Symmetric Polynomials 𝔊^(u,v)_λ: A Unified Approach

• The topic defines a two-parameter family of symmetric polynomials derived from fused-vertex lattice models that generalize q-Whittaker, Grothendieck, and related functions.
• They exhibit rich symmetry, duality, and involutive structures that bridge combinatorial, representation-theoretic, and probabilistic frameworks.
• Special parameter choices yield classical polynomials and enable explicit combinatorial expansions, Cauchy identities, and branching rules, offering versatile tools in Schubert calculus and integrable probability.

The family of symmetric polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$ is a two-parameter class of symmetric functions that generalizes and unifies several central objects in algebraic combinatorics and representation theory, including $q$-Whittaker polynomials, inhomogeneous $q$-Whittaker functions, Grothendieck polynomials, and their duals. These polynomials arise naturally from exactly solvable lattice models tied to the quantised loop algebra $\mathcal{U}_q(\mathfrak{sl}_{2}[z^{\pm}])$ and admit deep connections to Schubert calculus, integrable probability, and the theory of quantum groups. Their combinatorial, representation-theoretic, and probabilistic properties are delineated by various parameter specializations, symmetry operations, and branching rules, encompassing a substantial part of the modern landscape of symmetric function theory (Gunna et al., 4 Dec 2025, Garbali et al., 2016, Guo et al., 25 May 2025).

1. Definition and Lattice Model Construction

The polynomials $$\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda(x_1,\dots,x_n)$$ are defined for a partition $\lambda$ and parameter sequences $\mathbf{u}=(u_1,\dots,u_n)$, $\mathbf{v}=(v_1,\dots,v_n)$. In the lattice-model formulation, they are obtained as partition functions for a fused-vertex model with Boltzmann weights explicitly encoding the two parameter families. Let $\lambda'=(m_1(\lambda'),m_2(\lambda'),\dots)$ be the conjugate partition; the initial and final boundary conditions on vertical edges of the infinite row correspond to the occupation numbers given by $\lambda'$.

At each vertex with horizontal rapidity $x_i$ and local parameters $(u_i,v_i)$, the weight is

$$\mathbb{W}_{x;(u,v)}(a,b;c,d) = \delta_{a+b=c+d}\,x^d\,\sum_{p=0}^{\min(b,c)} (ux;q)_{c-p} (v/x;q)_p (x/v)^{p-b} \begin{bmatrix} c+d-p \ c-p \end{bmatrix}_q \begin{bmatrix} b \ p \end{bmatrix}_q,$$

with $a+b = c+d$ ("ice rule").

The row-operator $\mathbb{T}(x_i)$ acts on the infinite tensor product, and the full partition function is

$$\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_{\lambda/\mu}(x_1,\dots,x_n) = \langle \lambda'|\,\mathbb{T}(x_n) \cdots \mathbb{T}(x_1)\,|\mu' \rangle,$$

recovering $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_{\lambda}$ at $\mu=0$ (Gunna et al., 4 Dec 2025). This framework is paralleled by matrix-product formulas defining similarly unified families of symmetric functions, notably $G^{(u,v)}_\lambda(x_1,\dots,x_n;q,t)$, which interpolate between Macdonald, Grothendieck, and Hall–Littlewood genera (Garbali et al., 2016).

2. Symmetry and Involution Structures

The polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$ are symmetric in the $x$-variables for all parameter choices (Gunna et al., 4 Dec 2025, Garbali et al., 2016). The proof in the fused-vertex context utilizes Hecke algebra exchange relations and involutive symmetries at the level of the underlying lattice configurations.

A duality involution—termed the Hall-$q$ involution $\omega_q$—acts by $\omega_q(p_k) = (-1)^{k-1}p_k$ on power sums and sends

$$\omega_q\left(\mathfrak{G}_{\lambda}(x)\right) = \mathfrak{J}_{\lambda'}(x),$$

where $\mathfrak{J}^{(\mathbf{v},\mathbf{u})}_{\lambda'}$ is the "dual" polynomial family with parameters interchanged and conjugate partition. This duality generalizes the bar-involution in classical symmetric function theory, and organizes the $(\mathbf{u},\mathbf{v})$ families into a closed symmetry class under $\omega_q$ (Gunna et al., 4 Dec 2025).

3. Specializations and Connections to Well-Known Polynomials

Specializing the parameters produces a web of connections to classical families:

• $\mathbf{u}=\mathbf{v}=\mathbf{0}$: Ordinary $q$-Whittaker polynomials or Macdonald polynomials, depending on the detailed construction (Gunna et al., 4 Dec 2025, Garbali et al., 2016).
• $q\to 0$, $\mathbf{v}=\mathbf{0}$, $u_i=\beta$: Grothendieck polynomials $G_\lambda$, fundamental in $K$-theoretic Schubert calculus (Garbali et al., 2016, Guo et al., 25 May 2025).
• $q\to 0$, $\mathbf{u}=\mathbf{0}$, $v_i=\beta$: Dual Grothendieck polynomials $g_\lambda$.
• $q\to 0$, $u=v=s$: Inhomogeneous Hall–Littlewood/Borodin–Petrov polynomials (Garbali et al., 2016).
• $\mathbf{v}=\mathbf{0}$: Inhomogeneous $q$-Whittaker polynomials.
• $\mathbf{u}=\mathbf{0}$: Dual inhomogeneous $q$-Whittaker polynomials.
• All parameters zero: Schur functions.

The table below summarizes prominent cases:

Parameters	Specialization	Polynomial Family
$\mathbf{u}=\mathbf{v}=0$, $q=0$	Schur limit	Schur functions
$\mathbf{v}=\mathbf{0}$, $q=0$, $u_i=\beta$	Grothendieck regime	$G_\lambda$
$\mathbf{u}=\mathbf{0}$, $q=0$, $v_i=\beta$	Dual Grothendieck	$g_\lambda$
$q=0$, $u=v=s$	Inhomogeneous Hall–Littlewood	Borodin–Petrov $F_\lambda$
$u=v=0$	General Macdonald	$P_\lambda(x;q,t)$

These unifications provide a structural bridge between symmetric function theory, $K$-theoretic and cohomological Schubert calculus, and integrable probability (Gunna et al., 4 Dec 2025, Garbali et al., 2016, Guo et al., 25 May 2025).

4. Expansion Bases, Cauchy Identities, and Branching

Parameter specialization yields expansions of $q$-Whittaker, inhomogeneous $q$-Whittaker, and Grothendieck polynomials in each other's bases. The structure constants in these expansions are given by signed or unsigned sums over vertex/lattice path models, with the combinatorial weight at each vertex determined by parameter choices (Gunna et al., 4 Dec 2025). For example: $$W_\lambda(x) = \sum_{\mu} a_{\lambda,\mu}(q)\,\mathbb{F}_\mu(x),$$ where $a_{\lambda,\mu}(q) \in \mathbb{N}[q]$ counts path configurations in a "strip" lattice. The inverse expansion and the expansion from dual inhomogeneous $q$-Whittaker to ordinary $q$-Whittaker are analogous.

The family admits generalized Cauchy identities reflecting the integrable structure: $\sum_{\lambda} \mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda(x)\,\mathfrak{G}^{(\mathbf{v},\mathbf{u})}_\lambda(y) = \text{(explicit kernel in %%%%63%%%%, %%%%64%%%%, %%%%65%%%%)}.$ There are also Pieri-type branching rules with explicit single-variable adding/removal formulas that specialize to all classical cases (Schur, Hall–Littlewood, Grothendieck, etc.) for various parameter limits (Gunna et al., 4 Dec 2025, Garbali et al., 2016).

5. Schur Expansions and Crystal/Combinatorial Structures

For those specializations related to Grothendieck theory and its variants, there is a Schur expansion via crystal graphs associated to set-valued reverse plane partitions (SVRPPs). For a fixed $n$, the set of SVRPPs is endowed with raising/lowering crystal operators (of Bender–Knuth type), and the polynomials expand as

$$G_{\lambda/\mu}(\mathbf{x}_n; \mathbf{t}; \mathbf{w}) = \sum_{\gamma,\theta} \mathbf{t}^\gamma \mathbf{w}^\theta \sum_\nu H^{\nu,\gamma,\theta}_{\lambda/\mu,n} s_\nu(\mathbf{x}_n).$$

Each $H^{\nu,\gamma,\theta}_{\lambda/\mu,n}$ counts SVRPPs with specified statistics and crystal reading word properties. Letting $n \to \infty$ yields expansions in the infinite variable limit (Guo et al., 25 May 2025).

6. Saturated Newton Polytopes and Combinatorial Formulas

For straight shapes, the Schur supports of $\mathfrak{G}$-type polynomials form saturated intervals of dominance order on partitions. A general criterion then ensures saturated Newton polytopes: every lattice point in the Newton polytope corresponds to an actual monomial exponent of the polynomial, and the degree is determined by the maximal partition in this interval (Guo et al., 25 May 2025).

Further, action by omega-type involutions produce explicit combinatorial models: for specialization $t_i=\alpha$, $w_i=\beta$, the action of the Hall–Littlewood (bar) involution produces sums over marked multiset-valued tableaux, unifying weak set-valued tableaux and valued-set tableaux structures.

7. Algebraic and Probabilistic Connections

At the algebraic level, the construction of these polynomials is intimately tied to Fock space representations of quantum affine algebras, Drinfeld twists, and $R$-matrix stochasticization. In probabilistic applications, various specializations underpin exactly-solvable models in integrable probability, including Macdonald processes and stochastic vertex models in the KPZ universality class (Garbali et al., 2016).

The two-parameter families thereby provide not only unification of Schur, Hall–Littlewood, Macdonald, Grothendieck, and Whittaker/dual Whittaker polynomials, but also a flexible tool for interpolation between cohomological and $K$-theoretic contexts in Schubert calculus, as well as for the explicit computation of observables in integrable models.

---

References:

(Guo et al., 25 May 2025): Hybrid Grothendieck polynomials (Gunna et al., 4 Dec 2025): Inhomogeneous $q$-Whittaker Polynomials I: Duality and Expansions (Garbali et al., 2016): A new generalisation of Macdonald polynomials