论文标题
真正反思组的Eulerian表示
Eulerian representations for real reflection groups
论文作者
论文摘要
首先是针对对称组引入的欧拉群落，后来延伸到所有反射组，生成了一个称为Eulerian代表的一家族，将定期表示形式分解。在类型$ a $中，欧拉（Eulerian）的代表与与编织布置自然相关的戒指具有许多优雅但神秘的连接。 In this paper, we unify these results and show that they hold for any reflection group of coincidental type -- that is, $S_{n}$, $B_{n}$, $H_{3}$ or the dihedral group $I_{2}(m)$ -- by giving six characterizations of the Eulerian representations, including as components of the associated graded of the Varchenko-Gelfand ring $ \ MATHCAL {V} $。结果，我们表明所罗门的下降代数包含一个且仅当$ w $是偶然时，由具有相同下降数量的元素产生的交换子代数。更一般而言，当$ w $是任何有限的真实反射组时，我们会为欧拉（Eulerian）陈述的家庭提供无案例的构造，该家族通过ring $ \ mathcal {v} $的平坦分解来描述。
The Eulerian idempotents, first introduced for the symmetric group and later extended to all reflection groups, generate a family of representations called the Eulerian representations that decompose the regular representation. In Type $A$, the Eulerian representations have many elegant but mysterious connections to rings naturally associated with the braid arrangement. In this paper, we unify these results and show that they hold for any reflection group of coincidental type -- that is, $S_{n}$, $B_{n}$, $H_{3}$ or the dihedral group $I_{2}(m)$ -- by giving six characterizations of the Eulerian representations, including as components of the associated graded of the Varchenko-Gelfand ring $\mathcal{V}$. As a consequence, we show that Solomon's descent algebra contains a commutative subalgebra generated by sums of elements with the same number of descents if and only if $W$ is coincidental. More generally, when $W$ is any finite real reflection group, we give a case-free construction of a family of Eulerian representations described by a flat-decomposition of the ring $\mathcal{V}$.
