecules and viruses. A symmetry-breaking mechanism is applied to the family of polytopes {\cal V}_{H_{3}}(\lambda) constructed for each type of dominant point λ. Here a polytope {\cal V}_{H_{3}}(\lambda) is considered as a dual of a {\cal D}_{H_{3}}(\lambda) polytope obtained from the action of the Coxeter group H 3 on a single point \lambda\in{\bb R}^{3}. The H 3 symmetry is reduced to the symmetry of its two-dimensional subgroups H 2, A 1 × A 1 and A 2 that are used to examine the geometric structure of {\cal V}_{H_{3}}(\lambda) polytopes. The latter is presented as a stack of parallel circular/polygonal orbits known as the `pancake' structure of a polytope. Inserting more orbits into an orbit decomposition results in the extension of the {\cal V}_{H_{3}}(\lambda) structure into various nanotubes. Moreover, since a {\cal V}_{H_{3}}(\lambda) polytope may contain the orbits obtained by the action of H 3 on the seed points (a, 0, 0), (0, b, 0) and (0, 0, c) within its structure, the stellations of flat-faced {\cal V}_{H_{3}}(\lambda) polytopes are constructed whenever the radii of such orbits are appropriately scaled. Finally, since the fullerene C20 has the dodecahedral structure of {\cal V}_{H_{3}}(a,0,0), the construction of the smallest fullerenes C24, C26, C28, C30 together with the nanotubes C20+6N , C20+10N is presented.

2-dimensional Coxeter groups are biautomatic

Let W be a 2-dimensional Coxeter group, that is, one with 1/m st + 1/m sr + 1/m tr ≤ 1 for all triples of distinct s, t, r ∈ S. We prove that W is biautomatic. We do it by showing that a natural geodesic language is regular (for arbitrary W), and satisfies the fellow traveller property. As a consequence, by the work of Jacek Świątkowski, groups acting properly and cocompactly on buildings of type W are also biautomatic. We also show that the fellow traveller property for the natural language fails for $W=\widetilde {A}_3$ .

Rodier type theorem for generalized principal series

Given a regular supercuspidal representation $$\rho $$ ρ of the Levi subgroup M of a standard parabolic subgroup $$P=MN$$ P = M N in a connected reductive group G defined over a non-archimedean local field F, we serve you a Rodier type structure theorem which provides us a geometrical parametrization of the set $$JH(Ind^G_P(\rho ))$$ J H ( I n d P G ( ρ ) ) of Jordan–Hölder constituents of the Harish-Chandra parabolic induction representation $$Ind^G_P(\rho )$$ I n d P G ( ρ ) , vastly generalizing Rodier structure theorem for $$P=B=TU$$ P = B = T U Borel subgroup of a connected split reductive group about 40 years ago. Our novel contribution is to overcome the essential difficulty that the relative Weyl group $$W_M=N_G(M)/M$$ W M = N G ( M ) / M is not a coxeter group in general, as opposed to the well-known fact that the Weyl group $$W_T=N_G(T)/T$$ W T = N G ( T ) / T is a coxeter group. Along the way, we sort out all regular discrete series/tempered/generic representations for arbitrary G, generalizing Tadić’s work on regular discrete series representation for split $$(G)Sp_{2n}$$ ( G ) S p 2 n and $$SO_{2n+1}$$ S O 2 n + 1 , and also providing a new simple proof of Casselman–Shahidi’s theorem on generalized injectivity conjecture for regular generalized principal series. Indeed, such a beautiful structure theorem also holds for finite central covering groups.

Conjugacy classes and automorphisms of twin groups

The twin group {T_{n}} is a right-angled Coxeter group generated by {n-1} involutions, and the pure twin group {\mathrm{PT}_{n}} is the kernel of the natural surjection from {T_{n}} onto the symmetric group on n symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conjugacy classes of involutions in {T_{n}}, which, quite interestingly, is related to the well-known Fibonacci sequence. We also derive a recursive formula for the number of z-classes of involutions in {T_{n}}. We give a new proof of the structure of {\operatorname{Aut}(T_{n})} for {n\geq 3}, and show that {T_{n}} is isomorphic to a subgroup of {\operatorname{Aut}(\mathrm{PT}_{n})} for {n\geq 4}. Finally, we construct a representation of {T_{n}} to {\operatorname{Aut}(F_{n})} for {n\geq 2}.

Random coxeter groups

Much is known about random right-angled Coxeter groups (i.e., right-angled Coxeter groups whose defining graphs are random graphs under the Erdös–Rényi model). In this paper, we extend this model to study random general Coxeter groups and give some results about random Coxeter groups, including some information about the homology of the nerve of a random Coxeter group and results about when random Coxeter groups are [Formula: see text]-hyperbolic and when they have the FC-type property.

Local finiteness of the twisted Bruhat orders on affine Weyl groups

In this paper, we investigate various properties of strong and weak twisted Bruhat orders on a Coxeter group. In particular, we prove that any twisted strong Bruhat order on an affine Weyl group is locally finite, strengthening a result of Dyer [Quotients of twisted Bruhat orders, J. Algebra163 (1994), 3, 861–879]. We also show that, for a non-finite and non-cofinite biclosed set 𝐵 in the positive system of an affine root system with rank greater than 2, the set of elements having a fixed 𝐵-twisted length is infinite. This implies that the twisted strong and weak Bruhat orders have an infinite antichain in those cases. Finally, we show that twisted weak Bruhat order can be applied to the study of the tope poset of an infinite oriented matroid arising from an affine root system.

Some remarks on twin groups

The twin group [Formula: see text] is a right angled Coxeter group generated by [Formula: see text] involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this paper, we study some properties of twin groups whose analogues are well known for Artin braid groups. We give an algorithm for two twins to be equivalent under individual Markov moves. Further, we show that twin groups [Formula: see text] have [Formula: see text]-property and are not co-Hopfian for [Formula: see text].

Weak Generalized Lifting Property, Bruhat Intervals, and Coxeter Matroids

We provide a weaker version of the generalized lifting property that holds in complete generality for all Coxeter groups, and we use it to show that every parabolic Bruhat interval of a finite Coxeter group is a Coxeter matroid. We also describe some combinatorial properties of the associated polytopes.

The facial weak order in finite Coxeter groups

International audience We investigate a poset structure that extends the weak order on a finite Coxeter group W to the set of all faces of the permutahedron of W. We call this order the facial weak order. We first provide two alternative characterizations of this poset: a first one, geometric, that generalizes the notion of inversion sets of roots, and a second one, combinatorial, that uses comparisons of the minimal and maximal length representatives of the cosets. These characterizations are then used to show that the facial weak order is in fact a lattice, generalizing a well-known result of A. Bjo ̈rner for the classical weak order. Finally, we show that any lattice congruence of the classical weak order induces a lattice congruence of the facial weak order, and we give a geometric interpretation of its classes.