连接$\F_q$的置换的两种表示
Connecting two types of representations of a permutation of $\F_q$
摘要：
本文将有限域$\F_q$的置换$σ$的两种表示联系起来。一种类型是代数的，其中置换被表示为一次多项式和$x^{q-2}$的$k$副本的合成，对于$k$的某个规定值。另一种类型是组合的，其中置换被表示为一个一阶有理函数的组合，后跟$\bP^1(\F_q):=\F_q\cup\{\infty\}$上的$k$ $2$ -循环的乘积，其中每个$2$ -循环移动$\infty$。我们表明，在modding出明显的等价之间的代数表示，然后对于每个$k$有一个双射之间的代数表示的$σ$和组合表示的$σ$。我们还证明了类似的结果，置换的$\bP^1(\F_q)$。一个结果是一个新的表征的概念Carlitz秩的置换$\F_q$，我们使用其他地方提供一个明确的公式Carlitz秩。另一个结果涉及到一个经典的定理Carlitz，它说，如果$q>2$，那么$\F_q$的置换群是由一次多项式和$x^{q-2}$引起的置换生成的。我们的双射提供了一个新的视角，从这个视角可以看到文献中这个结果的两个证明是自然产生的，而不需要以前为了发现这些证明而需要的聪明技巧。
英文摘要：
In this paper, we connect two types of representations of a permutation $σ$ of the finite field $\F_q$. One type is algebraic, in which the permutation is represented as the composition of degree-one polynomials and $k$ copies of $x^{q-2}$, for some prescribed value of $k$. The other type is combinatorial, in which the permutation is represented as the composition of a degree-one rational function followed by the product of $k$ $2$-cycles on $\bP^1(\F_q):=\F_q\cup\{\infty\}$, where each $2$-cycle moves $\infty$. We show that, after modding out by obvious equivalences amongst the algebraic representations, then for each $k$ there is a bijection between the algebraic representations of $σ$ and the combinatorial representations of $σ$. We also prove analogous results for permutations of $\bP^1(\F_q)$. One consequence is a new characterization of the notion of Carlitz rank of a permutation on $\F_q$, which we use elsewhere to provide an explicit formula for the Carlitz rank. Another consequence involves a classical theorem of Carlitz, which says that if $q>2$ then the group of permutations of $\F_q$ is generated by the permutations induced by degree-one polynomials and $x^{q-2}$. Our bijection provides a new perspective from which the two proofs of this result in the literature can be seen to arise naturally, without requiring the clever tricks that previously appeared to be needed in order to discover those proofs.
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