Fundamental theorems for invariants of a class of pseudo-reflection groups

Harmonic differential forms for pseudo-reflection groups I. Semi-invariants

We give a type-independent construction of an explicit basis for the semi-invariant harmonic differential forms of an arbitrary pseudo-reflection group in characteristic zero. Our "top-down" approach uses the methods of Cartan's exterior calculus and is in some sense dual to related work of Solomon, Orlik--Solomon, and Shepler describing (semi-)invariant differential forms. We apply our results to a recent conjecture of Zabrocki which provides a representation theoretic-model for the Delta conjecture of Haglund--Remmel--Wilson in terms of a certain non-commutative coinvariant algebra for the s…

Mutations of reflections and existence of pseudo-acyclic orderings for type $A_n$

In a recent paper by K.-H. Lee, K. Lee and M. Mills, a mutation of reflections in the universal Coxeter group is defined in association with a mutation of a quiver. A matrix representation of these reflections is determined by a linear ordering on the set of vertices of the quiver. It was conjectured that there exists an ordering (called a pseudo-acyclic ordering in this paper) such that whenever two mutation sequences of a quiver lead to the same labeled seed, the representations of the associated reflections also coincide. In this paper, we prove this conjecture for every quiver mutation-equ…

Harmonic differential forms for pseudo-reflection groups II. Bi-degree bounds

This paper studies three results that describe the structure of the super-coinvariant algebra of pseudo-reflection groups over a field of characteristic $0$. Our most general result determines the top component in total degree, which we prove for all Shephard--Todd groups $G(m, p, n)$ with $m \neq p$ or $m=1$. Our strongest result gives tight bi-degree bounds and is proven for all $G(m, 1, n)$, which includes the Weyl groups of types $A$ and $B$/$C$. For symmetric groups (i.e. type $A$), this provides new evidence for a recent conjecture of Zabrocki related to the Delta Conjecture of Haglund--…

Generalized invariants of reflection groups in rank two

For each subgroup of GL_2(F_p) or order divisible by p, generated by (pseudo-)reflections, we compute the ideals of stable and generalized invariants. These groups and these ideals are related to the cohomology of compact Lie groups, Kac-Moody groups and p-compact groups.

Reflection groups acting on their hyperplanes

After having established elementary results on the relationship between a finite complex (pseudo-)reflection group W < GL(V) and its reflection arrangement A, we prove that the action of W on A is canonically related with other natural representations of W, through a `periodic' family of representations of its braid group. We also prove that, when W is irreducible, then the squares of defining linear forms for A span the quadratic forms on V, which imply |A| >= n(n+1)/2 for n = dim V, and relate the W-equivariance of the corresponding map with the period of our family.

On the smoothness of centres of rational Cherednik algebras in positive characteristic

In this article we study rational Cherednik algebras at $t = 1$ in positive characteristic. We study a finite dimensional quotient of the rational Cherednik algebra called the restricted rational Cherednik algebra. When the corresponding pseudo-reflection group belongs to the infinite series $G(m,d,n)$, we describe explicitly the block decomposition of the restricted algebra. We also classify all pseudo-reflection groups for which the centre of the corresponding rational Cherednik algebra is regular for generic values of the deformation parameter.

Angle measures of some cones associated with finite reflection groups

We give a generalization of "Curious Identity" of De Concini and Procesi. Our proof is based on the recent result of Waldspurger about the decomposition of the cone dual to the fundamental chamber of a finite reflection group as a disjoint union of some subcones.

Reflection positive one-parameter groups and dilations

The concept of reflection positivity has its origins in the work of Osterwalder--Schrader on constructive quantum field theory. It is a fundamental tool to construct a relativistic quantum field theory as a unitary representation of the Poincare group from a non-relativistic field theory as a representation of the euclidean motion group. This is the second article in a series on the mathematical foundations of reflection positivity. We develop the theory of reflection positive one-parameter groups and the dual theory of dilations of contractive hermitian semigroups. In particular, we connect r…

On tilings defined by discrete reflection groups

The recent articles of Waldspurger and Meinrenken contained the results of tilings formed by the sets of type $(1-w)C^\circ$, $w\in W$, where $W$ is a linear or affine Weyl group, and $C^\circ$ is an open kernel of a fundamental chamber $C$ of the group $W$. In this article we generalize these results to cocompact hyperbolic reflection groups. We also give more clear and simple proofs of the Waldspurger and Meinrenken theorems.

The freeness and trace conjectures for parabolic Hecke subalgebras

The two most fundamental conjectures on the structure of the generic Hecke algebra $\mathcal{H}(W)$ associated with a complex reflection group $W$ state that $\mathcal{H}(W)$ is a free module of rank $|W|$ over its ring of definition, and that $\mathcal{H}(W)$ admits a canonical symmetrising trace. The first conjecture has recently become a theorem, while the second conjecture, known to hold for real reflection groups, has only been proved for some exceptional non-real complex reflection groups (all of rank $2$ but one). The two most fundamental conjectures on the structure of the parabolic He…

A canonical system of basic invariants of a finite reflection group

A canonical system of basic invariants is a system of invariants satisfying a set of differential equations. The properties of a canonical system are related to the mean value property for polytopes. In this article, we naturally identify the vector space spanned by a canonical system of basic invariants with an invariant space determined by a fundamental antiinvariant. From this identification, we obtain explicit formulas of canonical systems of basic invariants. The construction of the formulas does not depend on the classification of finite irreducible reflection groups.

洛伦兹赝张量的一种便捷表示论：$\operatorname{O}(1,3)$ 中的 $\mathcal{P}$ 与 $\mathcal{T}$

A Convenient Representation Theory of Lorentzian Pseudo-Tensors: $\mathcal{P}$ and $\mathcal{T}$ in $\operatorname{O}(1,3)$

本文通过将整个洛伦兹群分解为单位连通分量与克莱因四元群的半直积，提出一种便捷的洛伦兹张量表示分类方法，并据此分析时空反射性质及电磁场时间反演。