Crystallographic Reflection Research Articles

Generalizations of Euler's Tonnetz on triangulated surfaces

We give a definition of a what we call a “tonnetz” on a triangulated surface, generalizing the famous Tonnetz of Euler (Euler, Leonhard. 1739. Tentamen novae theoriae musicae ex certissismis harmoniae principiis dilucide expositae. Saint Petersburg Academy, 147). In the modern interpretation of Euler’s Tonnetz, the vertices of a regular “ A 2 triangulation” of the plane are labelled with notes, or pitch-classes. In our generalization, we allow much more general labellings of triangulated surfaces. In particular, edge labellings turn out to lead to a rich set of examples. We construct natural examples that are related to crystallographic reflection groups and live on triangulations of tori. Underlying these, we observe a curious relationship between the mathematical Langlands duality and major/minor duality. We also construct “exotic” type- A 2 examples (different from Euler’s Tonnetz), and a tonnetz on a sphere that encodes all major ninth chords.

Steinberg's fixed point theorem for crystallographic complex reflection groups

Steinberg's fixed point theorem states that given a finite complex reflection group the stabilizer subgroup of a point is generated by reflections that fix this point. This statement is also true for affine Weyl groups. Of the infinite discrete complex reflection groups, it was shown that there are some infinite complex reflections groups that have non-trivial stabilizers that do not contain a single reflection, and therefore, these groups cannot satisfy the fixed point theorem. We thus classify the infinite discrete irreducible complex reflection groups of the infinite family which satisfy the statement of the fixed point theorem.

Exceptional moduli spaces for exceptional mathcal{N} = 3 theories

It is expected on general grounds that the moduli space of 4d mathcal{N} = 3 theories is of the form ℂ3r/Γ, with r the rank and Γ a crystallographic complex reflection group (CCRG). As in the case of Lie algebras, the space of CCRGs consists of several infinite families, together with some exceptionals. To date, no 4d mathcal{N} = 3 theory with moduli space labelled by an exceptional CCRG (excluding Weyl groups) has been identified. In this work we show that the 4d mathcal{N} = 3 theories proposed in [1], constructed via non-geometric quotients of type- mathfrak{e} 6d (2,0) theories, realize nearly all such exceptional moduli spaces. In addition, we introduce an extension of this construction to allow for twists and quotients by outer automorphism symmetries. This gives new examples of 4d mathcal{N} = 3 theories going beyond simple S-folds.

X-ray diffraction data from shock-compressed copper: Some consequences of metallurgical texture

We report the measurements of in situ Debye–Scherrer x-ray diffraction from copper foils shock compressed at the Orion laser facility to pressure in the range of 10–40 GPa. Our objective was to record distortion (variation of scattering angle at peak intensity, 2θ, with azimuthal position, φ, around the diffraction ring) of the Debye–Scherrer rings. We intended to measure the anisotropy of elastic strain and infer the effective strength of copper at a high strain rate. However, our measured diffraction data from all crystallographic reflection planes considered together are not consistent with a simple model that assumes homogeneous elastic strain. Consideration of both the β-fiber metallurgical texture of the rolled copper foil that we used as the sample material and the measured diffraction linewidths provides an empirical understanding of the data. We extend our understanding by using a Taylor-type, single-crystal plasticity model in which the total strain of each grain is assumed to be identical to that of the whole sample. This model reproduces many features of our experimental data and points to the importance of accounting for the plastic anisotropy of single-crystal grains, which can, in turn, lead to inter-grain elastic strain inhomogeneity and complex distortions of the diffraction rings.

Macdonald indices for four-dimensional N=3 theories

We brute-force evaluate the vacuum character for $\mathcal N=2$ vertex operator algebras labelled by crystallographic complex reflection groups $G(k,1,1)=\mathbb Z_k$, $k=3,4,6$, and $G(3,1,2)$. For $\mathbb Z_{3,4}$ and $G(3,1,2)$ these vacuum characters have been conjectured to respectively reproduce the Macdonald limit of the superconformal index for rank one and rank two S-fold $\mathcal N=3$ theories in four dimensions. For the $\mathbb Z_3$ case, and in the limit where the Macdonald index reduces to the Schur index, we find agreement with predictions from the literature.

Central Splitting of A2 Discrete Fourier–Weyl Transforms

Two types of bivariate discrete weight lattice Fourier–Weyl transforms are related by the central splitting decomposition. The two-variable symmetric and antisymmetric Weyl orbit functions of the crystallographic reflection group A2 constitute the kernels of the considered transforms. The central splitting of any function carrying the data into a sum of components governed by the number of elements of the center of A2 is employed to reduce the original weight lattice Fourier–Weyl transform into the corresponding weight lattice splitting transforms. The weight lattice elements intersecting with one-third of the fundamental region of the affine Weyl group determine the point set of the splitting transforms. The unitary matrix decompositions of the normalized weight lattice Fourier–Weyl transforms are presented. The interpolating behavior and the unitary transform matrices of the weight lattice splitting Fourier–Weyl transforms are exemplified.

N = 3 SCFTs in 4 dimensions and non-simply laced groups

In this paper we discuss various N = 3 SCFTs in 4 dimensions and in particular those which can be obtained as a discrete gauging of an N = 4 SYM theories with non- simply laced groups. The main goal of the project was to compute the Coulomb branch superconformal index and moduli space Hilbert series for the N = 3 SCFTs that are obtained from gauging a discrete subgroup of the global symmetry group of N = 4 Super Yang-Mills theory. The discrete subgroup contains elements of both SU(4) R-symmetry group and the S-duality group of N = 4 SYM. This computation was done for the simply laced groups (where the S-duality groups is SL(2, ℤ) and Langlands dual of the algebra {}^Lmathfrak{g} is simply mathfrak{g} ) by Bourton et al. [1], and we extended it to the non-simply laced groups. We also considered the orbifolding groups of the Coulomb branch for the cases when Coulomb branch is relatively simple; in particular, we compared them with the results of Argyres et al. [2], who classified all N≥ 3 moduli space orbifold geometries at rank 2 and with the results of Bonetti et al. [3], who listed all possible orbifolding groups for the freely generated Coulomb branches of N≥ 3 SCFTs. Finally, we have considered sporadic complex crystallographic reflection groups with rank greater than 2 and analyzed, which of them can correspond to an N = 3 SCFT with a principal Dirac pairing.

VOAs labelled by complex reflection groups and 4d SCFTs

We define and study a class of mathcal{N} = 2 vertex operator algebras {mathcal{W}}_{mathrm{G}} labelled by complex reflection groups. They are extensions of the mathcal{N} = 2 super Virasoro algebra obtained by introducing additional generators, in correspondence with the invariants of the complex reflection group G. If G is a Coxeter group, the mathcal{N} = 2 super Virasoro algebra enhances to the (small) mathcal{N} = 4 superconformal algebra. With the exception of G = ℤ2, which corresponds to just the mathcal{N} = 4 algebra, these are non-deformable VOAs that exist only for a specific negative value of the central charge. We describe a free-field realization of {mathcal{W}}_{mathrm{G}} in terms of rank(G) βγbc ghost systems, generalizing a construction of Adamovic for the mathcal{N} = 4 algebra at c = −9. If G is a Weyl group, {mathcal{W}}_{mathrm{G}} is believed to coincide with the mathcal{N} = 4 VOA that arises from the four-dimensional super Yang-Mills theory whose gauge algebra has Weyl group G. More generally, if G is a crystallographic complex reflection group, {mathcal{W}}_{mathrm{G}} is conjecturally associated to an mathcal{N} = 3 4d superconformal field theory. The free-field realization allows to determine the elusive “R-filtration” of {mathcal{W}}_{mathrm{G}} , and thus to recover the full Macdonald index of the parent 4d theory.

Steinberg's theorem for crystallographic complex reflection groups

Popov classified crystallographic complex reflection groups by determining lattices they stabilize. These analogs of affine Weyl groups have infinite order and are generated by reflections about affine hyperplanes; most arise as the semi-direct product of a finite complex reflection group and a full rank lattice. Steinberg's regularity theorem asserts that the regular orbits under the action of a reflection group are exactly the orbits lying off of reflecting hyperplanes. This theorem holds for finite reflection groups (real or complex) and also affine Weyl groups but fails for some crystallographic complex reflection groups. We determine when Steinberg's regularity theorem holds for the infinite family of crystallographic complex reflection groups. We include crystallographic groups built on finite Coxeter groups.

X-ray diffraction and X-ray standing-wave study of the lead stearate film structure

A new approach to the study of the structural quality of crystals is proposed. It is based on the use of X-ray standing-wave method without measuring secondary processes and considers the multiwave interaction of diffraction reflections corresponding to different harmonics of the same crystallographic reflection. A theory of multiwave X-ray diffraction is developed to calculate the rocking curves in the X-ray diffraction scheme under consideration for a long-period quasi-one-dimensional crystal. This phase-sensitive method is used to study the structure of a multilayer lead stearate film on a silicon substrate. Some specific structural features are revealed for the surface layer of the thin film, which are most likely due to the tilt of the upper layer molecules with respect to the external normal to the film surface.

Reflection groups and discrete integrable systems

We present a method of constructing discrete integrable systems with crystallographic reflection group (Weyl) symmetries, thus clarifying the relationship between different discrete integrable systems in terms of their symmetry groups. Discrete integrable systems are associated with space-filling polytopes arise from the geometric representation of the Weyl groups in the $n$-dimensional real Euclidean space $\mathbb{R}^n$. The ‘multi-dimensional consistency’ property of the discrete integrable system is shown to be inherited from the combinatorial properties of the polytope; while the dynamics of the system is described by the affine translations of the polytopes on the weight lattices of the Weyl groups. The connections between some well-known discrete systems such as the multi-dimensional consistent systems of quad-equations and discrete Painleve equations are obtained via the geometric constraints that relate the polytope of one symmetry group to that of another symmetry group, a procedure which we call geometric reduction.

Reflection group presentations arising from cluster algebras

We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system.

Lie theory and coverings of finite groups

We introduce the notion of an ‘inverse property’ (IP) quandle C which we propose as the right notion of ‘Lie algebra’ in the category of sets. For any IP-quandle we construct an associated group GC. For a class of IP-quandles which we call ‘locally skew’, and when GC is finite, we show that the noncommutative de Rham cohomology H1(GC) is trivial aside from a single generator θ that has no classical analogue. If we start with a group G then any subset C⊆G∖{e} which is ad-stable and inversion-stable naturally has the structure of an IP-quandle. If C also generates G then we show that GC↠G with central kernel, in analogy with the similar result for the simply-connected covering group of a Lie group. We prove that this ‘covering map’ GC↠G is an isomorphism for all finite crystallographic reflection groups W with C the set of reflections, and that C is locally skew precisely in the simply laced case. This implies that H1(W)=k when W is simply laced, proving in particular a conjecture for Sn in Majid (2004) [12]. We also consider C=ZP1∪ZP1 as a locally skew IP-quandle ‘Lie algebra’ of SL2(Z) and show that GC≅B3, the braid group on 3 strands. The map B3↠SL2(Z) which therefore arises naturally as a covering map in our theory, coincides with the restriction of the usual universal covering map SL2(R)˜→SL2(R) to the inverse image of SL2(Z).

Coxeter and crystallographic arrangements are inductively free

Using the classification of finite Weyl groupoids we prove that crystallographic arrangements, a large subclass of the class of simplicial arrangements which was recently defined, are hereditarily inductively free. In particular, all crystallographic reflection arrangements are hereditarily inductively free, among them the arrangement of type E 8 . With little extra work we prove that also all Coxeter arrangements are inductively free.

From m-clusters to m-noncrossing partitions via exceptional sequences

Let W be a finite crystallographic reflection group. The generalized Catalan number of W coincides both with the number of clusters in the cluster algebra associated to W, and with the number of noncrossing partitions for W. Natural bijections between these two sets are known. For any positive integer m, both m-clusters and m-noncrossing partitions have been defined, and the cardinality of both these sets is the Fuss–Catalan number C m (W). We give a natural bijection between these two sets by first establishing a bijection between two particular sets of exceptional sequences in the bounded derived category D b(H) for any finite-dimensional hereditary algebra H.

On elliptic Calogero–Moser systems for complex crystallographic reflection groups

To every irreducible finite crystallographic reflection group (i.e., an irreducible finite reflection group G acting faithfully on an abelian variety X), we attach a family of classical and quantum integrable systems on X (with meromorphic coefficients). These families are parametrized by G-invariant functions of pairs ( T , s ) , where T is a hypertorus in X (of codimension 1), and s ∈ G is a reflection acting trivially on T. If G is a real reflection group, these families reduce to the known generalizations of elliptic Calogero–Moser systems, but in the non-real case they appear to be new. We give two constructions of the integrals of these systems – an explicit construction as limits of classical Calogero–Moser Hamiltonians of elliptic Dunkl operators as the dynamical parameter goes to 0 (implementing an idea of V. Buchstaber, G. Felder and A. Veselov (1994) [BFV]), and a geometric construction as global sections of sheaves of elliptic Cherednik algebras for the critical value of the twisting parameter. We also prove algebraic integrability of these systems for values of parameters satisfying certain integrality conditions.

$m$-noncrossing partitions and $m$-clusters

Let $W$ be a finite crystallographic reflection group, with root system $\Phi$. Associated to $W$ there is a positive integer, the generalized Catalan number, which counts the clusters in the associated cluster algebra, the noncrossing partitions for $W$, and several other interesting sets. Bijections have been found between the clusters and the noncrossing partitions by Reading and Athanasiadis et al. There is a further generalization of the generalized Catalan number, sometimes called the Fuss-Catalan number for $W$, which we will denote $C_m(W)$. Here $m$ is a positive integer, and $C_1(W)$ is the usual generalized Catalan number. $C_m(W)$ counts the $m$-noncrossing partitions for $W$ and the $m$-clusters for $\Phi$. In this abstract, we will give an explicit description of a bijection between these two sets. The proof depends on a representation-theoretic reinterpretation of the problem, in terms of exceptional sequences of representations of quivers. Soit $W$ un groupe de réflexions fini et cristallographique, avec système de racines $\Phi$. Associé à $W$, il y a un entier positif, le nombre de Catalan généralisé, qui compte les amas dans l'algèbre amassée associée, les partitions non-croisées de $W$, et plusieurs autres ensembles intéressantes. Des bijections entre les amas et les partitions non-croisées ont été données par Reading et Athanasiadis et al. On peut encore généraliser le nombre de Catalan généralisé, obtenant le nombre Fuss-Catalan de $W$, que nous noterons $C_m(W)$. Ici $m$ est un entier positif, et $C_1(W)$ est le nombre Catalan généralisé standard. $C_m(W)$ compte les partitions $m$-non-croisées de $W$ et les $m$-amas de $\Phi$. Dans ce résumé, nous donnerons une bijection explicite entre ces deux ensembles. La démonstration dépend d'une réinterprétation des objets du point de vue des suites exceptionnelles de représentations de carquois.

Aqua-di-μ2-chloro-bis(diethylenetriamine)dicopper(II) dichloride

Reaction of CuCl2(H2O)2 with diethyl­enetri­amine (henceforth dien) in a 1:1 ratio has led to the formation of a non-centrosymmetric di-μ-chloro bibridged binuclear CuII species in which a water mol­ecule is coordinated axially to one of the CuII ions. [Cu2Cl2(C4H13N3)2(H2O)] contains dinuclear [(dien)2Cu2Cl2(H2O)]2+ cations and chloride ions. Both CuII ions in the dimer have a primary coordination sphere that is approximately planar, consisting of one dien mol­ecule and one chloride ion. These monomeric units are linked via semicoordinate Cu—Cl bonds to form dinuclear units. One CuII ion has a 4+1 coordination geometry while the second expands to a 4+2 geometry by the incorporation of a water mol­ecule. The dinuclear cation has crystallographic reflection symmetry, the mirror plane containing the central Cu2Cl2 unit and the central N atom of each dien ligand.

On the Poincaré series and cardinalities of finite reflection groups

Let W W be a crystallographic reflection group with length function ℓ ( ⋅ ) \ell (\cdot ) . We give a short and elementary derivation of the identity ∑ w ∈ W q ℓ ( w ) = ∏ ( 1 − q ht ⁡ ( α ) + 1 ) / ( 1 − q ht ⁡ ( α ) ) \sum _{w\in W}q^{\ell (w)}=\prod (1-q^{\operatorname {ht} (\alpha )+1})/(1-q^{\operatorname {ht}(\alpha )}) , where the product ranges over positive roots α \alpha , and ht ⁡ ( α ) \operatorname {ht} (\alpha ) denotes the sum of the coordinates of α \alpha with respect to the simple roots. We also prove that in the noncrystallographic case, this identity is valid in the limit q → 1 q\to 1 ; i.e., | W | = ∏ ( ht ⁡ ( α ) + 1 ) / ht ⁡ ( α ) |W|=\prod (\operatorname {ht} (\alpha )+1)/\operatorname {ht}(\alpha ) .

Presentations for crystallographic complex reflection groups

We find presentations for the irreducible crystallographic complex reflection groupsW whose linear part is not the complexification of a real reflection group. The presentations are given in the form of graphs resembling Dynkin diagrams and very similar to the presentations for finite complex reflection groups given in [2]. As in the case of affine Weyl groups, they can be obtained by adding a further node to the diagram for the linear part. We then classify the reflections in the groupsW and the minimal number of them needed to generateW, using the diagrams. Finally we show for more than half of the infinite series that a presentation for the fundamental group of the space of regular orbits ofW can be derived from our presentations.