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

Philip Tosteson

Vol. 16 (2022), No. 10, 2433–2491
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 modules. We obtain many constraints on the sequence of symmetric group representations underlying a finitely generated 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–Lusztig polynomial of the braid matroid.

Keywords
representation stability, surjections
Mathematical Subject Classification
Primary: 16G20
Secondary: 05E05, 06A07, 13P10, 20C30