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Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element

Theo Douvropoulos

Abstract

In a seminal work, Bessis gave a geometric interpretation of the noncrossing lattice $NC(W)$ associated to a well-generated complex reflection group $W$. Chief component of this was the trivialization theorem, a fundamental correspondence between families of chains of $NC(W)$ and the fibers of a finite quasi-homogeneous morphism, the $LL$ map. We consider a variant of the $LL$ map, prescribed by the trivialization theorem, and apply it to the study of finer enumerative and structural properties of $NC(W)$. In particular, we extend work of Bessis and Ripoll and enumerate the so-called "primitive factorizations" of the Coxeter element $c$. That is, length additive factorizations of the form $c=w\cdot t_1\cdots t_k$, where $w$ belongs to a given conjugacy class and the $t_i$'s are reflections.

Lyashko-Looijenga morphisms and primitive factorizations of the Coxeter element

Abstract

In a seminal work, Bessis gave a geometric interpretation of the noncrossing lattice associated to a well-generated complex reflection group . Chief component of this was the trivialization theorem, a fundamental correspondence between families of chains of and the fibers of a finite quasi-homogeneous morphism, the map. We consider a variant of the map, prescribed by the trivialization theorem, and apply it to the study of finer enumerative and structural properties of . In particular, we extend work of Bessis and Ripoll and enumerate the so-called "primitive factorizations" of the Coxeter element . That is, length additive factorizations of the form , where belongs to a given conjugacy class and the 's are reflections.

Paper Structure

This paper contains 27 sections, 36 theorems, 86 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Complex reflection groups and their braid groups
  3. The braid group B(W) and the discriminant hypersurface H
  4. The parabolic stratification of H
  5. Geometric factorizations of the Coxeter element
  6. The labeling map rlbl
  7. The LL map and the trivialization theorem
  8. Geometry of the LL map
  9. Transversality of the slice Ly
  10. Finiteness of the LL map and quasi-homogeneity
  11. Compatibilities between LL, rlbl, AW, and H
  12. The trivialization theorem
  13. The generic case
  14. The general case
  15. The ramification formula
  16. ...and 12 more sections

Key Result

Theorem 1.1

hurwitz-riemannsche-flachen In the symmetric group $S_n$, there are $n^{n-2}$ smallest length factorizations $t_1\cdots t_{n-1}=(12\cdots n)$ of the long cycle in transpositions $t_i$.

Figures (4)

  • Figure 1: Geometric factorizations via the slice $L_y$.
  • Figure 2: The Hurwitz action.
  • Figure 3: The lifted Lyashko- Looijenga morphism.
  • Figure 4: $\widehat{LL}\!=\! LL'\circ\rho_Z$.

Theorems & Definitions (78)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem : Section \ref{['Section: Primitive factorizations of a Cox elt']}
  • Example 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4: for real $W$ see Sec. 3 of saito-on-a-linear-structure, for the general case see Thm. 2.4 of bes-kpi1
  • Remark 2.5
  • Lemma 2.6
  • ...and 68 more