Groups With Pairings, Hall Modules, and Hall–Littlewood Polynomials

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 [6].