Inhomogeneous q-Whittaker Polynomials

• Inhomogeneous q-Whittaker polynomials are symmetric functions constructed using quantum loop algebra techniques and exactly solvable lattice models.
• They interpolate between different symmetric function families—q-Whittaker, Grothendieck, and dual Grothendieck—through additional inhomogeneity parameters.
• Novel duality and Cauchy identities emerge from transfer matrix methods and shuffle algebra, providing fresh insights into combinatorics and representation theory.

An inhomogeneous $q$-Whittaker polynomial is a symmetric function associated with the quantized loop algebra of type $A_1^{(1)}$ ($\mathcal{U}_q(\mathfrak{sl}_2[z^\pm])$), constructed through exactly solvable lattice models via the Yang–Baxter equation. This polynomial family, denoted $\mathfrak{G}_\lambda^{(\mathbf{u},\mathbf{v})}$, unifies $q$-Whittaker, inhomogeneous $q$-Whittaker, Grothendieck, and dual Grothendieck polynomials into a single integrable-theoretic framework by interpolating between them through additional inhomogeneity parameters $\mathbf{u}, \mathbf{v}$ (Gunna et al., 4 Dec 2025). The construction leverages quantum groups, their evaluation representations, transfer matrices, and combinatorial structures from Lyndon words and shuffle algebras, producing new duality and expansion identities previously unavailable in the context of symmetric functions.

1. Algebraic Foundations: Quantum Loop Algebra $U_q(\mathfrak{sl}_2[z^\pm])$

The algebraic underpinning of inhomogeneous $q$-Whittaker polynomials is the quantum loop algebra $U_q(\mathfrak{sl}_2[z^\pm])$. There are two canonical presentations:

• Drinfeld–Jimbo presentation: uses Chevalley generators $e_i$, $f_i$, $k_i^{\pm1}$ for $i=0,1$ and affine Cartan matrix $A = [[2,-2],[-2,2]]$, with $q$-commutation and $q$-Serre relations.
• Drinfeld “new” (current) presentation: organizes the algebra via generating series of currents $X^\pm(z)$ and Cartan currents $\varphi^\pm(z)$, allowing representation in terms of formal power series and enabling interaction with shuffle and Hall algebra perspectives (Neguţ et al., 2021, Nirov et al., 2016).

These two presentations are equivalent and yield the same Hopf algebra structure, whose key components (coproduct, counit, antipode) translate naturally between them.

2. Lattice Model Framework and Transfer Matrix Construction

The representation-theoretic realization of inhomogeneous $q$-Whittaker polynomials emerges from an exactly solvable lattice model governed by the $R$-matrix of $U_q(\mathfrak{sl}_2[z^\pm])$. In this model:

• Vertical lines carry spin-$\infty$ evaluation modules $\rho_{(u_j,v_j)}$ (parameterizing inhomogeneity).
• Horizontal lines are labeled by two-dimensional evaluation representations $\pi_{x_i}$.
• Vertex weights derive from the L-operator (a specialization of the universal $R$-matrix) and are encoded in a stochastic matrix satisfying the Yang–Baxter equation.

The row-to-row transfer matrices, built as products of $L$-operators and traced over the auxiliary space, commute pairwise due to the local RLL Yang–Baxter relation, ensuring the symmetry of the resulting polynomials in the spatial variables $x_1,\dots, x_n$ (Gunna et al., 4 Dec 2025).

3. Definition and Specializations of $\mathfrak{G}_\lambda^{(\mathbf{u},\mathbf{v})}$

The inhomogeneous $q$-Whittaker polynomial $\mathfrak{G}_\lambda^{(\mathbf{u},\mathbf{v})}$ is defined as a partition function for fixed boundary data in the aforementioned lattice model:

• States are labeled by partitions $\lambda$, $\mu$.
• Matrix elements correspond to sums over path configurations with specific start and end multiplicities ($\langle \lambda'|\ldots|\mu'\rangle$).

By varying $\mathbf{u},\mathbf{v}$, the same construction interpolates between families of symmetric polynomials:

Specialization	Resulting Family
$\mathbf{v}=0$	Inhomogeneous $q$-Whittaker
$t=0$ or $q=0$	Dual Grothendieck / Grothendieck
$\mathbf{u} = \mathbf{v} = \mathbf{1}$	Hall–Littlewood, $q$-Whittaker

This mechanism transforms the problem of analyzing symmetric functions into one of analyzing $R$-matrix solvable models and their transfer matrix spectra.

4. Duality and Cauchy Identities via Yang–Baxter Theory

Exploiting the integrable structure and commutation relations of transfer matrices, the model yields new Cauchy-type identities and dualities for $\mathfrak{G}_\lambda^{(\mathbf{u},\mathbf{v})}$. For example, Theorem 4.7 of (Gunna et al., 4 Dec 2025) establishes:

$$\sum_{\kappa\supset\lambda,\mu} c_{\kappa'}c_{\lambda'} \mathfrak{G}^{(\mathbf{v},\mathbf{u})}_{\kappa/\lambda}(x)\mathfrak{G}^{\mathbf{u}}_{\kappa/\mu}(y) = \prod_{i,j} \frac{1}{(x_i y_j;q)_\infty} \sum_{\kappa\subset\lambda,\mu} c_{\mu'}c_{\kappa'} \mathfrak{G}^{(\mathbf{v},\mathbf{u})}_{\mu/\kappa}(x)\mathfrak{G}^{\mathbf{u}}_{\lambda/\kappa}(y)$$

Such identities generalize classical symmetric function expansions (e.g., $q$-Whittaker or Schur functions) to the inhomogeneous and Grothendieck settings, illuminating new positive combinatorial expansions and algebraic relations.

5. Shuffle Algebra, Lyndon Words, and Combinatorics

The positive half $U_q^+(\mathfrak{sl}_2[z^\pm])$ possesses a shuffle algebra realization, where basis elements correspond to monomials in variables $z_1,\ldots,z_n$ subject to modified symmetries dictated by $q$-deformations (Neguţ et al., 2021, Neguţ et al., 2021). In type $A_1$, the combinatorics simplifies to Lyndon words in one letter, and PBW (Poincaré–Birkhoff–Witt) bases are naturally indexed by weakly decreasing sequences:

• The shuffle product on symmetric polynomials encodes the quadratic relations required by the quantum loop algebra.
• Enriquez's isomorphism between $U_q^+(\mathfrak{sl}_2[z^\pm])$ and the relevant shuffle algebra enables direct computation of combinatorial transition coefficients between symmetry bases (Neguţ et al., 2021).

6. Representation-Theoretic and Cluster Algebra Connections

Inhomogeneous $q$-Whittaker polynomials and their degenerations encode characters or partition functions associated to specific module categories of $U_q(L\mathfrak{sl}_2)$. At roots of unity, the Grothendieck ring of finite-dimensional representations acquires the structure of a generalized cluster algebra (of type $C_{l-1}$) (Gleitz, 2014), with cluster monomials corresponding to simple module classes and the combinatorics of polynomials (e.g., their exchange relations) mirroring T-systems for Kirillov–Reshetikhin modules.

7. Applications and Further Directions

Inhomogeneous $q$-Whittaker polynomials provide a unifying framework for several established families of symmetric functions and their applications in integrable probability, quantum integrable systems (e.g., via transfer matrices and $Q$-operators), and algebraic geometry (through Hall algebra and categorification approaches) (Dou et al., 2010, Nirov et al., 2016). They yield new combinatorial identities, expand representation-theoretic correspondences, and furnish explicit models for generalized symmetric function expansions parametrized by inhomogeneity data, with further implications for stochastic processes, quantum groups, and categorifications of cluster algebras.