Bijective proofs, algebraic structure, ubiquity, and quasi-open six-vertex asymptotics for the free-boundary q-Whittaker and Hall-Littlewood symmetry identities

The free boundary q-Whittaker and Hall-Littlewood processes introduced by He and Wheeler (arXiv:2512.02267, 2025) are measures on chains of partitions whose weights involve skew q-Whittaker or skew Hall-Littlewood polynomials together with four boundary parameters (a,b,c,d) entering through products of Rogers-Szego polynomials. The paper's central identity states that a natural observable $Z_n(x;q,t;a,b,c,d)$, a restricted Cauchy-type sum over skew q-Whittaker polynomials, is separately symmetric in the Macdonald parameters $(q,t)$ and in the four boundary parameters $(a,b,c,d)$; a companion result matches free-boundary Hall-Littlewood observables to a newly defined quasi-open stochastic six vertex model on a strip, and evaluates a bounded sum over skew Hall-Littlewood functions as a rectangular Koornwinder polynomial. All known proofs are analytic (contour-integral representations followed by residue analysis, and Yang-Baxter graphical arguments), and the paper states that they do not illuminate the nature of the identities, posing four open directions: extend the bijective (skew RSK dynamics) approach to these identities; find the algebraic or representation-theoretic structure behind the symmetries; determine how widespread such symmetry identities are among families of symmetric functions, for example at the full Macdonald level with nontrivial boundary weights; and obtain asymptotics (scaling limits, fluctuation theorems, KPZ-type limits) for the quasi-open and quasi-periodic six vertex models, whose current formulas are not amenable to asymptotic analysis.

Background

Integrable probability studies exactly solvable stochastic models through families of symmetric functions. A partition $\lambda=(\lambda_1\ge\lambda_2\ge\cdots)$ is a nonincreasing sequence of nonnegative integers, pictured as a Young diagram with $\lambda_i$ boxes in row $i$; the conjugate partition $\lambda'$ is its transpose. The Macdonald polynomials $P_\lambda(x;q,t)$ form a two-parameter basis of the ring of symmetric functions in an alphabet $x$; their specializations include Schur functions ($q=t$), Hall-Littlewood polynomials ($q=0$), and q-Whittaker polynomials ($t=0$). Macdonald processes, probability measures on chains of partitions built from these polynomials, encode random growth models, directed polymers, and interacting particle systems in the Kardar-Parisi-Zhang (KPZ) universality class, the class of random growth interfaces with correlated $t^{1/3}$-scale fluctuations. The stochastic six vertex model is a probability measure on arrow configurations on a lattice, with local weights satisfying the Yang-Baxter equation, an algebraic consistency relation that enables exact computations.

The Robinson-Schensted-Knuth (RSK) correspondence is a bijection between input data and chains of partitions recorded by pairs of semistandard Young tableaux (fillings of Young diagrams that increase weakly along rows and strictly down columns); deformed RSK maps are the combinatorial engine behind many exact identities between generating functions of symmetric polynomials. Imamura, Mucciconi and Sasamoto (2023) proved, via an intricate bijection they call skew RSK dynamics (a deterministic dynamics on pairs of skew semistandard tableaux built on affine crystal theory, the combinatorial skeleton of representations of quantum affine algebras), an identity equating a restricted Cauchy sum of q-Whittaker polynomials with a fermionic sum (one whose partition-function analogue has the determinantal form characteristic of free fermions), revealing that q-Whittaker measures are intimately related to fermionic measures; companion works recast the identity as one between restricted Cauchy sums for q-Whittaker and skew Schur polynomials, and He (2024) extended it with a random shift. He and Wheeler then recast these results as symmetry properties of the periodic q-Whittaker measure and, in the paper at issue, of the free boundary q-Whittaker measure, the q-Whittaker-level analogue of the free boundary Schur process of Betea, Bouttier, Nejjar and Vuletic, in which the boundary of the random Young diagram is itself random.

He and Wheeler define the free boundary q-Whittaker measure and prove their main theorem. Writing $(q;q)_m=\prod_{i=1}^m(1-q^i)$ for the q-Pochhammer product, $h_m(z;q)=\sum_k\binom{m}{k}_q z^k$ for the Rogers-Szego polynomials built from Gaussian q-binomial coefficients, and $h_\lambda(a,b;q)$ for Warnaar's products of these indexed by the part multiplicities of $\lambda$, the observable

$$Z_n(x;q,t;a,b,c,d)\;=\!\sum_{\mu\subseteq\lambda:\,\lambda_1\le n}\frac{h_{n-\lambda_1}(ab;q)\,h_{\lambda'}(a,b;q)}{(q;q)_{n-\lambda_1}\,\prod_{i\ge 1}(q;q)_{\lambda_i-\lambda_{i+1}}}\,P_{\lambda/\mu}(x;q,0)\;t^{|\mu|/2}\,h_{\mu'}(c/\sqrt{t},d/\sqrt{t};q)$$

is separately symmetric in the parameters $(q,t)$ and in $(a,b,c,d)$. Here $P_{\lambda/\mu}(x;q,0)$ is a skew q-Whittaker polynomial. This generalizes the Imamura-Mucciconi-Sasamoto identity and He's extension (recovered at special boundary parameters), and has a probabilistic meaning: after normalization, $Z_n$ is the probability that $\lambda_1+\chi\le n$ for $\lambda$ distributed by the free boundary q-Whittaker measure and $\chi$ an independent sum of two q-geometric random variables. The companion result matches observables of the free boundary Hall-Littlewood process to the quasi-open six vertex model, a new stochastic six vertex model on a strip assembled from alternating half-space triangles, with four boundary parameters and q-geometrically decaying rapidities; as $q\to 1$ it converges to the stationary measure of the open stochastic six vertex model on a strip. A related identity, essentially due to Finn and Vanicat, evaluates a bounded sum over skew Hall-Littlewood functions as a rectangular Koornwinder polynomial; Koornwinder polynomials are the affine type-BC Macdonald polynomials, connected to open-boundary integrable systems through q-Knizhnik-Zamolodchikov equations, which encode the action of the affine Hecke algebra.

All of these proofs are analytic: contour-integral representations followed by intricate residue analysis (in the free-boundary setting the direct analogue of the periodic argument produces a formula with none of the desired symmetries, and the symmetries are restored only after partial evaluation of the integrals), and Yang-Baxter graphical but non-bijective arguments. The authors state that their proof, while direct, does not seem to shed light on the nature of these identities, and pose the further directions that constitute this problem.

Problem Statement

The problem is to replace the analytic proofs of the free-boundary symmetry identities of He-Wheeler by structural understanding, to determine the reach of the phenomenon, and to convert the identities into probabilistic limit theorems. Four requested results, each stated in the natural generality of the source:

(1) Bijective proof. Construct a weight-preserving bijection, for example an extension of the skew RSK dynamics of Imamura-Mucciconi-Sasamoto, of the skew column RSK dynamics of Imamura-Mucciconi-Sasamoto-Scrimshaw, or any other combinatorial mechanism, that proves the $(q,t)$ and $(a,b,c,d)$ symmetry of the free-boundary $Z_n$ (Theorem 1.1 of the source), the corresponding identities for the free boundary Hall-Littlewood process, and the periodic identities of the same authors.

(2) Algebraic structure. Determine whether there is a deeper algebraic or representation-theoretic structure, such as an operator, an involution, a group action, or a framework connected to the Koornwinder and affine Hecke theory underlying the bounded Littlewood sums, behind the observed symmetries of the free-boundary $Z_n$.

(3) Ubiquity. Determine how widespread these types of symmetries and identities are: find additional families of symmetric functions or parameter generalizations for which analogous free-boundary symmetry identities hold, for instance at the full Macdonald level with nontrivial boundary weights, or establish a classification delimiting the phenomenon.

(4) Asymptotics. Obtain asymptotic results, including scaling limits, fluctuation theorems, and KPZ-type limits, for the quasi-open six vertex model, its quasi-periodic analogue, and related free-boundary measures, by whatever means (the source's contour-integral formulas, as they stand, do not seem amenable to asymptotic analysis).

These results jointly support one overarching objective: items (1)-(3) together answer where the identities come from and how far they go (the source itself observes that finding additional examples may also shed light on exactly where these identities come from), while item (4) harvests the expected payoff of the matching between free-boundary measures and integrable stochastic models, namely limit theorems for new boundary-type KPZ-class models. A complete resolution establishes the structural origin (items (1) and (2)), the scope (item (3)), and the asymptotic theory (item (4)); progress on any single item is meaningful partial progress toward the common objective.

References

1. Jimmy He and Michael Wheeler. Free boundary q-Whittaker and Hall-Littlewood processes. arXiv:2512.02267 (2025). https://arxiv.org/abs/2512.02267
2. Jimmy He and Michael Wheeler. Periodic q-Whittaker and Hall-Littlewood processes. Selecta Mathematica (N.S.) 32:11 (2026), doi:10.1007/s00029-025-01113-x; preprint arXiv:2310.03527 (2023). https://doi.org/10.1007/s00029-025-01113-x
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