Categorifications of rational Hilbert series and characters of $FS^{op}$ modules

TL;DR

This paper develops a method linking chain complexes to modules over combinatorial categories, establishing conditions for rational Hilbert series and applying these to analyze symmetric group representations and topological invariants.

Contribution

It introduces a novel approach to associate chain complexes with modules over combinatorial categories, leading to new homology vanishing results and character formulas for $FS^{op}$ modules.

Findings

Established homology-vanishing theorems for several combinatorial categories.

Derived constraints on symmetric group representations in finitely generated $FS^{op}$ modules.

Connected results to the homology of moduli spaces and Kazhdan-Lusztig polynomials.

Abstract

We introduce a method for associating a chain complex to a module over a combinatorial category, such that if the complex is exact then the module has a rational Hilbert series. We prove homology--vanishing theorems for these complexes for several combinatorial categories including: the category of finite sets and injections, the opposite of the category of finite sets and surjections, and the category of finite dimensional vector spaces over a finite field and injections. Our main applications are to modules over the opposite of the category of finite sets and surjections, known as $FS^{op}$ modules. We obtain many constraints on the sequence of symmetric group representations underlying a finitely generated $FS^{op}$ module. In particular, we describe its character in terms of functions that we call character exponentials. Our results have new consequences for the character of the homology of the moduli space of stable marked curves, and for the equivariant Kazhdan-Luzstig polynomial of the braid matroid.