The equivariant Euler characteristic of real Coxeter toric varieties

Rational cohomology of the real Coxeter toric variety of type A

The toric variety corresponding to the Coxeter fan of type A can also be described as a De Concini-Procesi wonderful model. Using a general result of Rains which relates cohomology of real De Concini-Procesi models to poset homology, we give formulas for the Betti numbers of the real toric variety, and the symmetric group representations on the rational cohomologies. We also show that the rational cohomology ring is not generated in degree 1.

Equivariant multiplicities of Coxeter arrangements and invariant bases

Let $\A$ be an irreducible Coxeter arrangement and $W$ be its Coxeter group. Then $W$ naturally acts on $\A$. A multiplicity $\bfm : \A\rightarrow \Z$ is said to be equivariant when $\bfm$ is constant on each $W$-orbit of $\A$. In this article, we prove that the multi-derivation module $D(\A, \bfm)$ is a free module whenever $\bfm$ is equivariant by explicitly constructing a basis, which generalizes the main theorem of \cite{T02}. The main tool is a primitive derivation and its covariant derivative. Moreover, we show that the $W$-invariant part $D(\A, \bfm)^{W}$ for any multiplicity $\bfm$ is …

A remark on descent for Coxeter groups

Let $Γ$ be a finite Coxeter group with reflection representation $R$. We show that a $Γ$-equivariant quasicoherent sheaf on $R$ descends to the quotient space $R//Γ$ if it descends to the quotient space $R//\langle s_i\rangle$ for every simple reflection $s_i\in Γ$.

Coxeter transformations of the derived categories of coherent sheaves

In this paper, we study the Coxeter transformation of the derived categories of coherent sheaves on smooth complete varieties. We first obtain that if the rank of the Grothendieck group is finite, say $m$, then its characteristic polynomials is $(λ+(-1)^{n})^m$, where $n$ is dimension of the variety. We then study the Jordan canonical forms of the Coxeter transformations for rational surfaces, smooth complete toric varieties with ample canonical or anticanonical bundles, and prove that their Jordan canonical forms can determine and be determined by the Betti numbers of these varieties. As an a…

Coxeter 排列与建筑上的等变 perverse 层

Equivariant perverse sheaves on Coxeter arrangements and buildings

本文描述了有限 Coxeter 群作用下反射超平面排列上的等变 perverse 层范畴，并利用 Kapranov–Schechtman 分析得到其与显式代数模的等价；同时推广到仿射建筑，建立与深度零表示的联系。

Toric reflection groups

Several finite complex reflection groups have a braid group which is isomorphic to a torus knot group. The reflection group is obtained from the torus knot group by declaring meridians to have order $k$ for some $k\geq 2$, and meridians are mapped to reflections. We study all possible quotients of torus knot groups obtained by requiring meridians to have finite order. Using the theory of $J$-groups of Achar and Aubert, we show that these groups behave like (in general infinite) complex reflection groups of rank two. The large family of "toric reflection groups" which we obtain includes, among …