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

Jiahe Shen, Roger Van Peski

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.

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

Abstract

We relate the combinatorics of Hall-Littlewood polynomials to that of abelian -groups with alternating or Hermitian perfect pairings. Our main result is an analogue of the classical relationship between the Hall algebra of abelian -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 . 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.

Paper Structure

This paper contains 18 sections, 27 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. The Hall algebra and modules over it
  3. Random modules with pairings
  4. Plan of paper
  5. Preliminaries
  6. Hall algebra
  7. Hecke ring
  8. Hall-Littlewood polynomials and the ring of symmetric functions
  9. The Alternating Hall Module
  10. Alternating module background
  11. The alternating Hecke module $H(G^{\mathop{\mathrm{alt}}\nolimits},K)$
  12. The alternating Hall polynomial
  13. The Hermitian Hall Module
  14. Hermitian module background
  15. The Hermitian Hecke module $H(G^{\mathop{\mathrm{her}}\nolimits},K)$
  16. ...and 3 more sections

Key Result

Theorem 1.1

In the above setting, we have: Moreover, the mappings $(\phi,\phi^{\mathop{\mathrm{alt}}\nolimits})$ (resp. $(\phi,\phi^{\mathop{\mathrm{her}}\nolimits})$) give a module isomorphism from the $H(\mathfrak{o})\otimes_\mathbb{Z}\mathbb{Q}$-module $H^{\mathop{\mathrm{alt}}\nolimits}(\mathfrak{o})\otimes_\mathbb{Z}\mathbb{Q}$ (resp. $

Theorems & Definitions (57)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Proposition 1.3
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.1
  • ...and 47 more