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Symmetric functions and the explicit moment problem for abelian groups

Roger Van Peski

TL;DR

The paper addresses the explicit moment problem for random abelian groups by connecting the moment method to Macdonald symmetric-function theory. It demonstrates an explicit inversion formula that expresses the distribution of the group from its $H$-moments using Hall-Littlewood polynomial identities, and it specializes to the Hall-Littlewood and $q$-Whittaker cases. The main contributions include a concrete inversion theorem for $p$-groups and its extension to multiple primes and to modules over complete DVRs with finite residue fields. This work provides a unifying algebraic-combinatorial framework that links probabilistic group models with symmetric-function techniques, with potential implications for number theory and related areas.

Abstract

Recently, Sawin and Wood (arXiv:math/2210.06279) proved a formula for the distribution of a random abelian group $G$ in terms of its $H$-moments $\mathbb{E} [\#\operatorname{Sur}(G,H)]$. We show that properties of Macdonald polynomials yield an alternate proof.

Symmetric functions and the explicit moment problem for abelian groups

TL;DR

The paper addresses the explicit moment problem for random abelian groups by connecting the moment method to Macdonald symmetric-function theory. It demonstrates an explicit inversion formula that expresses the distribution of the group from its -moments using Hall-Littlewood polynomial identities, and it specializes to the Hall-Littlewood and -Whittaker cases. The main contributions include a concrete inversion theorem for -groups and its extension to multiple primes and to modules over complete DVRs with finite residue fields. This work provides a unifying algebraic-combinatorial framework that links probabilistic group models with symmetric-function techniques, with potential implications for number theory and related areas.

Abstract

Recently, Sawin and Wood (arXiv:math/2210.06279) proved a formula for the distribution of a random abelian group in terms of its -moments . We show that properties of Macdonald polynomials yield an alternate proof.
Paper Structure (9 sections, 8 theorems, 37 equations, 3 figures)

This paper contains 9 sections, 8 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Macdonald symmetric functions
  3. Partitions and symmetric polynomials.
  4. Symmetric functions.
  5. Macdonald symmetric functions and polynomials.
  6. Specializations.
  7. Formulas in the Hall-Littlewood and $q$-Whittaker cases.
  8. Proofs and extensions
  9. Multiple primes.

Key Result

Theorem 1.1

Fix $p$ prime and let $G$ be a random finite abelian $p$-group with finite $H$-moments $M_H := \mathbb{E}[\#\operatorname{Sur}(G,H)]$ for each finite abelian $p$-group $H$. Then for any $\nu \in \mathbb{Y}$, provided that the sum on the right hand side converges absolutely. Furthermore, the law of $G$ is the unique probability measure with moments $M_H$.

Figures (3)

  • Figure : $\lambda = (5,2,2,1)$
  • Figure : $\lambda = (5,2,2,1)$
  • Figure : $\lambda' = (4,3,1,1,1)$

Theorems & Definitions (24)

  • Theorem 1.1
  • Definition 1
  • Definition 2
  • Definition 3
  • Remark 1
  • Definition 4
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Remark 2
  • ...and 14 more