Groups with pairings, Hall modules, and Hall-Littlewood polynomials

2025/04/16 by Shen, Jiahe, Van Peski, Roger · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Probability (math.PR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2504.12405

Abstract

We relate the combinatorics of Hall-Littlewood polynomials to that of abelian p-groups with alternating or Hermitian perfect pairings. Our main result is an analogue of the classical relationship between the Hall algebra of abelian p-groups (without pairings) and Hall-Littlewood polynomials. Specifically, we define a module over the classical Hall algebra with basis indexed by groups with pairings, and explicitly relate its structure constants to Hall-Littlewood polynomials at different values of the parameter t. We also show certain expectation formulas with respect to Cohen-Lenstra type measures on groups with pairings. In the alternating case this gives a new and simpler proof of previous results of Delaunay-Jouhet.

Citations

• Non-Archimedean GUE corners and Hecke modules
• Local and global universality of random matrix cokernels
• Universality for cokernels of random matrix products
• Limits and fluctuations of p-adic random matrix products
• Random Partitions and Cohen-Lenstra Heuristics
• Macdonald-positive specializations of the algebra of symmetric functions: Proof of the Kerov conjecture
• Cohen-Lenstra heuristics and local conditions
• 2^∞-Selmer groups, 2^∞-class groups, and Goldfeld's conjecture
• KPZ and Airy limits of Hall-Littlewood random plane partitions
• Hall-Littlewood polynomials and Cohen-Lenstra heuristics for Jacobians\n of random graphs
• Modeling the distribution of ranks, Selmer groups, and Shafarevich-Tate groups of elliptic curves
• p^ℓ-Torsion Points In Finite Abelian Groups And Combinatorial Identities
• Macdonald processes
• New combinatorial formula for modified Hall-Littlewood polynomials

Related

• On Hall-Littlewood polynomials
• Hall-Littlewood Polynomials, Alcove Walks, and Fillings of Young Diagrams
• Hall polynomials, inverse Kostka polynomials and puzzles
• Honeycombs for Hall polynomials
• Non-Symmetric Hall-Littlewood Polynomials