论文标题
pólya枚举定理在代数几何形状中
Pólya enumeration theorems in algebraic geometry
论文作者
论文摘要
我们将由于麦克唐纳（MacDonald）引起的公式概括，该公式将$ x^{n}/g $的单数betti编号与$ x $的近似值相关联，其中$ x $是一种紧凑的歧管，$ g $是对称组$ s_ {n} $的任何子组，以$ x^{n} $ concorting couordinates concortuting couordinates concorting coordinates concerting copertrut。我们的结果是完全公理的：在一般环境中，给定对同胞学的内态性$ h^{\ bullet}（x）$，它解释了我们如何明确地将lefschetz系列的系列诱导的内态性相关联于$ h^{\ bulter}（\ bulter}（x^{n}）$ $ h^{\ bullet}（x）$在künneth公式的存在下，相对于杯子产品。例如，当$ x $是一种紧凑的歧管时，我们将单数共同体提供的Lefschetz系列具有合理的系数。另一方面，当$ x $是有限字段$ \ mathbb {f} _ {q} $的投影变化时，我们使用$ l $ - ad-adicétalecoomology，可以选择质量数量$ l $。我们还解释了我们的公式如何概括pólya枚举定理，这是一种经典的组合定理，将图形的颜色计为给定的对称性，其中$ x $被视为有限的颜色集。当$ x $是超过$ \ mathbb {c} $的平稳的投射品种时，我们的公式还概括了Cheah的结果，将$ x^{n}/g $与$ x $的hodge数字相关联。我们还将看到我们的结果概括了以下事实：1。紧凑型歧管$ x $的对称能力的庞加莱多项式的生成函数是合理的； 2。光滑的投影型品种的对称功能的hodge-deligne多项式的生成函数$ x $ over $ \ mathbb {c} $是理性的； 3。投影品的Zeta系列$ x $超过$ \ mathbb {f} _ {q} $是理性的。当我们将$ s_ {n} $替换为$ a_ {n} $，交替的组时，我们还证明了类似的合理性结果。
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 $l$-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 $\mathbb{C}$, 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 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 $S_{n}$ with $A_{n}$, alternating groups.
