Related papers: Logarithmic forms and anti-invariant forms of refl…

We study the space of multilinear invariants \( V_f \) of degree \( f \) for a specified finite unitary reflection group. A subspace \( W_f \) of typical invariants is also introduced. We note that the dimension of \( W_f \) is given by…

Let $x$ be an eigenvector for an element of a finite irreducible reflection group $W$. Let $W_x$ denote the subgroup of $W$ which stabilises $x$. We provide an upper bound for the number of roots in the root system of $W_x$ . This…

We consider the set $\Irr(W)$ of (complex) irreducible characters of a finite Coxeter group $W$. The Kazhdan--Lusztig theory of cells gives rise to a partition of $\Irr(W)$ into "families" and to a natural partial order $\leq_{\cLR}$ on…

In [APS], the authors characterize the partitions of $n$ whose corresponding representations of $S_n$ have nontrivial determinant. The present paper extends this work to all irreducible finite Coxeter groups $W$. Namely, given a nontrivial…

Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of…

We determine the Waring rank of the fundamental skew invariant of any complex reflection group whose highest degree is a regular number. This includes all irreducible real reflection groups.

Hilbert--Lie groups are Lie groups whose Lie algebra is a real Hilbert space whose scalar product is invariant under the adjoint action. These infinite-dimensional Lie groups are the closest relatives to compact Lie groups. Here we study…

A strongly reflective modular form with respect to an orthogonal group of signature (2,n) determines a Lorentzian Kac--Moody algebra. We find a new geometric application of such modular forms: we prove that if the weight is larger than n…

We introduce in this paper a hypercohomology version of the resonance varieties and obtain some relations to the characteristic varieties of rank one local systems on a smooth quasi-projective complex variety $M$, see Theorem (3.1) and…

For any Coxeter group W, we define a filtration of H^*(W;ZW) by W-submodules and then compute the associated graded terms. More generally, if U is a CW complex on which W acts as a reflection group we compute the associated graded terms for…

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)…

Let Y be a divisor on a smooth algebraic variety X. We investigate the geometry of the Jacobian scheme of Y, homological invariants derived from logarithmic differential forms along Y, and their relationship with the property that Y is a…

We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schr\"odinger equations admit separation of variables in polar coordinates and are exactly…

Let $ \mathbb{A}$ be a cellular algebra over a field $\mathbb{F}$ with a decomposition of the identity $ 1_{\mathbb{A}} $ into orthogonal idempotents $ e_i$, $i \in I$ (for some finite set $I$) satisfying some properties. We describe the…

We consider ideals of polynomials vanishing on the W-orbits of the intersections of mirrors of a finite reflection group W. We determine all such ideals which are invariant under the action of the corresponding rational Cherednik algebra…

We derive some local properties of abstract crystals with logarithmic poles over a smooth base in positive characteristic and obtain the existence of the canonical coordinates of certain ordinary crystals. We then apply the results to…

We consider cones of real forms which are sums of squares forms and invariant by a (finite) reflection group. We show how the representation theory of these groups allows to use the symmetry inherent in these cones to give more efficient…

We review the properties of the finite Coxeter groups which are most useful for applications to cohomological invariants, namely their classes of involutions and their "cubes" (abelian subgroups generated by reflections).

We describe a collection of computer scripts written in PARI/GP to compute, for reflection groups determined by finite-volume polyhedra in $\mathbb{H}^3$, the commensurability invariants known as the invariant trace field and invariant…

In this note, we study the polynomial representation of the quantum Olshanetsky-Perelomov system for a finite reflection group $W$ of type $B_n$. We endow the polynomial ring ${\mathbb C} [x_1,\ldots\\\ldots, x_n]$ with a structure of…