%0 Generic %1 cheong2020polya %A Cheong, Gilyoung %D 2020 %K enumeration %T Pólya enumeration theorems in algebraic geometry %U http://arxiv.org/abs/2003.04825 %X 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 $F_q$, we use the $l$-adic étale cohomology with a suitable choice of prime number $l$. 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 will also see that our result 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 $C$ is rational; 3. the zeta series of a projective variety $X$ over $F_q$ is rational. We also prove analogous rationality results when we replace $S_n$ with $A_n$, alternating groups.

@misc{cheong2020polya, 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 $H^{\bullet}(X)$, 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 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 $\mathbb{C}$, 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 $\mathbb{C}$ 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 $S_{n}$ with $A_{n}$, alternating groups.}, added-at = {2020-05-05T09:30:43.000+0200}, author = {Cheong, Gilyoung}, biburl = {https://www.bibsonomy.org/bibtex/224ab6350ebbf8299739733b0cb1d60b9/simonechiarello}, description = {P\'olya enumeration theorems in algebraic geometry}, interhash = {bf4f37c2999fb33ebb7bf602e95c7cec}, intrahash = {24ab6350ebbf8299739733b0cb1d60b9}, keywords = {enumeration}, note = {cite arxiv:2003.04825Comment: 20 pages. We have reorganized the introduction. Comments are always welcome!}, timestamp = {2020-05-05T09:30:43.000+0200}, title = {P\'olya enumeration theorems in algebraic geometry}, url = {http://arxiv.org/abs/2003.04825}, year = 2020 }