Symmetric Polynomials 𝒢^(u,v)_λ: Two-Parameter Functions

• Symmetric Polynomials 𝒢^(u,v)_λ are a two-parameter family of symmetric functions defined via exactly solvable lattice models and combinatorial tableau methods.
• The construction leverages fused Boltzmann weights, partition functions, and specialized parameters to interpolate between classical bases like Schur, Macdonald, and Grothendieck functions.
• These polynomials bridge integrable probability, representation theory, and quantum integrable systems, providing actionable insights into combinatorial identities and expansion formulas.

The symmetric polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$ form a two-parameter family of symmetric functions in several variables, defined and studied in the context of exactly solvable lattice models and algebraic combinatorics. These polynomials unify and interpolate between classical symmetric function bases—including Schur, Macdonald, $q$-Whittaker, ordinary Grothendieck, dual Grothendieck, and inhomogeneous $q$-Whittaker polynomials—by introducing infinite parameter sequences $\mathbf{u} = (u_1,u_2,\ldots)$ and $\mathbf{v} = (v_1,v_2,\ldots)$, in addition to underlying deformation parameters such as $q$ and $t$. Distinct specialisations of these parameters recover several fundamental families in algebraic combinatorics, symmetric functions, and representation theory. The theory of $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$ is situated at the juncture of symmetric function theory, integrable probability, and the theory of quantum integrable systems.

1. Definitions and Lattice Model Construction

The polynomials $$\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda(x_1,\dots,x_n)$$ are naturally encoded as partition functions of exactly solvable lattice models. Fix two infinite sequences $\mathbf{u}=(u_1,u_2,\ldots)$ and $q$0. For a partition $q$1 and its conjugate $q$2, associate a basis vector $q$3 in a Fock space $q$4 with $q$5. The row-transfer matrix $q$6 involves fused Boltzmann weights $q$7 that depend rationally on $q$8, $q$9, $q$0, $q$1, and the involved spins. For partitions $q$2, define the polynomial as

$q$3

with the non-skew case $q$4 (Gunna et al., 4 Dec 2025).

A related class, with homogeneous parameters $q$5 (i.e., $q$6), also arises from combinatorial formulas using set-valued reverse plane partitions (SVRPPs) and various statistics encoding irredundant content, column-equalities, and excess (Guo et al., 25 May 2025). Another generalisation, $q$7, is realised via a matrix-product solution to the Zamolodchikov–Faddeev algebra and non-symmetric building blocks $q$8, which are then symmetrised over the Weyl group (Garbali et al., 2016).

2. Specialisations and Interpolating Properties

Distinct choices of the parameters yield classical, dual, and inhomogeneous bases:

• $q$9: Macdonald (when $\mathbf{u} = (u_1,u_2,\ldots)$0 are present) or $\mathbf{u} = (u_1,u_2,\ldots)$1-Whittaker ($\mathbf{u} = (u_1,u_2,\ldots)$2) polynomials.
• $\mathbf{u} = (u_1,u_2,\ldots)$3: Inhomogeneous $\mathbf{u} = (u_1,u_2,\ldots)$4-Whittaker polynomials.
• $\mathbf{u} = (u_1,u_2,\ldots)$5, $\mathbf{u} = (u_1,u_2,\ldots)$6: Grothendieck polynomials $\mathbf{u} = (u_1,u_2,\ldots)$7.
• $\mathbf{u} = (u_1,u_2,\ldots)$8, $\mathbf{u} = (u_1,u_2,\ldots)$9, $\mathbf{v} = (v_1,v_2,\ldots)$0: Dual Grothendieck polynomials $\mathbf{v} = (v_1,v_2,\ldots)$1.
• For the homogeneous parameter case $\mathbf{v} = (v_1,v_2,\ldots)$2, the Schur function $\mathbf{v} = (v_1,v_2,\ldots)$3 is recovered (Gunna et al., 4 Dec 2025, Garbali et al., 2016, Guo et al., 25 May 2025).

This unification results from the interplay of weights in the underlying lattice models and in the combinatorial tableau statistics.

3. Symmetry, Cauchy Identities, and Dualities

The polynomials $\mathbf{v} = (v_1,v_2,\ldots)$4 are symmetric in the $\mathbf{v} = (v_1,v_2,\ldots)$5 variables by construction. The underlying lattice model satisfies the Yang–Baxter equation, yielding functional Cauchy identities of the form

$\mathbf{v} = (v_1,v_2,\ldots)$6

with suitable normalisation constants $\mathbf{v} = (v_1,v_2,\ldots)$7 (Gunna et al., 4 Dec 2025).

A canonical involutive duality exchanges $\mathbf{v} = (v_1,v_2,\ldots)$8, mapping $\mathbf{v} = (v_1,v_2,\ldots)$9, where $q$0 is a Hall–Littlewood–type symmetric function, induced by a local gauge-transformation of the lattice weights and corresponding to a $q$1-deformation of the standard $q$2 involution (Gunna et al., 4 Dec 2025).

4. Combinatorial and Algebraic Structures

The SVRPP model of hybrid Grothendieck polynomials encodes $q$3 as sums over fillings of $q$4 by sets of positive integers, subject to weak increase row and column constraints:

$q$5

where $q$6 counts irredundant content (column support), $q$7 counts redundant cells, and $q$8 counts excess (additional elements per row) (Guo et al., 25 May 2025).

Kashiwara operators $q$9 equip the combinatorial model with a $t$0-crystal structure, preserving the relevant statistics and enabling a Schur function expansion indexed by highest-weight vectors. Furthermore, the Newton polytope of these polynomials for straight shapes is saturated, as all partitions between the minimal and maximal content occur with nonzero coefficients (Guo et al., 25 May 2025).

Explicit algebraic representations for $t$1 leverage a matrix-product ansatz involving $t$2-boson algebras and higher-rank $t$3-matrices, parameterising the polynomials by normalization constants $t$4, which coincide with those of the Macdonald polynomials when $t$5 (Garbali et al., 2016). The corresponding non-symmetric functions $t$6 provide a systematic route to triangular expansions in monomial and power-sum bases.

5. Transition Coefficients and Expansion Formulas

The expansion of $t$7 and related symmetric polynomials in various classical bases is governed by explicit positive combinatorial formulas. There exist lattice-path and tableau models for the transition coefficients $t$8 between the $t$9-Whittaker, inhomogeneous, and Grothendieck bases: $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$0 with each coefficient given as an exactly solvable lattice model partition function with local weights and positivity properties $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$1, $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$2, $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$3 (Gunna et al., 4 Dec 2025).

Branching rules, such as one-variable skew formulas, precisely control the one-variable specialisation and allow recursive construction of higher-rank polynomials. Explicit factorised formulas govern these expansions in terms of the parameters $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$4, $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$5, $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$6, and row multiplicities (Gunna et al., 4 Dec 2025).

6. Connections to Integrable Probability and Future Directions

$\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$7 and their degenerations have applications in integrable probability, particularly in connection to particle systems, stochastic vertex models, and geometric representation theory. The matrix-product constructions and Yang–Baxter solvability give rise to identities and positive combinatorial formulas reminiscent of those arising in quantum integrable systems.

Several structural conjectures remain open, such as full Cauchy and Pieri rules for general $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$8, explicit orthogonality relations and norms beyond the known $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$9 settings, and the eigenoperator structure for $$\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda(x_1,\dots,x_n)$$0. The connections of these deformations with deeper aspects of Schur positivity, crystal commutor combinatorics, and representation theory continue to motivate ongoing research (Garbali et al., 2016, Gunna et al., 4 Dec 2025, Guo et al., 25 May 2025).

7. Summary of Key Parameter Actions and Specialisations

Parameter Regime	Specialisation	Classical Object
$$\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda(x_1,\dots,x_n)$$1, $$\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda(x_1,\dots,x_n)$$2	Macdonald $$\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda(x_1,\dots,x_n)$$3 Schur	Schur function $$\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda(x_1,\dots,x_n)$$4
$$\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda(x_1,\dots,x_n)$$5, $$\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda(x_1,\dots,x_n)$$6	Grothendieck	$$\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda(x_1,\dots,x_n)$$7
$$\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda(x_1,\dots,x_n)$$8, $$\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda(x_1,\dots,x_n)$$9, $\mathbf{u}=(u_1,u_2,\ldots)$0	Dual Grothendieck	$\mathbf{u}=(u_1,u_2,\ldots)$1
$\mathbf{u}=(u_1,u_2,\ldots)$2, $\mathbf{u}=(u_1,u_2,\ldots)$3	Borodin–Petrov rational inhomogeneous	$\mathbf{u}=(u_1,u_2,\ldots)$4
$\mathbf{u}=(u_1,u_2,\ldots)$5, $\mathbf{u}=(u_1,u_2,\ldots)$6	Macdonald/$\mathbf{u}=(u_1,u_2,\ldots)$7-Whittaker	$\mathbf{u}=(u_1,u_2,\ldots)$8
General $\mathbf{u}=(u_1,u_2,\ldots)$9	Two-parameter interpolation	$q$00

These interpolations, together with algebraic and combinatorial tools, underscore the centrality of $q$01 in modern developments at the intersection of combinatorics, integrable systems, and symmetric function theory (Garbali et al., 2016, Gunna et al., 4 Dec 2025, Guo et al., 25 May 2025).