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Search: All articles in the CMB digital archive with keyword reflection group

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1. CMB Online first

Nakashima, Norihiro; Terao, Hiroaki; Tsujie, Shuhei
Canonical systems of basic invariants for unitary reflection groups
It has been known that there exists a canonical system for every finite real reflection group. The first and the third authors obtained an explicit formula for a canonical system in the previous paper. In this article, we first define canonical systems for the finite unitary reflection groups, and then prove their existence. Our proof does not depend on the classification of unitary reflection groups. Furthermore, we give an explicit formula for a canonical system for every unitary reflection group.

Keywords:basic invariant, invariant theory, finite unitary reflection group
Categories:13A50, 20F55

2. CMB 2000 (vol 43 pp. 496)

Xu, Yuan
Harmonic Polynomials Associated With Reflection Groups
We extend Maxwell's representation of harmonic polynomials to $h$-harmonics associated to a reflection invariant weight function $h_k$. Let $\CD_i$, $1\le i \le d$, be Dunkl's operators associated with a reflection group. For any homogeneous polynomial $P$ of degree $n$, we prove the polynomial $|\xb|^{2 \gamma +d-2+2n}P(\CD)\{1/|\xb|^{2 \gamma +d-2}\}$ is a $h$-harmonic polynomial of degree $n$, where $\gamma = \sum k_i$ and $\CD=(\CD_1,\ldots,\CD_d)$. The construction yields a basis for $h$-harmonics. We also discuss self-adjoint operators acting on the space of $h$-harmonics.

Keywords:$h$-harmonics, reflection group, Dunkl's operators
Categories:33C50, 33C45

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