Pólya enumeration theorems in algebraic geometry

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 $H^\bullet (X)$, it explains how we can explicitly relate the Lefschetz series of the induced endomorphism on $H^\bullet (X^n)^G$ to that of the given endomorphism on $H^\bullet (X)$ in the presence of the Künneth 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 $\ell$-adic étale cohomology with a suitable choice of prime number $\ell$. We also explain how our formula generalizes the Pólya 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 $C$, our formula also generalizes a result of Cheah that relates the Hodge numbers of $X^n / G$ to those of $X$. We also discuss how the generating function for the Lefschetz series of the endomorphisms on $H^\bullet (X^n)^{S_n}$ is rational, and this generalizes the following facts: 1. the generating function of the Poincaré 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 $\mathbb{C}$ 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 $S_n$ by the alternating groups $A_n$.

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Received 15 June 2020

Received revised 20 December 2020

Accepted 15 February 2022

Published 21 April 2023