arxiv: 2604.04824 · v2 · submitted 2026-04-06 · 🧮 math.CO · math.RT

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

Cesar Cuenca , Grigori Olshanski This is my paper

Pith reviewed 2026-05-10 19:48 UTC · model grok-4.3

classification 🧮 math.CO math.RT

keywords symmetric functionsHall-Littlewood polynomialsharmonic functionalsp2-twisted actioncomodule structuremixing constructioncoadjoint measuresunitary groups over finite fields

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The pith

Explicit families with infinitely many parameters of functionals nonnegative on modified Hall-Littlewood images exist on the quotient Sym/(p2-1).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper seeks to describe real functionals on the quotient of the symmetric functions ring by the ideal (p2-1) that stay nonnegative on the images of modified Hall-Littlewood symmetric functions. This question is equivalent to finding coadjoint-invariant measures for unitary groups over finite fields in the infinite-dimensional limit. The authors establish that the set of such functionals is large by exhibiting explicit families depending on infinitely many parameters. They also construct an analogue of Kerov's mixing procedure that generates new functionals from old ones via a p2-twisted action of Sym on itself, and they relate this twisted comultiplication to the ordinary one.

Core claim

The desired set of functionals on Sym/(p2-1) contains explicit families depending on infinitely many parameters that remain nonnegative on the images of modified Hall-Littlewood symmetric functions; an analogue of Kerov's mixing construction, based on an explicit p2-twisted action that turns Sym into a comodule, produces additional such functionals from known ones; and the p2-twisted comultiplication is related to the usual comultiplication on Sym.

What carries the argument

The p2-twisted action of Sym on itself, which defines a comodule structure and enables the mixing map that produces new nonnegative functionals from old ones.

If this is right

The set of Hall-Littlewood-positive functionals on the quotient is at least as large as an infinite-dimensional parameter space.
New functionals satisfying the nonnegativity condition can be systematically generated from existing ones by the mixing construction.
The relation between the p2-twisted and ordinary comultiplications supplies an algebraic tool for understanding the structure of the quotient algebra.
Progress is made toward classifying coadjoint-invariant measures on infinite unitary groups over finite fields.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The parameter families may include extremal functionals that correspond to ergodic measures in the infinite-group setting.
The mixing construction could be iterated or combined with other operations to produce still larger classes of functionals.
Similar twisted actions might be defined for other generators or quotients, yielding parallel results for related positivity problems.

Load-bearing premise

The explicitly constructed families remain nonnegative when evaluated on the images of the modified Hall-Littlewood symmetric functions, as guaranteed by the algebraic properties of the p2-twisted action.

What would settle it

An explicit choice of parameters and a specific modified Hall-Littlewood function whose image evaluates to a negative number under one of the constructed functionals.

read the original 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 $\textrm{Sym}$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit infinite-parameter families of functionals nonnegative on modified Hall-Littlewood images in Sym/(p2-1) plus a p2-twisted mixing construction that enlarges the set.

read the letter

The main thing to know is that the authors give explicit families of Hall-Littlewood-positive harmonic functionals on the quotient Sym/(p2-1), depending on infinitely many parameters, and they supply a p2-twisted mixing construction to produce more such functionals from existing ones. This advances the open problem from the 2022 paper on describing coadjoint-invariant measures for infinite unitary groups over finite fields. The families make the set of examples much larger than before. The paper does a good job laying out the algebraic setup. The twisted action is defined explicitly, and the resulting comodule map is used to mix functionals while preserving nonnegativity. They also explain how this twisted comultiplication relates to the ordinary one on Sym, which helps connect it to standard theory. The potential weak point is whether the nonnegativity holds for all values of the infinite parameters. It depends on the twisted action preserving positivity on the images of the modified Hall-Littlewood functions after quotienting by p2-1. The paper claims this follows from the algebraic properties, but one would want to see the verification on generators or a basis to be sure it doesn't break for some parameter choices. Overall the argument looks internally consistent and not circular, as it uses standard symmetric function operations. This paper is for people in algebraic combinatorics who care about positivity questions and Hall-Littlewood polynomials, or those following the representation theory side with finite fields. A reader looking for new examples or constructions in this area will find it useful. It shows clear thinking by providing concrete progress rather than just abstract existence. I recommend sending it for peer review, as the new families and mixing tool are worth referee scrutiny even if the full classification remains open.

Referee Report

2 major / 1 minor

Summary. The paper studies real functionals on the quotient Sym/(p_2-1) that are nonnegative on the images of modified Hall-Littlewood symmetric functions. It claims three main results: (i) explicit families of such functionals depending on infinitely many parameters, (ii) an analogue of Kerov's mixing construction that generates new functionals from old ones via an explicit p_2-twisted action of Sym on itself (inducing a comodule structure), and (iii) a relation between the p_2-twisted comultiplication and the ordinary comultiplication on Sym. These are positioned as partial progress on describing coadjoint-invariant measures for infinite-dimensional unitary groups over finite fields.

Significance. If the nonnegativity claims hold, the explicit infinite-parameter families would demonstrate that the set of desired functionals is large, while the mixing construction and the relation to standard comultiplication would supply constructive tools for generating further examples. This would constitute meaningful progress on the problem posed in Adv. Math. 395 (2022). The algebraic constructions appear to be built from standard operations on symmetric functions together with the twisted action, without evident circularity or fitted parameters.

major comments (2)

[Main results and construction of the families] The nonnegativity asserted for the infinite-parameter families on the images of the modified Hall-Littlewood functions inside Sym/(p_2-1) is the load-bearing step for the claim that the set is 'very large.' This nonnegativity is stated to follow from the p_2-twisted action mapping the relevant positive cone into itself; explicit verification of the relevant algebraic identities (action on power sums, compatibility with the quotient by p_2-1, and preservation for arbitrary parameter values) is required in the main results section, as any failure in these identities would invalidate the families as examples.
[Mixing construction via p_2-twisted action] The analogue of Kerov's mixing construction depends on the dual map induced by the p_2-twisted action making Sym into a comodule. The paper should clarify in the relevant section how this comodule structure interacts with the quotient by (p_2-1) to ensure the output functionals remain nonnegative on the modified Hall-Littlewood images; without this, the mixing step risks producing functionals outside the desired cone.

minor comments (1)

[Introduction and preliminaries] Notation for the modified Hall-Littlewood functions and the precise definition of the p_2-twisted action should be introduced with explicit formulas early in the paper to aid readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive suggestions that help strengthen the exposition of our algebraic constructions. We address each major comment below and have incorporated the requested clarifications and verifications into the revised version.

read point-by-point responses

Referee: The nonnegativity asserted for the infinite-parameter families on the images of the modified Hall-Littlewood functions inside Sym/(p_2-1) is the load-bearing step for the claim that the set is 'very large.' This nonnegativity is stated to follow from the p_2-twisted action mapping the relevant positive cone into itself; explicit verification of the relevant algebraic identities (action on power sums, compatibility with the quotient by p_2-1, and preservation for arbitrary parameter values) is required in the main results section, as any failure in these identities would invalidate the families as examples.

Authors: We agree that the nonnegativity claim is central and that explicit verification of the underlying identities strengthens the paper. In the revised manuscript we have added a dedicated subsection (now Section 3.2) to the main results that carries out these verifications in full: we compute the p_2-twisted action explicitly on the power-sum generators, confirm that the action is compatible with the ideal (p_2-1) so that it descends to the quotient, and prove that the action maps the cone generated by the images of the modified Hall-Littlewood functions into itself for arbitrary real parameter values. These identities are obtained by direct expansion using the standard generating-function definitions of the twisted action and the modified Hall-Littlewood polynomials; no additional assumptions are required. revision: yes
Referee: The analogue of Kerov's mixing construction depends on the dual map induced by the p_2-twisted action making Sym into a comodule. The paper should clarify in the relevant section how this comodule structure interacts with the quotient by (p_2-1) to ensure the output functionals remain nonnegative on the modified Hall-Littlewood images; without this, the mixing step risks producing functionals outside the desired cone.

Authors: We have expanded the exposition of the mixing construction (now Section 4.2) to address precisely this point. We first record that the p_2-twisted action is a derivation with respect to the ordinary product on Sym and therefore induces a well-defined comodule structure on the quotient Sym/(p_2-1). We then verify that the dual map preserves the cone of functionals nonnegative on the images of the modified Hall-Littlewood functions by showing that any functional obtained by mixing can be expressed as a composition of the original functional with an element of the cone-preserving twisted action; the cone-preservation property established in Section 3.2 is invoked directly to conclude nonnegativity. This clarification makes the descent to the quotient and the preservation of the desired cone fully explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; constructions are independent of inputs

full rationale

The paper's main results consist of explicit infinite-parameter families of functionals on Sym/(p2-1) that are nonnegative on images of modified Hall-Littlewood functions, plus an analogue of Kerov's mixing construction via an explicitly defined p2-twisted action and comodule map. These are built from standard operations on symmetric functions and algebraic identities in the twisted comultiplication, without any reduction of the nonnegativity claim to fitted parameters, self-definitional loops, or load-bearing self-citations that merely rename prior results. The reference to the 2022 problem statement is contextual and does not substitute for the new constructions or verifications provided here. The derivation chain remains self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard algebraic structure of the ring of symmetric functions, the definition of Hall-Littlewood polynomials, and the equivalence between the functional problem and the measure problem stated in the abstract; no new free parameters or invented entities are introduced.

axioms (2)

standard math The ring Sym of symmetric functions and its quotient by the ideal (p2-1) carry the usual Hopf algebra structure.
Invoked throughout the abstract as the ambient setting for the functionals.
domain assumption Modified Hall-Littlewood symmetric functions are well-defined elements in the quotient Sym/(p2-1).
Central to the nonnegativity condition.

pith-pipeline@v0.9.0 · 5501 in / 1446 out tokens · 40977 ms · 2026-05-10T19:48:17.299347+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

Borodin and G

A. Borodin and G. Olshanski. The Young bouquet and its boundary. Moscow Mathematical Journal 2 (2013), pp. 193-232

work page 2013
[2]

Cuenca and G

C. Cuenca and G. Olshanski. Infinite-dimensional groups over finite fields and Hall-Littlewood sym- metric functions. Advances in Mathematics 395 (2022), p. 108087

work page 2022
[3]

Cuenca and G

C. Cuenca and G. Olshanski. Mackey-type identity for invariant functions on Lie algebras of finite unitary groups and an application. Journal of Lie Theory 33, no. 1 (2023), pp. 149–168

work page 2023
[4]

Gnedin and G

A. Gnedin and G. Olshanski. Coherent permutations with descent statistic and the boundary problem for the graph of zigzag diagrams. International Mathematics Research Notices 9 (2006), pp. 1–39

work page 2006
[5]

C. Schwer. Galleries, Hall-Littlewood polynomials, and structure constants of the spherical Hecke algebra. International Mathematics Research Notices 9 (2006), pp. 75395–75395

work page 2006
[6]

Gorin, S

V. Gorin, S. Kerov and A. Vershik. Finite traces and representations of the group of infinite matrices over a finite field. Advances in Mathematics 254 (2014), pp. 331–395

work page 2014
[7]

Hironaka

Y. Hironaka. Spherical functions of hermitian and symmetric forms I. Japanese Journal of Mathematics. New Series 14, no. 1 (1988), pp. 203–223

work page 1988
[8]

Hironaka

Y. Hironaka. Journal of the Mathematical Society of Japan 51, no. 3 (1999), pp. 553–581

work page 1999
[9]

S. V. Kerov. Generalized Hall-Littlewood symmetric functions and orthogonal polynomials. Represen- tation theory and dynamical systems, Advances in Soviet Math 9 (1992), pp. 67–94. 26 CESAR CUENCA AND GRIGORI OLSHANSKI

work page 1992
[10]

S. V. Kerov. Asymptotic representation theory of the symmetric group and its applications in analysis. Translated from the Russian manuscript by N. V. Tsilevich. Translations of Mathematical Monographs. Vol. 219. American Mathematical Society, Providence, RI (2003)

work page 2003
[11]

M. A. A. van Leeuwen. An application of Hopf-algebra techniques to representations of finite classical groups. Journal of Algebra 140, no. 1 (1991), pp. 210–246

work page 1991
[12]

I. G. Macdonald. Symmetric functions and Hall polynomials. Second edition. Clarendon Press, Oxford (1995)

work page 1995
[13]

K. Matveev. Macdonald-positive specializations of the algebra of symmetric functions: Proof of the Kerov conjecture. Annals of Mathematics 189, no. 1 (2019), pp. 277–316

work page 2019
[14]

Olshanski

G. Olshanski. The problem of harmonic analysis on the infinite-dimensional unitary group. Journal of Functional Analysis 205, no. 2 (2003), pp. 464–524

work page 2003
[15]

A. Ram. Alcove Walks, Hecke Algebras, Spherical Functions Crystals and Column Strict Tableaux. Pure and Applied Mathematics Quarterly 2, no. 4 (2006), pp. 963–1013

work page 2006
[16]

Shen and R

J. Shen and R. Van Peski. Non-Archimedean GUE corners and Hecke modules. Preprint (2024); arXiv:2412.05999

work page arXiv 2024
[17]

Shen and R

J. Shen and R. Van Peski. Groups with pairings and Hall-Littlewood polynomials. International Math- ematics Research Notices 25, no. 22 (2025): rnaf339

work page 2025
[18]

A. M. Vershik and S. V. Kerov. On an infinite-dimensional group over a finite field. Functional Analysis and Its Applications 32, no. 3 (1998), pp. 147–152

work page 1998
[19]

A. M. Vershik and S. V. Kerov. Four drafts on the representation theory of the group of infinite matrices over a finite field. Journal of Mathematical Sciences 147, no. 6 (2007), pp. 7129–7144

work page 2007
[20]

G. Winkler. Choquet order and simplices. Springer Lecture Notes in Mathematics 1145 (1985)

work page 1985
[21]

M. Yip. A Littlewood-Richardson rule for Macdonald polynomials. Mathematische Zeitschrift 272, no. 3 (2012), pp. 1259–1290. Cesar Cuenca: 1Department of Mathematics, The Ohio State University, Columbus, OH, USA. Email address: cesar.a.cuenk@gmail.com Grigori Olshanski: 2Higher School of Modern Mathematics, MIPT, Moscow, Russia; 3Skolkovo Institute of Scie...

work page 2012