Polya 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-center dot(X), it explains how we can explicitly relate the Lefschetz series of the induced endomorphism on H-center dot(Xn)(G) to that of the given endomorphism on H-center dot(X) in the presence of the Kunneth 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 etale cohomology with a suitable choice of prime number l. We also explain how our formula generalizes the Polya 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-center dot(X-n)(Sn) is rational, and this generalizes the following facts: 1. the generating function of the Poincare 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 by the alternating groups A(n).