Polya enumeration theorems in algebraic geometry

DOI10.4310/MRL.2022.V29.N5.A2zbMATH Open1523.14077arXiv2003.04825OpenAlexW3011589540MaRDI QIDQ6042851

Gilyoung Cheong

Publication date: 4 May 2023

Published in: Mathematical Research Letters (Search for Journal in Brave)

Abstract: We generalize a formula due to Macdonald that relates the singular Betti numbers of $X^n/G$ to those of $X$, where $X$ is a compact manifold and $G$ is any subgroup of the symmetric group $S_n$ acting on $X^n$ by permuting coordinates. Our result is completely axiomatic: in a general setting, given an endomorphism on the cohomology $$, it explains how we can explicitly relate the Lefschetz series of the induced endomorphism on $$ to that of the given endomorphism on $$ in the presence of the K"unneth formula with respect to a cup product. For example, when $X$ is a compact manifold, we take the Lefschetz series given by the singular cohomology with rational coefficients. On the other hand, when $X$ is a projective variety over a finite field $mathbb{F}_q$, we use the $l$-adic 'etale cohomology with a suitable choice of prime number $l$. We also explain how our formula generalizes the P'olya enumeration theorem, a classical theorem in combinatorics that counts colorings of a graph up to given symmetries, where $X$ is taken to be a finite set of colors. When $X$ is a smooth projective variety over $mathbbC$, our formula also generalizes a result of Cheah that relates the Hodge numbers of $X^n/G$ to those of $X$. We will also see that our result generalizes the following facts: 1. the generating function of the Poincar'e polynomials of symmetric powers of a compact manifold $X$ is rational; 2. the generating function of the Hodge-Deligne polynomials of symmetric powers of a smooth projective variety $X$ over $mathbbC$ is rational; 3. the zeta series of a projective variety $X$ over $mathbb{F}_q$ is rational. We also prove analogous rationality results when we replace $S_n$ with $A_n$, alternating groups.

Full work available at URL: https://arxiv.org/abs/2003.04825

Mathematics Subject Classification ID

Group actions on varieties or schemes (quotients) (14L30) Finite ground fields in algebraic geometry (14G15)

Related Items (6)

Arithmetics and combinatorics of tropical Severi varieties of univariate polynomials ⋮ Twisted genera of symmetric products ⋮ PBW degeneration: algebra, geometry, and combinatorics ⋮ Title not available (Why is that?) ⋮ Ensembles de polynômes irréductibles et théorèmes de densité ⋮ Title not available (Why is that?)

This page was built for publication: Polya enumeration theorems in algebraic geometry

Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6042851)