Classification of Vertex-Transitive Zonotopes

We begin this paper with a short survey on finite reflection groups. First we establish what a reflection in Euclidean space is. Then we introduce a root system, which is then partitioned into two sets: one of positive roots and one with negative roots. Th is articulates our understanding of groups generated by simple reflections. Furthermore, we develop our insight to Weyl groups and crystallographic groups before exploring crystallographic root systems. The section section of this paper examines the twisted group algebra along with the Demazure element Xi and the Demazure-Lusztig element Ti. Lastly, the third section of this paper computes such Xi and Ti in the case of S3 where S3 is the symmetric group Sn, n=3.

Let $W$ be a finite reflection group associated with a root system $R$ in $\mathbb R^d$. Let $C_+$ denote a positive Weyl chamber. Consider an open subset $Ω$ of $\mathbb R^d$, symmetric with respect to reflections from $W$. Let $Ω_+=Ω\cap C_+$ be the positive part of $Ω$. We define a family $\{-Δ_η^+\}$ of self-adjoint extensions of the Laplacian $-Δ_{Ω_+}$, labeled by homomorphisms $η\colon W\to \{1,-1\}$. In the construction of these $η$-Laplacians $η$-symmetrization of functions on $Ω$ is involved. The Neumann Laplacian $-Δ_{N,Ω_+}$ is included and corresponds to $η\equiv1$. If $H^{1}(Ω)=H^{1}_0(Ω)$, then the Dirichlet Laplacian $-Δ_{D,Ω_+}$ is either included and corresponds to $η={\rm sgn}$; otherwise the Dirichlet Laplacian is considered separately. Applying the spectral functional calculus we consider the pairs of operators $Ψ(-Δ_{N,Ω})$ and $Ψ(-Δ_η^+)$, or $Ψ(-Δ_{D,Ω})$ and $Ψ(-Δ_{D,Ω_+})$, where $Ψ$ is a Borel function on $[0,\infty)$. We prove relations between the integral kernels for the operators in these pairs, which are given in terms of symmetries governed by $W$.

A finite group W generated by reflections is called a finite reflection group and have been studied deeply in different important contexts such as geometry and group theory, matrix algebra and Lie Theory. In this paper we study a class of fuzzy subgroups of finite reflection groups which are called parabolic fuzzy subgroups using preferential equality. In the classical case parabolic subgroups are associated with a set of generators, each a simple reflection, indexed by elements of subsets of a fixed root system Δ . We establish a one-to-one correspondence between the class of fuzzy parabolic subgroups of W and the class of fuzzy subsets of S where S is a set of simple reflections associated with Δ .

On elliptic Calogero–Moser systems for complex crystallographic reflection groups

Variants of some of the Brauer-Fowler theorems

In this paper, we study $k$-parabolic arrangements, a generalization of $k$-equal arrangements for finite real reflection groups. When $k=2$, these arrangements correspond to the well-studied Coxeter arrangements. Brieskorn (1971) showed that the fundamental group of the complement, over $\mathbb {C}$, of the type $W$ Coxeter arrangement is isomorphic to the pure Artin group of type $W$. Khovanov (1996) gave an algebraic description for the fundamental group of the complement, over $\mathbb {R}$, of the $3$-equal arrangement. We generalize Khovanov’s result to obtain an algebraic description of the fundamental groups of the complements of $3$-parabolic arrangements for arbitrary finite reflection groups. Our description is a real analogue to Brieskorn’s description.

Test sets for nonnegativity of polynomials invariant under a finite reflection group

Using Cohen’s classification of symplectic reflection groups, we prove that the parabolic subgroups, that is, stabilizer subgroups, of a finite symplectic reflection group, are themselves symplectic reflection groups. This is the symplectic analog of Steinberg’s Theorem for complex reflection groups.Using computational results required in the proof, we show the nonexistence of symplectic resolutions for symplectic quotient singularities corresponding to three exceptional symplectic reflection groups, thus reducing further the number of cases for which the existence question remains open.Another immediate consequence of our result is that the singular locus of the symplectic quotient singularity associated to a symplectic reflection group is pure of codimension two.

Finite complex reflection groups have the remarkable property that the character field k of their reflection representation is a splitting field, that is, every irreducible complex representation can be realized over k. Here we show that this statement remains true for extensions of finite complex reflection groups by elements in their normalizer. Also, we generalize the corresponding result for cyclotomic Hecke algebras to Hecke algebras attached to extended finite complex reflection groups.

In this talk we introduce generalised Calogero–Moser models and demonstrate their integrability by constructing universal Lax pair operators. These include models based on non-crystallographic root systems, that is the root systems of the finite reflection groups, H 3, H 4, and the dihedral group I 2(m), besides the well-known ones based on crystallographic root systems, namely those associated with Lie algebras. Universal Lax pair operators for all of the generalised Calogero–Moser models and for any choices of the potentials are linear combinations of the reflection operators. The equivalence of the Lax pair with the equations of motion is proved by decomposing the root system into a sum of two-dimensional sub-root systems, A 2, B 2, G 2, and I 2(m). The root type and the minimal type Lax pairs, derived in our previous papers, are given as the simplest representations. The spectral parameter dependence plays an important role in the Lax pair operators, which bear a strong resemblance to the Dunkl operators.

The Dunkl differential-difference operator associated with a finite reflection group is used to extend the Weyl–Heisenberg algebra. The subalgebra of mastersymmetries graded by the characters of the Coxeter group W is constructed. It provides us with a new class of intertwining operators for appropriate W-invariant differential equations. These intertwining operators turn out to be suitable for constructing fundamental solutions. The remarkable example is given by the iterated wave operator with a Calogero-type potential for which the fundamental solution can be derived through the relevant intertwining operator in an explicit form. It is found that under appropriate conditions such an equation satisfies Huygens’s principle in the sense that its fundamental solution possesses a nontrivial (inner) lacuna.

In 1960, Sobolev proved that for a finite reflection group G, a G-invariant cubature formula is of degree t if and only if it is exact for all G-invariant polynomials of degree at most t . In this paper, we make some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and, moreover, gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007), which classifies tight Euclidean designs invariant under the Weyl group of type B, to other finite reflection groups.

We study the counterpart of a direct sum decomposition of L2(Rd) into subspaces which are invariant under the classical Fourier transform for the Dunkl transform associated with a family of weight functions hκ which keep invariant under a given finite reflection group. The explicit Dunkl transform formula for each component of a function in L2(Rd; h2 κ) is established, which generalizes Bochner’s formulae for spherical harmonics and spherical monogenics in Dunkl case.

G-majorization with applications to matrix orderings

For a family of weight functions invariant under a finite reflection group, we prove a transference theorem between the L p multiplier of h-harmonic expansions on \(\mathbb{S}^{d}\) and that of the Dunkl transform. This theorem is stated together with some related definitions and notations in Section 7.1. The proof of this transference theorem is, however, rather long, so we split it into three parts, which are given in the Sections 7.2, 7.3, and 7.4, respectively.