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Monodromy Representations of Mixed Braid Groups

Chitrabhanu Chaudhuri, Saswati Mukherjee

Abstract

We explicitly describe unitary representations of mixed braid groups on the cohomology of Abelian branched covers of $\mathbf{CP}^1$ . We show that the image of the representation is generated by complex reflections and relate it to the multivariate Burau representation.

Monodromy Representations of Mixed Braid Groups

Abstract

We explicitly describe unitary representations of mixed braid groups on the cohomology of Abelian branched covers of . We show that the image of the representation is generated by complex reflections and relate it to the multivariate Burau representation.

Paper Structure

This paper contains 5 sections, 9 theorems, 131 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Topological Construction of the Monodromy
  3. Eigenspace Decomposition of the Cohomology
  4. The Hermitian Intersection Form
  5. Monodromy Action of the Mixed Braid Group

Key Result

Proposition 3.1

As $\mathbb{C}[\Gamma]$ modules we have isomorphisms Hence $\blacktriangleleft$$\blacktriangleleft$

Figures (4)

  • Figure 1: Generators of $\mathcal{B}_{n,\Lambda}$ seen as mapping classes.
  • Figure 2: Arrangement of the disks when $n=16$, $m=4$, and all parts have size $4$.
  • Figure 3: The curves $s_i$ and $t_{j,k}$.
  • Figure 4: The first two rows show the intersections of the generating curves. The third row shows the curve $u$ in blue when all parts $\Lambda_j$ have size at least 2.

Theorems & Definitions (21)

  • Proposition 3.1
  • proof
  • Remark 3.2
  • Lemma 4.1
  • Remark 4.2
  • proof
  • Lemma 4.3
  • proof
  • Theorem 4.4
  • proof
  • ...and 11 more