Row–column transfer-matrix inversion relations and inhomogeneous generalizations for vertex models of canonical Grothendieck polynomials and their duals

Gunna and Zinn-Justin (Algebraic Combinatorics, 2023) constructed row and column integrable vertex models whose transfer matrices generate Yeliussizov's two-parameter canonical Grothendieck polynomials $G_\lambda^{(\alpha,\beta)}$ and their duals $g_\lambda^{(\alpha,\beta)}$, and proved a Baxter-type inversion relation between the row and column transfer matrices of the $G$-family at arbitrary $(\alpha,\beta)$. For the dual family, the analogous relation was established only between the row model of $g_\lambda^{(1,0)}$ and the column model of $g_\lambda^{(0,1)}$, the authors stating that within their construction it 'does not exist for general $\alpha$ and $\beta$'; moreover, inhomogeneous (vertical-variable) generalizations, which require $R$-matrices with the difference property (entries depending on the spectral parameters only through their ratio), could be constructed only for the one-parameter specializations $G_\lambda^{(\alpha,0)}$ and $g_\lambda^{(\alpha,0)}$ (row models) and $G_\lambda^{(0,\beta)}$ and $g_\lambda^{(0,\beta)}$ (column models). The problem is to complete this framework: determine whether an inversion relation between row and column transfer matrices of the dual canonical family exists at general independent $(\alpha,\beta)$ — constructing one together with its Cauchy-type consequences, or rigorously ruling it out within a defined class of models — and determine whether Yang–Baxter-compatible $R$-matrices or L-operators with the difference property and both parameters independent exist, which would yield inhomogeneous generalizations of the full two-parameter canonical families. Both questions are currently unaddressed in the literature.

Background

Symmetric Grothendieck polynomials $G_\lambda$ are the stable (many-variable) limits of the Grothendieck polynomials representing K-theory classes of Schubert varieties on Grassmannians; they are inhomogeneous ($G_\lambda=s_\lambda$ plus higher-degree terms, where $s_\lambda$ is the Schur polynomial) and admit a combinatorial model via set-valued Young tableaux, in which each box of a Young diagram is filled with a finite set of labels rather than a single label (Buch). Lam and Pylyavskyy introduced the dual Grothendieck polynomials $g_\lambda$ as generating functions of reverse plane partitions (fillings of a Young diagram that are weakly increasing along each row and column); $\{g_\lambda\}$ is dual to $\{G_\lambda\}$ under the Hall inner product on the ring of symmetric functions (the standard bilinear form in which the Schur basis is orthonormal), and $g_\lambda=s_\lambda$ plus lower-degree terms. The involution $\omega$ on symmetric functions, which exchanges elementary and complete symmetric polynomials and sends $s_\lambda$ to $s_{\lambda'}$ (the conjugate partition), maps these bases to the weak and weak dual Grothendieck families. Yeliussizov introduced the two-parameter refinement called canonical Grothendieck polynomials $G_\lambda^{(\alpha,\beta)}$ and dual canonical Grothendieck polynomials $g_\lambda^{(\alpha,\beta)}$, characterized by $\omega(G_\lambda^{(\alpha,\beta)})=G_{\lambda'}^{(\beta,\alpha)}$ and similarly for $g$; they degenerate at $\alpha=0$ to the Grothendieck and dual Grothendieck families and at $\beta=0$ to the weak Grothendieck and weak dual Grothendieck families, and were motivated by canonical (Kazhdan–Lusztig-type) bases of Hecke algebras.

A parallel line of work expresses such symmetric functions as partition functions of exactly solvable lattice models: local Boltzmann weights (the weight attached to each allowed configuration of the four edges meeting at a vertex) are assigned to vertices of a square lattice, and the $R$-matrix — the operator that intertwines two crossing lines — is required to satisfy the Yang–Baxter equation, a local relation on three crossing lines that guarantees commutativity of row transfer matrices (the operators that build the partition function line by line) and hence symmetry of the partition function in the horizontal spectral variables. For one-parameter Grothendieck polynomials, Wheeler and Zinn-Justin used such a model to compute Littlewood–Richardson-type structure constants. Gunna and Zinn-Justin built, for each canonical family, two vertex models — a row model and a column model, according to whether the partition is encoded through row or column multiplicities — with fermionic or bosonic auxiliary lines (horizontal lines carrying a two-state or a countably infinite-state space), and derived Cauchy identities pairing the two families at matched parameters.

Two structural features of their construction are incomplete. First, following Baxter's tradition of inversion relations in exactly solved models, they proved $T(-x)\,\widetilde T\!\left(\frac{x}{1+(\alpha-\beta)x}\right)=1$ between the row and column transfer matrices of the $G$-family at arbitrary $(\alpha,\beta)$ (Proposition 5.1), and this relation is the engine behind their Cauchy identity at general $(\alpha,\beta)$ (Theorem 6.5). For the dual family they could prove only $t(-x)\,\widetilde t(x)=1$, connecting the row model of $g_\lambda^{(1,0)}$ with the column model of $g_\lambda^{(0,1)}$ (Proposition 5.2), and they state that in their construction such an inversion relation 'does not exist for general $\alpha$ and $\beta$'; notably, their dual Cauchy identity for the generalized (inhomogeneous) polynomials (Theorem 6.7) has to be routed through this special case plus a matching parameter specialization. Second, to generalize the polynomials by attaching independent variables ('inhomogeneities') to the vertical lines of the lattice, the $R$-matrix must satisfy the difference property — its entries must depend on the spectral parameters only through their ratio. In their models this property holds, in the natural spectral variables, only in the $\beta=0$ specializations of the row models and the $\alpha=0$ specializations of the column models; the $G$-family $R$-matrices can be cast in difference form at general $(\alpha,\beta)$ only after reparametrizing the spectral variables ($x\mapsto x/(1-\beta x)$ for rows, $x\mapsto x/(1+\alpha x)$ for columns), and even in this form the authors obtain no inhomogeneous two-parameter model. They accordingly summarize that inhomogeneous generalization is possible only for $G_\lambda^{(\alpha,0)}$ and $g_\lambda^{(\alpha,0)}$ (row models) and for $G_\lambda^{(0,\beta)}$ and $g_\lambda^{(0,\beta)}$ (column models), so the full two-parameter families admit no known inhomogeneous integrable model.

Problem Statement

Complete the row–column duality (inversion-relation) framework of the Gunna–Zinn-Justin vertex models for the two-parameter canonical Grothendieck families. Both parts below serve this single objective; a resolution of either part constitutes clear progress, and full resolution brings the integrable treatment of the two-parameter canonical families to the same level of completeness as the six-vertex/Schur and one-parameter Grothendieck settings.

(a) Dual-family inversion relation. Determine whether there exists an inversion relation between the row-model and column-model transfer matrices of the dual canonical Grothendieck family $g_\lambda^{(\alpha,\beta)}$ valid for arbitrary independent parameters $\alpha,\beta$ — possibly after modifying the auxiliary spaces, the Boltzmann weights, the normalization, or the rescaling of spectral parameters. If it exists, prove it and derive the consequences it entails (commutation relations, dual Cauchy-type identities, and duality statements under the Hall inner product at general $(\alpha,\beta)$, extending those identities presently provable only through the special case connecting the row model of $g_\lambda^{(1,0)}$ with the column model of $g_\lambda^{(0,1)}$). If no such relation exists within a precisely specified class of models, prove that non-existence and characterize the algebraic obstruction. The source establishes the analogous relation for the $G$-family at general $(\alpha,\beta)$ and asserts only that within its own constructions the dual-family relation 'does not exist for general $\alpha$ and $\beta$'.

(b) Inhomogeneous two-parameter models. Determine whether $R$-matrix parametrizations, alternative L-operators (the matrices of Boltzmann weights of a single horizontal line, from which the transfer matrices are built), or deformations exist that satisfy the Yang–Baxter equation and the difference property (entries depending on spectral parameters only through their ratio) while keeping $\alpha$ and $\beta$ independent, so that the full two-parameter families $G_\lambda^{(\alpha,\beta)}$ and $g_\lambda^{(\alpha,\beta)}$ admit consistent inhomogeneous (vertical-spectral-variable) generalizations. If they exist, construct them, verify integrability, identify the resulting generalized polynomials, and check that their degenerations reproduce the known generalized families obtainable at $\beta=0$ from row models and at $\alpha=0$ from column models; if they do not, prove the obstruction within a defined class.

An answer of constructive type must concern the genuine two-parameter canonical families at independent $\alpha,\beta$, not a disguised one-parameter specialization; an answer of negative type must delimit the class of admissible models with enough precision that the impossibility claim is not an artifact of an artificially narrow class.

References

1. Ajeeth Gunna and Paul Zinn-Justin, 'Vertex models for Canonical Grothendieck polynomials and their duals', Algebraic Combinatorics 6 (2023), no. 1, 109–163, DOI 10.5802/alco.235, arXiv:2009.13172, https://doi.org/10.5802/alco.235.
2. Damir Yeliussizov, 'Duality and deformations of stable Grothendieck polynomials', Journal of Algebraic Combinatorics 45 (2017), no. 1, 295–344, DOI 10.1007/s10801-016-0708-4, arXiv:1601.01581, https://doi.org/10.1007/s10801-016-0708-4.
3. Thomas Lam and Pavlo Pylyavskyy, 'Combinatorial Hopf algebras and K-homology of Grassmanians', International Mathematics Research Notices 2007 (2007), no. 24, Article rnm125, DOI 10.1093/imrn/rnm125, arXiv:0705.2189, https://doi.org/10.1093/imrn/rnm125.
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