The Hodge filtration and the contact-order filtration of derivations of Coxeter arrangements

Bases of the contact-order filtration of derivations of Coxeter arrangements

In [5] (=Terao, H.: Multiderivations of Coxeter arrangements. Inventiones math., 148 (2002) 659--674), we constructed a basis for the contact-order filtration of the module of derivations on the orbit space of a finite real reflection group acting on an $\ell$-dimensional Euclidean space. Recently M. Yoshinaga constructed another basis for the contact-order filtration in [7] (=Yoshinaga, M.: The primitive derivation and freeness of multi-Coxeter arrangements. preprint 2002). In this note we give an explicit formula relating Yoshinaga's basis to the basis given in [5]. The two bases turn out to…

A primitive derivation and logarithmic differential forms of Coxeter arrangements

Let $W$ be a finite irreducible real reflection group, which is a Coxeter group. We explicitly construct a basis for the module of differential 1-forms with logarithmic poles along the Coxeter arrangement by using a primitive derivation. As a consequence, we extend the Hodge filtration, indexed by nonnegative integers, into a filtration indexed by all integers. This filtration coincides with the filtration by the order of poles. The results are translated into the derivation case.

Primitive filtrations of the modules of invariant logarithmic forms of Coxeter arrangements

We define {\bf primitive derivations} for Coxeter arrangements which may not be irreducible. Using those derivations, we introduce the {\bf primitive filtrations} of the module of invariant logarithmic differential forms for an arbitrary Coxeter arrangement with an arbitrary multiplicity. In particular, when the Coxeter arrangement is irreducible with a constant multiplicity, the primitive filtration has already been studied, which generalizes the Hodge filtration introduced by K. Saito.

Coxeter combinatorics and spherical Schubert geometry

For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are spherical for the action of a Levi subgroup. We evidence the conjecture, employing the combinatorics of Demazure modules, and work of R. Avdeev-A. Petukhov, M. Can-R. Hodges, R. Hodges-V. Lakshmibai, P. Karuppuchamy, P. Magyar-J. Weyman-A. Zelevinsky, N. Perrin, J. Stembridge, and B. Tenner. In type A, we establish connections with the key polynomials of A. …

良生成复反射群的对数向量场的霍奇滤过

A Hodge filtration of logarithmic vector fields for well-generated complex reflection groups

针对不可约良生成复反射群，研究者通过分析作用于本原向量场的平坦联络，构造出其反射构形上对数向量场模的显式基，得到该模的霍奇滤过，推广并统一了实反射群的相关结果。

从驯顺函数的 Hodge 理论到多面体的 Ehrhart 理论

From Hodge theory for tame functions to Ehrhart theory for polytopes

本文通过分析 Sabbah 混合 Hodge 结构中 Hodge 滤过的 Poincaré 多项式性质，建立正则函数 Hodge 理论与多面体 Ehrhart 理论之间的联系。