Marzena Szajewska

Four types of special functions of G 2 and their discretization

Properties of four infinite families of special functions of two real variables, based on the compact simple Lie group G 2, are compared and described. Two of the four families (called here C- and S-functions) are well known. New results of the paper are in description of two new families of G 2-functions not found in the literature. They are denoted as S L - and S S -functions. It is shown that all four families have analogous useful properties. In particular, they are orthogonal when integrated over a finite region F of the Euclidean space. They are also orthogonal as discrete functions when their values are sampled at the lattice points F M ⊂F and added up with appropriate weight function. The weight functions are determined for the new families. Products of 10 types among the four families of functions, namely CC, CS, SS, SS L , CS S , SS L , SS S , S S S S , S L S S and S L S L , are completely decomposable into the finite sum of the functions belonging to just one of the families. Uncommon arithmetic properties of the functions are pointed out and questions about numerous other properties are brought forward.

Breaking of icosahedral symmetry: C60 to C70.

We describe the existence and structure of large fullerenes in terms of symmetry breaking of the molecule. Specifically, we describe the existence of in terms of breaking of the icosahedral symmetry of by the insertion into its middle of an additional decagon. The surface of is formed by 12 regular pentagons and 25 regular hexagons. All 105 edges of are of the same length. It should be noted that the structure of the molecules is described in exact coordinates relative to the non-orthogonal icosahedral bases. This symmetry breaking process can be readily applied, and could account for and describe other larger cage cluster fullerene molecules, as well as more complex higher structures such as nanotubes.

C70, C80, C90 and carbon nanotubes by breaking of the icosahedral symmetry of C60.

The icosahedral symmetry group H3 of order 120 and its dihedral subgroup H2 of order 10 are used for exact geometric construction of polytopes that are known to exist in nature. The branching rule for the H3 orbit of the fullerene C60 to the subgroup H2 yields a union of eight orbits of H2: four of them are regular pentagons and four are regular decagons. By inserting into the branching rule one, two, three or n additional decagonal orbits of H2, one builds the polytopes C70, C80, C90 and nanotubes in general. A minute difference should be taken into account depending on whether an even or odd number of H2 decagons are inserted. Vertices of all the structures are given in exact coordinates relative to a non-orthogonal basis naturally appropriate for the icosahedral group, as well as relative to an orthonormal basis. Twisted fullerenes are defined. Their surface consists of 12 regular pentagons and 20 hexagons that have three and three edges of equal length. There is an uncountable number of different twisted fullerenes, all with precise icosahedral symmetry. Two examples of the twisted C60 are described.

Orthogonal polynomials of compact simple Lie groups: branching rules for polynomials

Polynomials in this paper are defined starting from a compact semisimple Lie group. A known classification of maximal, semisimple subgroups of simple Lie groups is used to select the cases to be considered here. A general method is presented and all the cases of rank ≤3 are explicitly studied. We derive the polynomials of simple Lie groups B3 and C3 as they are not available elsewhere. The results point to far reaching Lie theoretical connections to the theory of multivariable orthogonal polynomials.

Faces of Platonic solids in all dimensions

This paper considers Platonic solids/polytopes in the real Euclidean space R(n) of dimension 3 ≤ n < ∞. The Platonic solids/polytopes are described together with their faces of dimensions 0 ≤ d ≤ n - 1. Dual pairs of Platonic polytopes are considered in parallel. The underlying finite Coxeter groups are those of simple Lie algebras of types A(n), B(n), C(n), F4, also called the Weyl groups or, equivalently, crystallographic Coxeter groups, and of non-crystallographic Coxeter groups H3, H4. The method consists of recursively decorating the appropriate Coxeter-Dynkin diagram. Each recursion step provides the essential information about faces of a specific dimension. If, at each recursion step, all of the faces are in the same Coxeter group orbit, i.e. are identical, the solid is called Platonic. The main result of the paper is found in Theorem 2.1 and Propositions 3.1 and 3.2.

Icosahedral symmetry breaking: C60 to C84, C108 and to related nanotubes

This paper completes the series of three independent articles [Bodner et al. (2013). Acta Cryst. A69, 583-591, (2014), PLOS ONE, 10.1371/journal.pone.0084079] describing the breaking of icosahedral symmetry to subgroups generated by reflections in three-dimensional Euclidean space {\bb R}^3 as a mechanism of generating higher fullerenes from C60. The icosahedral symmetry of C60 can be seen as the junction of 17 orbits of a symmetric subgroup of order 4 of the icosahedral group of order 120. This subgroup is noted by A1 × A1, because it is isomorphic to the Weyl group of the semi-simple Lie algebra A1 × A1. Thirteen of the A1 × A1 orbits are rectangles and four are line segments. The orbits form a stack of parallel layers centered on the axis of C60 passing through the centers of two opposite edges between two hexagons on the surface of C60. These two edges are the only two line segment layers to appear on the surface shell. Among the 24 convex polytopes with shell formed by hexagons and 12 pentagons, having 84 vertices [Fowler & Manolopoulos (1992). Nature (London), 355, 428-430; Fowler & Manolopoulos (2007). An Atlas of Fullerenes. Dover Publications Inc.; Zhang et al. (1993). J. Chem. Phys. 98, 3095-3102], there are only two that can be identified with breaking of the H3 symmetry to A1 × A1. The remaining ones are just convex shells formed by regular hexagons and 12 pentagons without the involvement of the icosahedral symmetry.

Reduction of orbits of finite Coxeter groups of non-crystallographic type

A reduction of orbits of finite reflection groups to their reflection subgroups is produced by means of projection matrices, which transform points of the orbit of any group into points of the orbits of its subgroup. Projection matrices and branching rules for orbits of finite Coxeter groups of non-crystallographic type are presented. The novelty in this paper is producing the branching rules that involve non-crystallographic Coxeter groups. Moreover, these branching rules are relevant to any application of non-crystallographic Coxeter groups including molecular crystallography and encryption.

TWO-DIMENSIONAL HYBRIDS WITH MIXED BOUNDARY VALUE PROBLEMS

Boundary value problems are considered on a simplex F in the real Euclidean space R 2 . The recent discovery of new families of special functions, orthogonal on F , makes it possible to consider not only the Dirichlet or Neumann boundary value problems on F , but also the mixed boundary value problem which is a mixture of Dirichlet and Neumann type, ie. on some parts of the boundary of F a Dirichlet condition is fulfilled and on the other Neumann’s works.

Faces of root polytopes in all dimensions

In this paper the root polytopes of all finite reflection groups W with a connected Coxeter-Dynkin diagram in {\bb R}^n are identified, their faces of dimensions 0 ≤ d ≤ n - 1 are counted, and the construction of representatives of the appropriate W-conjugacy class is described. The method consists of recursive decoration of the appropriate Coxeter-Dynkin diagram [Champagne et al. (1995). Can. J. Phys. 73, 566-584]. Each recursion step provides the essentials of faces of a specific dimension and specific symmetry. The results can be applied to crystals of any dimension and any symmetry.

Icosahedral symmetry breaking: C60to C84, C108and to related nanotubes

This paper completes the series of three independent articles [Bodner et al. (2013). Acta Cryst. A69, 583–591, (2014), PLOS ONE, 10.1371/journal. pone.0084079] describing the breaking of icosahedral symmetry to subgroups generated by reflections in three-dimensional Euclidean space R as a mechanism of generating higher fullerenes from C60. The icosahedral symmetry of C60 can be seen as the junction of 17 orbits of a symmetric subgroup of order 4 of the icosahedral group of order 120. This subgroup is noted by A1 A1, because it is isomorphic to the Weyl group of the semi-simple Lie algebra A1 A1. Thirteen of the A1 A1 orbits are rectangles and four are line segments. The orbits form a stack of parallel layers centered on the axis of C60 passing through the centers of two opposite edges between two hexagons on the surface of C60. These two edges are the only two line segment layers to appear on the surface shell. Among the 24 convex polytopes with shell formed by hexagons and 12 pentagons, having 84 vertices [Fowler & Manolopoulos (1992). Nature (London), 355, 428–430; Fowler & Manolopoulos (2007). An Atlas of Fullerenes. Dover Publications Inc.; Zhang et al. (1993). J. Chem. Phys. 98, 3095–3102], there are only two that can be identified with breaking of the H3 symmetry to A1 A1. The remaining ones are just convex shells formed by regular hexagons and 12 pentagons without the involvement of the icosahedral symmetry.