Strong Lefschetz elements of the coinvariant rings of finite Coxeter groups

The strong Lefschetz property of the coinvariant ring of the Coxeter group of type H4

We prove that the coinvariant ring of the irreducible Coxeter group of type H4 has the strong Lefschetz property.

Lefschetz theory for exterior algebras and fermionic diagonal coinvariants

Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^*)$ as well as its quotient $DR_W := \wedge (V \oplus V^*)/ \langle \wedge (V \oplus V^*)^{W}_+ \rangle$ by the ideal generated by its homogeneous $W$-invariants with vanishing constant term. We describe the bigraded isomorphism type of $DR_W$; when $W = \mathfrak{S}_n$ is the symmetric group, the answer is a difference of Kronecker products of hook-shaped $\mathfrak{S}_n$-modules. We relate the Hilbert series of…

Rational Cherednik algebras and diagonal coinvariants of G(m,p,n)

We construct a quotient ring of the ring of diagonal coinvariants of the complex reflection group $W=G(m,p,n)$ and determine its graded character. This generalises a result of Gordon for Coxeter groups. The proof uses a study of category $\cO$ for the rational Cherednik algebra of $W$.

Hessenberg varieties and hyperplane arrangements

Given a semisimple complex linear algebraic group $G$ and a lower ideal $I$ in positive roots of $G$, three objects arise: the ideal arrangement $\mathcal{A}_I$, the regular nilpotent Hessenberg variety $\mbox{Hess}(N,I)$, and the regular semisimple Hessenberg variety $\mbox{Hess}(S,I)$. We show that a certain graded ring derived from the logarithmic derivation module of $\mathcal{A}_I$ is isomorphic to $H^*(\mbox{Hess}(N,I))$ and $H^*(\mbox{Hess}(S,I))^W$, the invariants in $H^*(\mbox{Hess}(S,I))$ under an action of the Weyl group $W$ of $G$. This isomorphism is shown for general Lie type, an…

四元数商奇点的零纤维

Zero fibers of quaternionic quotient singularities

本文将 Haiman 关于实反射群对角 coinvariant 环的猜想推广到四元数反射群，研究商映射零纤维的函数环，证明其在高秩情形具有来自辛反射代数不可约表示的 $(g+1)^n$ 维商，并猜想该结论普遍成立。