Some Finite Abelian Group Theory and Some $${q}$$q-Series Identities
Some Finite Abelian Group Theory and Some $${q}$$q-Series Identities
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一些有限阿贝尔群论和一些 $${q}$$q 级数恒等式
DOI：
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发表时间：
2014
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通讯作者：
D. Garton
中科院分区：
文献类型：
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作者：
D. Garton
本文的目的是计算有限交换 $${\ell}$$ℓ 群的同构类偏序集的各个子集的某些加权和，其中 $${\ell}$$ℓ 是固定的奇素数。这些总和与 Hall 和 Cohen-Lenstra 之前计算的总和类似。该计算扩展了 Cohen-Lenstra 之前的分析，同时还使用了超几何函数和 $${q}$$q 级数工具。本文中计算的恒等式虽然本身美观，但也适用于构建 Cohen-Lenstra-Martinet 启发式的随机矩阵版本。
The purpose of this article is to compute certain weighted sums over various subposets of the poset of isomorphism classes of finite abelian $${\ell}$$ℓ-groups, where $${\ell}$$ℓ is a fixed odd prime. These sums are similar to previous sums computed by Hall and Cohen-Lenstra. The computation expands upon the previous analysis of Cohen-Lenstra while also using the tools of hypergeometric functions and $${q}$$q-series. The identities computed in this article, while aesthetically pleasing in their own right, also turn out to be applicable to the construction of a random matrix version of the Cohen-Lenstra-Martinet heuristics.