---
title: "Symmetric Polynomials \U0001D50A^(u,v)_λ: A Unified Approach"
url: https://www.emergentmind.com/topics/symmetric-polynomials-mathfrak-g-mathbf-u-mathbf-v-_
type: topic
---

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

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 [2512.04468][1605.07200][2505.19072].

## 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 $\mu=0$ [2512.04468]. 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 [1605.07200].

## 2. Symmetry and Involution Structures

The polynomials $\mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda$ are symmetric in the $x$-variables for all parameter choices [2512.04468][1605.07200]. 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 $\omega_q$ [2512.04468].

## 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 [2512.04468][1605.07200].
- $q\to 0$, $\mathbf{v}=\mathbf{0}$, $u_i=\beta$: Grothendieck polynomials $G_\lambda$, fundamental in $K$-theoretic Schubert calculus [1605.07200][2505.19072].
- $q\to 0$, $\mathbf{u}=\mathbf{0}$, $v_i=\beta$: Dual Grothendieck polynomials $g_\lambda$.
- $q\to 0$, $u=v=s$: Inhomogeneous Hall–Littlewood/Borodin–Petrov polynomials [1605.07200].
- $\mathbf{v}=\mathbf{0}$: 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 [2512.04468][1605.07200][2505.19072].

## 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 [2512.04468]. 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:
\[
\sum_{\lambda} \mathfrak{G}^{(\mathbf{u},\mathbf{v})}_\lambda(x)\,\mathfrak{G}^{(\mathbf{v},\mathbf{u})}_\lambda(y) = \text{(explicit kernel in $x,y$, $q$, $\mathbf{u},\mathbf{v}$)}.
\]
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 [2512.04468][1605.07200].

## 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 [2505.19072].

## 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 [2505.19072].

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 [1605.07200].

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.

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**References:**  
[2505.19072]: Hybrid Grothendieck polynomials  
[2512.04468]: Inhomogeneous $q$-Whittaker Polynomials I: Duality and Expansions  
[1605.07200]: A new generalisation of Macdonald polynomials

Source: https://www.emergentmind.com/topics/symmetric-polynomials-mathfrak-g-mathbf-u-mathbf-v-_