Coxeter元的Lyashko-Looijenga态射和次极大因子分解
Lyashko-Looijenga morphisms and submaximal factorisations of a Coxeter element
摘要：
当W是有限反射群时，W型非交叉分划格NCP_W是一个丰富的组合对象，推广了n边形非交叉分划的概念.一个公式（唯一已知的证明是逐例的）将NCP_W中给定长度的多重链的个数表示为广义的Fuss-Catalan数，取决于W的不变度。我们描述了如何理解这个公式的一些规格，在一个无案例的方式，使用的解释的链NCP_W的Lyashko-Looijenga覆盖（LL）的纤维，构造从W的判别超曲面的几何。我们研究了代数映射LL，描述了它的判别式和雅可比矩阵的分解。作为副产品，我们推广了K. Saito关于实反射群的计数公式，并推导出W的Coxeter元的某些因子分解的计数公式。
英文摘要：
When W is a finite reflection group, the noncrossing partition lattice NCP_W of type W is a rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in NCP_W as a generalised Fuss-Catalan number, depending on the invariant degrees of W. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of NCP_W as fibers of a Lyashko-Looijenga covering (LL), constructed from the geometry of the discriminant hypersurface of W. We study algebraically the map LL, describing the factorisations of its discriminant and its Jacobian. As byproducts, we generalise a formula stated by K. Saito for real reflection groups, and we deduce new enumeration formulas for certain factorisations of a Coxeter element of W.
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