A geometrical construction for the polynomial invariants of some reflection groups

A geometric version of BGP reflection functors

Quiver Grassmannians and quiver flags are natural generalisations of usual Grassmannians and flags. They arise in the study of quiver representations and Hall algebras. In general, they are projective varieties which are neither smooth nor irreducible. We use a scheme theoretic approach to calculate their tangent space and to obtain a dimension estimate similar to one of Reineke. Using this we can show that if there is a generic representation, then these varieties are smooth and irreducible. If we additionally have a counting polynomial we deduce that their Euler characteristic is positive an…

Separating algebras and finite reflection groups

A separating algebra is, roughly speaking, a subalgebra of the ring of invariants whose elements distinguish between any two orbits that can be distinguished using invariants. In this paper, we introduce a geometric notion of separating algebra. This allows us to prove that only groups generated by reflections may have polynomial separating algebras, and only groups generated by bireflections may have complete intersection separating algebras.

On the cohomology of the Milnor fibre of a hyperplane arrangement

We investigate the cohomology of the Milnor fibre of a reflection arrangement as a module for the group $Γ$ generated by the reflections, together with the cyclic monodromy. Although we succeed completely only for unitary reflection groups of rank two, we establish some general results which relate the isotypic componenents of the monodromy on the cohomology, to the Hodge structure and to the cohomology degree. Using eigenspace theory for reflection groups, we prove some sum formulae for additive functions such as the equivariant weight polynomial and certain polynomials related to the Euler c…

Geometrically Nilpotent Subvarieties

We construct some examples of polynomial maps over finite fields that admit subvarieties with a peculiar property: every geometric point is mapped to a fixed point by some iteration of the map, while the whole subvariety is not. Several related open questions are stated and discussed.

完备的单纯形扇，Stanley-Reisner环和等变h多项式

Complete simplicial fans, Stanley--Reisner rings, and equivariant h-polynomials

本文推导了有限群作用于完备单纯形扇的Stanley-Reisner环的Artinian约简的分层特征公式，基于等变h多项式，并研究了该群在关联的典范流形上同调的表示。

Flat Bases of Invariant Polynomials and P-matrices of E7 and E8

Let $G$ be a compact group of linear transformations of an Euclidean space $V$. The $G$-invariant $C^\infty$ functions can be expressed as $C^\infty$ functions of a finite basic set of $G$-invariant homogeneous polynomials, called an integrity basis. The mathematical description of the orbit space $V/G$ depends on the integrity basis too: it is realized through polynomial equations and inequalities expressing rank and positive semi-definiteness conditions of the $P$-matrix, a real symmetric matrix determined by the integrity basis. The choice of the basic set of $G$-invariant homogeneous polyn…