Two Littlewood identities for fully inhomogeneous spin Hall-Littlewood symmetric rational functions

Abstract: Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $F_λ$ arise as partition functions of certain path configurations in the $\mathfrak{sl}2$ higher spin six vertex models. They are multiparameter generalizations of the classical Hall-Littlewood symmetric polynomials. We establish two new generalizations of the classical Littlewood identity, where we express a weighted sum of $Fλ$'s over all partitions $λ$ as a product of the Littlewood kernel and another simple product in one case, and a product of the Littlewood kernel and a Pfaffian in the other case. As a corollary we obtain a novel Littlewood identity for Hall-Littlewood symmetric polynomials. We also elaborate on the newly established connection between the fully inhomogeneous spin Hall-Littlewood symmetric rational functions $F_λ$ and the modified Robbins polynomials, the latter being multivariate generating functions for alternating sign matrices. This connection allowed us to discover the two generalizations of the Littlewood identity and we provide a bijection between the underlying combinatorial models in the case where $λ$ is strictly decreasing.

• The paper establishes new Littlewood-type identities that generalize classical symmetric function formulas to fully inhomogeneous spin Hall-Littlewood functions.
• It employs combinatorial recursion and Pfaffian techniques to derive exact formulas for partition functions in integrable lattice models.
• The work bridges algebraic combinatorics and statistical mechanics by linking spin Hall-Littlewood functions with alternating sign matrix generating polynomials.

Two Littlewood Identities for Fully Inhomogeneous Spin Hall-Littlewood Symmetric Rational Functions

Introduction and Motivation

The paper "Two Littlewood identities for fully inhomogeneous spin Hall-Littlewood symmetric rational functions" (2603.29836) presents significant advances in the study of symmetric functions associated with integrable lattice models. The central focus is the generalization of classical Littlewood identities from Schur and Hall-Littlewood symmetric polynomials to a family of highly inhomogeneous, multiparameter symmetric rational functions, $F_\lambda$, arising in the context of $\mathfrak{sl}_2$ higher spin six vertex models.

The classical Littlewood identity is a foundational result in the algebraic combinatorics of symmetric functions, expressing a sum over Schur functions as a factorized product. Its numerous generalizations, including associated identities for Hall-Littlewood polynomials and connections to combinatorial models like alternating sign matrices (ASMs), have deep implications in representation theory and integrable probability. The present work extends these identities to the setting of fully inhomogeneous spin Hall-Littlewood symmetric rational functions, which retain integrability and rich parameter dependence.

Main Results: Two New Littlewood Identities

The First Identity: Product Formula Generalization

For the spin Hall-Littlewood functions $F_\lambda(u_1,\ldots,u_n \mid s_0, s_1, \ldots)$, the authors establish a Littlewood-type identity that generalizes the classical sum over partitions $\lambda$ of Schur polynomials. Explicitly, for $u_i$ and $s_j$ specializations linked to vertex model parameters, they show: $$\sum_{\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n \geq 0} \left[ \prod_{r\geq 0} \frac{(-s_r;q)_{m_r(\lambda)}}{(q;q)_{m_r(\lambda)}} F_\lambda(u_1,\ldots,u_n|s_0,s_1,\ldots) \right] = \prod_{i=1}^{n} \frac{1}{1-u_i} \prod_{1 \leq i < j \leq n} \frac{1-q u_i u_j}{1-u_i u_j}$$ under designated parameterizations. This identity recovers the original Littlewood formula for Schur functions as $q \to 0$, $s_j = 0$.

The proof employs recursion relations derived from the path model combinatorics (reflecting the integrability of the vertex model), and the identity is reduced to a sum over subsets and an explicit symmetrization formula, leveraging the algebraic Bethe Ansatz.

The Second Identity: A Pfaffian Generalization

The second principal identity involves the same spin Hall-Littlewood functions but expresses the partition sum as a product times a Pfaffian. This result generalizes another variant of the Littlewood identity to the inhomogeneous spin setting: $$\sum_{\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n \geq 0} \left[ \cdots \right] F_\lambda(u_1,\ldots,u_n|s_0,s_1,\ldots) = \left(\text{explicit product factors}\right) \times \operatorname{Pf} M$$ where the matrix $\mathfrak{sl}_2$0 is antisymmetric with entries depending rationally on the spectral parameters $\mathfrak{sl}_2$1, the spin parameters $\mathfrak{sl}_2$2, and deformation parameters $\mathfrak{sl}_2$3, $\mathfrak{sl}_2$4. The right-hand side factorizes except for the Pfaffian. For Hall-Littlewood specializations, this reduces to a new Littlewood identity for Hall-Littlewood polynomials expressed as a product times a Pfaffian. Notably, Kawanaka's identity emerges as a particular case of this general statement.

The proof uses intricate recursion, sign manipulations, and inductive Pfaffian expansions, carefully managing parameter specializations so that the combinatorics correspond with path crossings and the antisymmetry of the matrix.

Combinatorial and Vertex Model Interpretation

The functions $\mathfrak{sl}_2$5 are realized as partition functions of ensembles of non-intersecting lattice paths on the square grid, with weights given by Boltzmann factors tied to the spin, spectral, and deformation parameters. The fully inhomogeneous case admits arbitrary inhomogeneities in both the horizontal (spectral) and vertical (spin) directions. The polynomial weights on the path ensembles translate to polynomial weights on monotone triangles, connecting directly with the theory of alternating sign matrices.

A highlight is the explicit, combinatorial construction linking the $\mathfrak{sl}_2$6 with modified Robbins polynomials, which generalize the generating polynomials of ASMs. This connection is made precise for strictly decreasing partition $\mathfrak{sl}_2$7 via a bijective correspondence between certain weighted path ensembles (in the higher spin six vertex model) and combinatorial objects parameterizing ASMs (down-arrowed monotone triangles and decorated monotone triangles).

Proof Techniques and Algebraic Structure

The central algebraic tool is the derivation of recursion relations (“branching rules”), which encode how the $\mathfrak{sl}_2$8-variable functions decompose in terms of their “smaller” analogues. This recursion is compatible with both the path model and the polynomial expressions for $\mathfrak{sl}_2$9. By translating the sum over all integer partitions into an equivalent sum over subsets with subset-dependent weights and by symbolically symmetrizing or antisymmetrizing key factors, the authors reduce the main identities to polynomial or Pfaffian identities over function spaces. These are then proven by induction and combinatorial arguments manipulating signs, factoring, and reductions via specializations.

In the more technical portions, the authors exploit properties of the Pfaffian (antisymmetry, expansion along rows/columns, and multiplicative properties under diagonal conjugation). Careful combinatorial counting via the “sign of a subset” encapsulates the structure of involutive cancellations arising from the presence of identical rows/columns in the antisymmetric matrices when variables are specialized.

Implications and Future Directions

These generalized Littlewood identities provide new, highly-parameterized exact formulas for the partition function of broad classes of integrable vertex models. The formalism covers, as special cases, all classical Littlewood identities and many other symmetric polynomial summation results (Cauchy identities, Kawanaka identity, and others). Notably, the appearance of the Pfaffian in the second identity relates to deeper properties of the path model and the underlying quantum group structure.

The connection established between the spin Hall-Littlewood system and modified Robbins/ASM-generating polynomials enhances the dictionary between algebraic combinatorics, symmetric functions, and exactly solvable lattice models. The authors also point to ongoing investigations regarding bijective “vertex model” proofs inspired by the Yang-Baxter equation, with a possible bridge to probabilistic bijections between ASM-type objects and plane partitions.

Future directions will likely include:

• Bijective/probabilistic interpretations: Systematic development of a bijective proof using vertex model moves, possibly generalizing RSK or other combinatorial correspondences.
• Extension to higher rank and different quantum algebras: Exploring analogous identities for different classes of solvable models.
• Analytic applications: Application of the identities to the computation of correlation functions or structure constants in integrable probability and representation theory.
• Refinements and deformation: Analysis of $F_\lambda(u_1,\ldots,u_n \mid s_0, s_1, \ldots)$0 and the identities under further parameter specialization, including links with Macdonald polynomials, Koornwinder polynomials, and connections to supersymmetric gauge theory partition functions.

Conclusion

The article provides a rigorous extension of Littlewood-type summation identities to the domain of rational symmetric functions arising from fully inhomogeneous, higher spin integrable lattice models. The combination of algebraic and combinatorial methods, particularly the exploitation of the rich parameter space, produces identities with far-reaching implications for the intersection of combinatorics, representation theory, and statistical mechanics. The connection to ASM generating functions and Pfaffian structures further illustrates the depth of these results (2603.29836).