Reflection Groups and Differential Forms

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--…

非本原复反射群G(r,p,n)上不变微分算子的一致分解定理

A uniform decomposition theorem for invariant differential operators on imprimitive complex reflection groups G(r,p,n)

该论文证明非本原复反射群G(r,p,n)上不变微分算子的一致分解定理，给出单分量的显式生成元，推广实反射群相关结果，还通过伽罗瓦下降给出单分量的第二种描述。

Differential operators on toric varieties and Fourier transform

We show that Fourier transforms on the Weyl algebras have a geometric counterpart in the framework of toric varieties, namely they induce isomorphisms between twisted rings of differential operators on regular toric varieties, whose fans are related to each other by reflections of one-dimensional cones. The simplest class of examples is provided by the toric varieties related by such reflections to projective spaces. It includes the blow-up at a point in affine space and resolution of singularities of varieties appearing in the study of the minimal orbit of sl(n+1).

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.

Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism

To any finite group G of automorphisms of a symplectic vector space V we associate a new multi-parameter deformation, H_k, of the smash product of G with the polynomial algebra on V. The algebra H_k, called a symplectic reflection algebra, is related to the coordinate ring of a universal Poisson deformation of the quotient singularity V/G. If G is the Weyl group of a root system in a vector space h and V=h\oplus h^*, then the algebras H_k are `rational' degenerations of Cherednik's double affine Hecke algebra. Let G=S_n, the Weyl group of g=gl_n. We construct a 1-parameter deformation of the H…

Semiinvariants of Finite Reflection Groups

Let G be a finite group of complex n by n unitary matrices generated by reflections acting on C^n. Let R be the ring of invariant polynomials, and χbe a multiplicative character of G. Let Ω^χbe the R-module of χ-invariant differential forms. We define a multiplication in Ω^χand show that under this multiplication Ω^χhas an exterior algebra structure. We also show how to extend the results to vector fields, and exhibit a relationship between χ-invariant forms and logarithmic forms.

通过表示论实现偏微分方程的并行化

Parallelisation of partial differential equations via representation theory

本文通过表示论构造对称适应基，将偏微分方程离散化解耦为独立并行系统，并以薛定谔方程为例展示其在高维和低对称性情形下的显著效率提升。

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.

Quasi-Invariants of Complex Reflection Groups

We introduce quasi-invariant polynomials for an arbitrary finite complex reflection group W. Unlike in the Coxeter case, the space Q_k of quasi-invariants of a given multiplicity is not, in general, an algebra but a module over the coordinate ring of some (singular) affine variety X_k. We extend the main results of Etingof, Ginzburg and the first author (see [BEG]) to this setting: in particular, we show that the variety X_k and the module Q_k are Cohen-Macaulay, and the rings of differential operators on X_k and Q_k are simple rings, Morita equivalent to the Weyl algebra A_n(C), where n = dim…