Hall-Littlewood polynomials and Cohen-Lenstra heuristics for Jacobians of random graphs

TL;DR

This paper explores the connection between Cohen-Lenstra heuristics for Jacobians of random graphs and Hall-Littlewood polynomials, providing combinatorial proofs and an algorithm for generating related random partitions.

Contribution

It establishes a novel link between random graph Jacobians and symmetric functions, offering new combinatorial insights and a Markov chain-based sampling method.

Findings

Connected Cohen-Lenstra heuristics to Hall-Littlewood polynomials

Provided combinatorial proofs of properties of random partitions

Developed a Markov chain algorithm for generating partitions

Abstract

Cohen-Lenstra heuristics for Jacobians of random graphs give rise to random partitions. We connect these random partitions to the Hall-Littlewood polynomials of symmetric function theory, and use this connection to give combinatorial proofs of properties of these random partitions. In addition, we use Markov chains to give an algorithm for generating these partitions.