Topology of character varieties and representations of quivers

TL;DR

This paper discusses a conjecture related to the topology of character varieties and quiver representations, connecting mixed Hodge polynomials of surface representation varieties with Macdonald polynomial formulas, supported by new results extending previous work.

Contribution

It introduces new results that follow from a conjecture linking character varieties, quiver representations, and Macdonald polynomials, expanding understanding of their topological and algebraic structures.

Findings

Support for the conjecture through new results

Connections between mixed Hodge polynomials and Macdonald formulas

Extensions of previous theoretical frameworks

Abstract

In arXiv:0810.2076 we presented a conjecture generalizing the Cauchy formula for Macdonald polynomials. This conjecture encodes the mixed Hodge polynomials of the representation varieties of Riemann surfaces with semi-simple conjugacy classes at the punctures. We proved several results which support this conjecture. Here we announce new results which are consequences of those of arXiv:0810.2076.