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

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 $\textrm{Sym}$.

• The paper introduces explicit embeddings showing that the cone Ψ(-t) is as large as its p1 counterpart and generally infinite-dimensional.
• It develops a novel p2-twisted mixing construction that preserves HL-positivity via a binary operation under a modified comultiplication.
• The work establishes structural links between p2-twisted and standard comultiplication, shedding light on invariant measure classification for infinite unitary groups.

Hall-Littlewood-Positive Harmonic Functionals on the Algebra of Symmetric Functions

Background and Problem Statement

This paper addresses the intricate structure of positive harmonic linear functionals on certain quotients of the algebra $Sym$ of symmetric functions, subject to harmonicity and Hall-Littlewood-positivity constraints, particularly those associated with a $p_2$-twisted action. The focus is on describing the set $\Psi(-t)$ of linear functionals on $Sym$ that are both $p_2$-harmonic and $(-t)$-Hall-Littlewood-positive (HL-positive), i.e., non-negative on a specific family of (modifed) Hall-Littlewood functions indexed by all partitions, under the relation $p_2f \mapsto f$ for harmonicity.

This question generalizes the classical harmonic analysis and invariant measure problems for infinite-dimensional finite or unitary groups, extending the structures considered in previous works regarding $GL(\infty, \mathbb{F}_q)$ and its unitary analogues. Characterizing these cones of functionals forms a boundary problem analogous to the well-studied case for usual Hall-Littlewood-positive harmonic functionals ($p_1$-harmonic) but incorporating a $p_2$-twist that fundamentally complicates the analytic structure.

Structure of the Cones and Description of Functionals

The basic objects are:

• The algebra $p_2$0 of symmetric functions with standard generators $p_2$1.
• The Hall-Littlewood basis $p_2$2, which interpolates between Schur and monomial symmetric functions and is especially significant at $p_2$3 for finite field applications.

For $p_2$4, the set $p_2$5 consists of functionals $p_2$6 on $p_2$7 that:

1. Satisfy $p_2$8-harmonicity: $p_2$9 for all $\Psi(-t)$0;
2. Are $\Psi(-t)$1-HL-positive: $\Psi(-t)$2 for all $\Psi(-t)$3.

The geometric structure of $\Psi(-t)$4 as a convex cone aligns with advanced Choquet theory: extreme points generate all functionals via mixing (convex combinations and limits), and the cone itself is a simplex isomorphic to probability measures on its set of extreme rays.

By contrast, in the classical ($\Psi(-t)$5-harmonic, $\Psi(-t)$6-HL-positive) setting, all extreme points and mixing constructions are fully characterized using the result of Kerov–Matveev, realized in the context of Macdonald positivity and leading to complete parameterizations of the associated convex cones. However, the $\Psi(-t)$7-twisted setup, central in the analysis of infinite unitary groups, resists such an explicit description by classical methods, as the combinatorics and positivity constraints are incompatible with naïve extensions of the $\Psi(-t)$8-harmonic situation.

Main Results

(1) Large Explicit Family of Embeddings

The authors construct explicit embeddings

$\Psi(-t)$9

where $Sym$0 is the cone of $Sym$1-harmonic, $Sym$2-HL-positive linear functionals. These embeddings are constructed via a plethystic map $Sym$3 sending $Sym$4, and its dual. The embeddings demonstrate that $Sym$5 is at least as large as $Sym$6, and generically much larger, thus showing the convex cone is genuinely infinite-dimensional and highly nontrivial.

(2) $Sym$7-Twisted Mixing Construction (Kerov-Type)

A central contribution is an analogue of Kerov’s mixing construction in this $Sym$8-twisted context. Utilizing recent structure constant results from Shen and Van Peski (Shen et al., 2024), the authors construct a binary operation

$Sym$9

mixing a $p_2$0-HL-positive, $p_2$1-harmonic functional $p_2$2 with a $p_2$3-HL-positive, $p_2$4-harmonic functional $p_2$5, controlling the degrees of homogeneous components via parameters $p_2$6 and $p_2$7. The operation is shown to preserve HL-positivity and the respective harmonicities, producing new functionals in $p_2$8. This construction is facilitated by a $p_2$9-twisted comultiplication structure ($(-t)$0), distinct from the standard Hopf algebra comultiplication, and crucial for retaining positivity due to specific structure constant nonnegativity.

(3) Structural Relationship Between $(-t)$1-Twisted and Standard Comultiplication

A further result is an explicit algebraic relation (Theorem 6.A) between the $(-t)$2-twisted action, the twisted comultiplication, and the standard comultiplication on $(-t)$3, valid at the level of symmetric functions and their module structures. This theorem provides an exact formula for the interplay between $(-t)$4 (plethysm) and iterated comultiplication, further elucidating the combinatorial framework underlying the constructions.

Numerical and Nonnegativity Results

A foundational technical advance leveraged is the explicit verification of nonnegativity for the $(-t)$5-twisted structure constants, established via calculations in the non-Archimedean Hecke algebra context by Shen and Van Peski (Shen et al., 2024). This result is indispensable for obtaining large subsets of positive functionals by mixing, since without positivity of structure constants, HL-positivity would not be preserved under the operation.

Implications and Perspectives

On the representation-theoretic side, the description of $(-t)$6 solves, in principle, the problem of classifying $(-t)$7-invariant measures on the space of infinite Hermitian matrices over finite fields—an analogue of the celebrated Thoma classification for the infinite symmetric group and Olshanski–Vershik boundary theory for $(-t)$8. The construction directly ties to Mackey-type formulas and positivity phenomena in infinite symmetric and unitary group harmonic analysis.

Combinatorially and algebraically, the $(-t)$9-twisted setup and its mixing construction highlight the intricate ways Hopf algebraic and representation-theoretic concepts interact with positivity in symmetric functions. The presence of an explicit but highly nontrivial algebraic structure, which cannot be replicated by naïve use of the standard comultiplication, underscores the uniqueness of the infinite-dimensional unitary setting.

Despite progress, the authors explicitly note that a full classification of all extreme rays (i.e., all indecomposable positive $p_2f \mapsto f$0-harmonic functionals) remains open, and preliminary evidence suggests that new, as yet unknown, combinatorial and structural ideas will be required to mirror the efficacy of Kerov–Matveev–Matveev-type parameterizations from the $p_2f \mapsto f$1-harmonic setting.

Future Directions

Immediate avenues of investigation involve:

• Refining the structure of the cones $p_2f \mapsto f$2, $p_2f \mapsto f$3, and identifying their extreme rays, possibly through further branching graph analysis or new combinatorial invariants.
• Exploring the potential for a characteristic map or representation-theoretic geometric approach analogous to those in [CO2] and van Leeuwen’s work, perhaps enabling a cohomological or categorical rephrasing of the $p_2f \mapsto f$4-twisted harmonicity problem.
• Investigating connections to the theory of Hecke algebras, categorified Hall algebras, and emerging positive characteristic phenomena in symmetric and unitary group theory, in light of the non-Archimedean GUE corner results of Shen–Van Peski.

Conclusion

This paper establishes explicit embeddings, positivity-preserving mixing constructions, and structural results enabling partial description of the cone $p_2f \mapsto f$5 of $p_2f \mapsto f$6-harmonic, $p_2f \mapsto f$7-Hall-Littlewood-positive functionals on symmetric functions. These results represent an important step in infinite-dimensional harmonic analysis for unitary groups, demonstrating both the rich algebraic complexity and the need for further conceptual innovation to achieve a Kerov–Matveev-level classification in the $p_2f \mapsto f$8-twisted regime.