Coxeter-Dynkin Diagram Research Articles

FCC, BCC and SC Lattices Derived from the Coxeter-Weyl groups and quaternions

We construct the fcc (face centered cubic), bcc (body centered cubic) and sc (simple cubic) lattices as the root and the weight lattices of the affine extended Coxeter groups W(A3) and W(B3)=Aut(A3). It is naturally expected that these rank-3 Coxeter-Weyl groups define the point tetrahedral symmetry and the octahedral symmetry of the cubic lattices which have extensive applications in material science. The imaginary quaternionic units are used to represent the root systems of the rank-3 Coxeter-Dynkin diagrams which correspond to the generating vectors of the lattices of interest. The group elements are written explicitly in terms of pairs of quaternions which constitute the binary octahedral group. The constructions of the vertices of the Wigner-Seitz cells have been presented in terms of quaternionic imaginary units. This is a new approach which may link the lattice dynamics with quaternion physics. Orthogonal projections of the lattices onto the Coxeter plane represent the square and honeycomb lattices.

A geometric definition of Gabrielov numbers

Gabrielov numbers describe certain Coxeter–Dynkin diagrams of the 14 exceptional unimodal singularities and play a role in Arnold’s strange duality. In a previous paper, the authors defined Gabrielov numbers of a cusp singularity with an action of a finite abelian subgroup $$G$$ of $$\mathrm{SL}(3,{\mathbb {C}})$$ using the Gabrielov numbers of the cusp singularity and data of the group $$G$$ . Here we consider a crepant resolution $$Y \rightarrow {\mathbb {C}}^3/G$$ and the preimage $$Z$$ of the image of the Milnor fibre of the cusp singularity under the natural projection $${\mathbb {C}}^3 \rightarrow {\mathbb {C}}^3/G$$ . Using the McKay correspondence, we compute the homology of the pair $$(Y,Z)$$ . We construct a basis of the relative homology group $$H_3(Y,Z;{\mathbb {Q}})$$ with a Coxeter–Dynkin diagram where one can read off the Gabrielov numbers.

Affine extensions of non-crystallographic Coxeter groups induced by projection

In this paper, we show that affine extensions of non-crystallographic Coxeter groups can be derived via Coxeter-Dynkin diagram foldings and projections of affine extended versions of the root systems E8, D6, and A4. We show that the induced affine extensions of the non-crystallographic groups H4, H3, and H2 correspond to a distinguished subset of those considered in [P.-P. Dechant, C. Bœhm, and R. Twarock, J. Phys. A: Math. Theor. 45, 285202 (2012)]. This class of extensions was motivated by physical applications in icosahedral systems in biology (viruses), physics (quasicrystals), and chemistry (fullerenes). By connecting these here to extensions of E8, D6, and A4, we place them into the broader context of crystallographic lattices such as E8, suggesting their potential for applications in high energy physics, integrable systems, and modular form theory. By inverting the projection, we make the case for admitting different number fields in the Cartan matrix, which could open up enticing possibilities in hyperbolic geometry and rational conformal field theory.

SNUB 24-CELL DERIVED FROM THE COXETER–WEYL GROUP W(D4)

Snub 24-cell is the unique uniform chiral polytope in four dimensions consisting of 24 icosahedral and 120 tetrahedral cells. The vertices of the four-dimensional semiregular polytope snub 24-cell and its symmetry group (W(D4)/C2): S3 of order 576 are obtained from the quaternionic representation of the Coxeter–Weyl group W(D4). The symmetry group is an extension of the proper subgroup of the Coxeter–Weyl group W(D4) by the permutation symmetry of the Coxeter–Dynkin diagram D4. The 96 vertices of the snub 24-cell are obtained as the orbit of the group when it acts on the vector Λ = (τ, 1, τ, τ) or on the vector Λ = (σ, 1, σ, σ) in the Dynkin basis with [Formula: see text] and [Formula: see text]. The two different sets represent the mirror images of the snub 24-cell. When two mirror images are combined it leads to a quasiregular four-dimensional polytope invariant under the Coxeter–Weyl group W(F4). Each vertex of the new polytope is shared by one cube and three truncated octahedra. Dual of the snub 24-cell is also constructed. Relevance of these structures to the Coxeter groups W(H4) and W(E8) has been pointed out.

A geometric construction of Coxeter-Dynkin diagrams of bimodal singularities

We consider the Berglund-Hubsch transpose of a bimodal invertible polynomial and construct a triangulated category associated to the compactification of a suitable deformation of the singularity. This is done in such a way that the corresponding Grothendieck group with the (negative) Euler form can be described by a graph which corresponds to the Coxeter-Dynkin diagram with respect to a distinguished basis of vanishing cycles of the bimodal singularity.

Delaunay and voronoi polytopes of the root lattice E 7 and of the dual lattice E*7

We give a detailed geometrically clear description of all faces of the Delaunay and Voronoi polytopes of the root lattice E7 and the dual lattice E*7. Here three uniform polytopes related to the Coxeter-Dynkin diagram of the Lie algebra E7 play a special role. These are the Gosset polytope PGos = 321, which is a Delaunay polytope, the contact polytope 231 (both for the lattice E7), and the Voronoi polytope PV(E*7) = 132 of the dual lattice E*7. This paper can be considered as an illustration of the methods for studying Delaunay and Voronoi polytopes.

4d-polytopes described by Coxeter diagrams and quaternions

4D-polytopes and their dual polytopes can be described as the orbits of the rank-4 Coxeter-Weyl groups. Their symmetries follow from the quaternionic descriptions of the rank-4 Coxeter-Dynkin diagrams. There exists a one to one correspondence between the finite subgroups of quaternions and the rank-4 Coxeter-Weyl groups.

Catalan solids derived from three-dimensional-root systems and quaternions

Catalan solids are the duals of the Archimedean solids, the vertices of which can be obtained from the Coxeter–Dynkin diagrams A3, B3, and H3 whose simple roots can be represented by quaternions. The respective Weyl groups W(A3), W(B3), and W(H3) acting on the highest weights generate the orbits corresponding to the solids possessing these symmetries. Vertices of the Platonic and Archimedean solids result from the orbits derived from fundamental weights. The Platonic solids are dual to each other; however, the duals of the Archimedean solids are the Catalan solids whose vertices can be written as the union of the orbits, up to some scale factors, obtained by applying the above Weyl groups on the fundamental highest weights (100), (010), and (011) for each diagram. The faces are represented by the orbits derived from the weights (010), (110), (101), (011), and (111), which correspond to the vertices of the Archimedean solids. Representations of the Weyl groups W(A3), W(B3), and W(H3) by the quaternions simplify the calculations with no reference to the computer calculations.

Polyhedra obtained from Coxeter groups and quaternions

We note that all regular and semiregular polytopes in arbitrary dimensions can be obtained from the Coxeter-Dynkin diagrams. The vertices of a regular or semiregular polytope are the weights obtained as the orbit of the Coxeter-Weyl group acting on the highest weight representing a selected irreducible representation of the Lie group. This paper, in particular, deals with the determination of the vertices of the Platonic and Archimedean solids from the Coxeter diagrams A3, B3, and H3 in the context of the quaternionic representations of the root systems and the Coxeter-Weyl groups. We use Lie algebraic techniques in the derivation of vertices of the polyhedra and show that the polyhedra possessing the tetrahedral, octahedral, and icosahedral symmetries are related to the Coxeter-Weyl groups representing the symmetries of the diagrams of A3, B3, and H3, respectively. This technique leads to the determination of the vertices of all Platonic and Archimedean solids except two chiral polyhedra, snubcuboctahedron and snubicosidodecahedron.

On Polynomials Interpolating Between the Stationary State of a O(n) Model and a Q.H.E. Ground State

We obtain a family of polynomials defined by vanishing conditions and associated to tangles. We study more specifically the case where they are related to a O(n) loop model. We conjecture that their specializations at z i = 1 are positive in n. At n = 1, they coincide with the Razumov-Stroganov integers counting alternating sign matrices. We derive the CFT modular invariant partition functions labelled by Coxeter-Dynkin diagrams using the representation theory of the affine Hecke algebras.

Antisymmetric Orbit Functions

In the paper, properties of antisymmetric orbit functions are reviewed and further developed. Antisymmetric orbit functions on the Euclidean space $E_n$ are antisymmetrized exponential functions. Antisymmetrization is fulfilled by a Weyl group, corresponding to a Coxeter-Dynkin diagram. Properties of such functions are described. These functions are closely related to irreducible characters of a compact semisimple Lie group $G$ of rank $n$. Up to a sign, values of antisymmetric orbit functions are repeated on copies of the fundamental domain $F$ of the affine Weyl group (determined by the initial Weyl group) in the entire Euclidean space $E_n$. Antisymmetric orbit functions are solutions of the corresponding Laplace equation in $E_n$, vanishing on the boundary of the fundamental domain $F$. Antisymmetric orbit functions determine a so-called antisymmetrized Fourier transform which is closely related to expansions of central functions in characters of irreducible representations of the group $G$. They also determine a transform on a finite set of points of $F$ (the discrete antisymmetric orbit function transform). Symmetric and antisymmetric multivariate exponential, sine and cosine discrete transforms are given.

Quantum affine Cartan matrices, Poincaré series of binary polyhedral groups, and reflection representations

We first review some invariant theoretic results about the finite subgroups of SU(2) in a quick algebraic way by using the McKay correspondence and quantum affine Cartan matrices. By the way it turns out that some parameters (a, b, h; p, q, r) that one usually associates with such a group and hence with a simply-laced Coxeter–Dynkin diagram have a meaningful definition for the non-simply-laced diagrams, too, and as a byproduct we extend Saito’s formula for the determinant of the Cartan matrix to all cases. Returning to invariant theory we show that for each irreducible representation i of a binary tetrahedral, octahedral, or icosahedral group one can find a homomorphism into a finite complex reflection group whose defining reflection representation restricts to i.

Coxeter-Dynkin diagrams of some space curve singularities

The so called wedge singularities, that consist of a plane curve singularity C and a line transverse to the plane of C, are the simplest space curve singularities which are not a complete intersection. We show that for every wedge singularity X there is an isolated complete intersection singularity Y related to X and we describe the discriminant of X in terms of Y. We also show that the monodromy group of X corresponds to the one of Y.

The Automorphisms of Affine Fusion Rings

The fusion rings associated to affine Kac-Moody algebras appear in several different contexts in math and mathematical physics. In this paper we find all automorphisms of all affine fusion rings, or equivalently the symmetries of the corresponding fusion coefficients. Most of these are directly related to symmetries of the corresponding Coxeter-Dynkin diagram. We also find all pairs of isomorphic affine fusion rings.

Algebraic and Geometric Isomonodromic Deformations

Using the Gauss-Manin connection (Picard-Fuchs differential equation) and a result of Malgrange, a special class of algebraic solutions to isomonodromic deformation equations, the geometric isomonodromic deformations, is defined from "families of families" of algebraic varieties. Geometric isomonodromic deformations arise naturally from combinatorial strata in the moduli spaces of elliptic surfaces over ℙ1. The complete list of geometric solutions to the Painlevé VI equation arising in this way is determined. Motivated by this construction, we define another class of algebraic isomonodromic deformations whose monodromy preserving families arise by "pullback" from (rigid) local systems. Using explicit methods from the theory of Hurwitz spaces, all such algebraic Painlevé VI solutions coming from arithmetic triangle groups are classified.

A'Campo-Gusein-Zade diagrams as partially ordered sets

The real analogues of many results about complex monodromies of singularities can be formulated and proved in terms of partial orderings on A'Campo-Gusein-Zade diagrams, the real versions of Coxeter-Dynkin diagrams of singularities. In this paper it is proved that the only diagrams among the A'Campo-Gusein-Zade diagrams of singularities that determine partially ordered sets of finite type (in the sense of representations of a quiver) are the diagrams of simple singularities. To encode the real decompositions of a singularity the analogue of Vasilev invariants turn out to be surjections of a partially ordered set onto a chain. Formulae are proved for Arnold (mod 2)-invariants of plane curves in terms of the corresponding A'Campo-Gusein-Zade diagrams. We define, in the context of higher Bruhat orders, higher partially ordered sets and we describe their connection with the higher M-Morsifications An. We also consider certain previously known results about real singularities from the point of view of partially ordered sets.

Coxeter-Dynkin diagrams of fat points in ?2 and of their stabilizations

Coxeter-Dynkin diagrams of fat points in ?2 and of their stabilizations

On -monodromy groups of singularities

In this paper some facts are proved concerning the -analogue of symmetric and skew-symmetric monodromy groups of singularities. The image of the Burau representation of the group of braids on three strings is described. For a root of 1 the image of the group of -coloured braids of a singularity in the skew-symmetric monodromy group is studied. The latter is distinguished among the -monodromy groups for the roots of 1. The orbits of the braid monodromy group over are described. A connection is established between the spectra of the generalized Cartan -matrix and the Coxeter -operator for Coxeter-Dynkin diagrams without cycles of odd length.

On a certain type of Coxeter-Dynkin diagrams of plane curve singularities

On a certain type of Coxeter-Dynkin diagrams of plane curve singularities

Bouquets of Baer modules

In this paper some transcendental numbers are used to construct infinite-dimensional indecomposable Baer modules. Let R be a ring whose category of modules has a torsion theory. An R-module, M, is Baer if every extension of M by any torsion R-module splits. In this paper, R will be a path algebra, i.e., an algebra whose basis over a field K are the vertices and paths of a directed graph. Multiplication is given by path composition. When R is a path algebra obtained from an extended Coxeter-Dynkin diagram with no oriented cycles, we characterize Baer modules of countable rank. This characterization is used to show that modules constructed from Liouville sequences yield a family, ▪ = { B n } ∞ n=0 , of Baer modules satisfying the following conditions: every extension of B m by B n splits for every pair ( m,n); if m≠ n, B m is not isomorphic to B n , while automorphisms of B n are given by multiplications by nonzero elements of K. Each B n is shown to be a submodule of a rank-one module. Another application of our characterization is the determination of the rank-one modules with the property that every submodule of infinite rank has a nonzero direct summand that is Baer. In analogy with ℵ r -free modules, we define ℵ r -Baer modules and give an example of an ℵ 1-Baer module that is not Baer. The existence of a Baer module, M, that is not a direct sum of Baer modules of countable rank is also proved. However every nonzero submodule of M has a nonzero direct summand. A problem suggested by these results is the existence and structure of indecomposable Baer modules of uncountable rank.