Hall-Littlewood-positive harmonic functionals on the algebra of symmetric functions

TL;DR

This paper investigates positive harmonic functionals on symmetric functions related to Hall-Littlewood functions, providing explicit examples, a new construction method, and insights into their algebraic structure.

Contribution

It introduces a large family of positive functionals, an analogue of Kerov's mixing construction, and clarifies the relation between twisted and usual comultiplications.

Findings

The set of positive functionals is very large, with explicit parameter-dependent examples.

An analogue of Kerov's mixing construction is developed for these functionals.

The relation between $p_2$-twisted and usual comultiplication on Sym is explained.

Abstract

We study the problem of describing the set of real functionals on the quotient $\textrm{Sym}/(p_2-1)$ of the ring of symmetric functions that are nonnegative on the images of certain modified Hall-Littlewood symmetric functions. This question is equivalent to the problem, posed in [Adv Math 395, p.108087 (2022)], of describing the set of coadjoint-invariant measures for unitary groups over a finite field in the infinite-dimensional setting. Our main results constitute partial progress towards this problem. Firstly, we show that the desired set of functionals is very large, in the sense that it contains explicit families of examples depending on infinitely many parameters. Secondly, we provide an analogue of Kerov's mixing construction that produces new sought after functionals from known old ones. This construction depends on an explicit "$p_2$-twisted action" of $\textrm{Sym}$ on itself and the resulting dual map that makes $\textrm{Sym}$ into a comodule. Finally, our third main result explains the relation between the $p_2$-twisted comultiplication and the usual comultiplication on $\textrm{Sym}$.