划分函子与普适指数关系
Partition functors and universal exponential relations
AI总结：
本文引入划分函子并构造单子PD，建立了乘法与加法幂运算间的普适指数关系，统一了表示环中对称幂与Adams运算及Morava E-理论中对称幂与Hecke算子的经典关系。
AI中文摘要：
我们引入划分函子：一种以有限集合的划分为指标、并配备沿细化方向的限制映射与转移映射的代数结构。我们在划分函子范畴以及若干带有额外乘法结构的划分函子范畴上构造了一个单子PD。对于划分环，其相关的完备对称函数环中的指数元素在应用PD后获得典范的对数。这产生了乘法幂运算与加法幂运算之间的普适指数关系。对于表示环，这可用于恢复对称幂与Adams运算之间的经典关系；而对于Morava E-理论，它可用于恢复Ganter关于对称幂与Hecke算子之间的指数关系。我们证明了对称群乘积的表示环构成初始划分环，并且对于对称幺半划分函子，单子PD与对称不变张量和分幂包络密切相关。我们还构造了一个携带普适指数元素的对称幺半划分环，使得其在PD下的像携带普适指数关系。
英文摘要：
We introduce partition functors: algebraic structures indexed by partitions of finite sets and equipped with restriction and transfer maps along refinements. We construct a monad PD on the category of partition functors and on several categories of partition functors with additional multiplicative structure. For a partition ring, exponential elements in its associated completed ring of symmetric functions acquire canonical logarithms after applying PD. This gives rise to a universal exponential relation between multiplicative and additive power operations. For representation rings this can be used to recover the classical relation between symmetric powers and Adams operations, while for Morava E-theory it can be used to recover Ganter's exponential relation between symmetric powers and Hecke operators. We show that the representation rings of products of symmetric groups form the initial partition ring and that, for symmetric monoidal partition functors, the monad PD is closely related to symmetric invariant tensors and the divided power envelope. We also construct a symmetric monoidal partition ring carrying the universal exponential element, so that its image under PD carries the universal exponential relation.
