A weighted Mobius function

作者：Garton Derek^*

来源：Discrete Mathematics, 2017, 340(9): 2200-2207.

DOI：10.1016/j.disc.2017.04.014

• Cohen-Lenstra heuristics
• Mobius functions
• Posets
• Discrete probability distributions
• Moments

Fix an odd prime l and let G be the poset of isomorphism classes of finite abelian l-groups, ordered by inclusion. If xi: G -> R(>= 0)is a discrete probability distribution on G and A is an element of G, define the Ath moment of xi to be Sigma(B is an element of G) vertical bar Surj (B, A)vertical bar xi (B). The question of determining conditions that ensure xi is completely determined by its moments has been of recent interest in many problems of Cohen-Lenstra type. Furthermore, recovering xi from its moments requires a new Mobius-type inversion formula on G. In this paper, we define this function, relate it to the classical Mobius function on the poset of subgroups of a fixed group, and prove two theorems about when it vanishes. As one corollary of these theorems, we obtain an analog of Hall's theorem on the vanishing of the classical Mobius function. As another, we obtain an infinite family of pairs of groups on which the classical Mobius function vanishes; obtaining such pairs is a group-theoretic topic of recent interest.

• 出版日期2017-9