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Explicit formulas for mixed Hodge polynomials of character varieties of nilpotent groups

Ruoxi Li, Rahul Singh

Abstract

Let $Hom^0(Γ,G)$ be the path-connected component of the identity representation of the variety of representations of a finitely generated nilpotent group $Γ$ into a connected reductive complex affine algebraic group $G$. With the formulas given by Florentino, Lawton and Silva, we provide explicit partition type formulas for the mixed Hodge polynomials of character varieties $Hom^0(Γ,G)// G$ when $G=Sp_{2n}$ and $G=SO_{n}$.

Explicit formulas for mixed Hodge polynomials of character varieties of nilpotent groups

Abstract

Let be the path-connected component of the identity representation of the variety of representations of a finitely generated nilpotent group into a connected reductive complex affine algebraic group . With the formulas given by Florentino, Lawton and Silva, we provide explicit partition type formulas for the mixed Hodge polynomials of character varieties when and .

Paper Structure

This paper contains 9 sections, 6 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. Character varieties
  5. Mixed Hodge polynomials
  6. Symplectic and odd orthogonal groups
  7. Symplectic groups
  8. Odd orthogonal groups
  9. Even orthogonal groups

Key Result

Proposition 3.6

Let $\mathbf{a}, \mathbf{b}\in C_2\wr S_n$. Then $\mathbf{a}$ and $\mathbf{b}$ are conjugate in $C_2\wr S_n$ if and only if the associated signed partitions are equal.

Theorems & Definitions (23)

  • Remark 2.1
  • Definition 2.4
  • Remark 2.5
  • Example 2.7
  • Remark 2.10
  • Definition 3.1
  • Definition 3.2
  • Example 3.3
  • Definition 3.4
  • Proposition 3.6
  • ...and 13 more