Orthogonal Dualities of Dynamic Stochastic Higher Spin Vertex Models, using the Drinfeld Twister

Technical Summary: Orthogonal Dualities of Dynamic Stochastic Higher Spin Vertex Models, using the Drinfeld Twister

Problem Statement
The paper addresses the construction of duality functions for integrable dynamic stochastic models, specifically dynamic stochastic higher spin vertex models. While Markov duality is a powerful tool for analyzing interacting particle systems, discovering duality functions often relies on ad-hoc methods or "black art." Previous algebraic approaches for non-dynamic models have been extended to dynamic models (introduced by Borodin using the Felder-Varchenko elliptic quantum group), but a unified algebraic framework for constructing orthogonal polynomial dualities in dynamic settings has remained elusive. The specific challenge is to establish a duality between dynamic stochastic higher spin vertex models and their non-dynamic counterparts, and to identify these duality functions explicitly as orthogonal polynomials.

Methodology
The authors employ a novel algebraic framework based on the universal Drinfeld twister of the quantum group $U_q(\mathfrak{sl}_2)$, treating it as a quasi-triangular quasi-Hopf algebra. This approach diverges from previous works that utilized the Felder-Varchenko elliptic quantum group or relied on fusion procedures to derive weights.

The core methodology involves:

1. Universal R-Matrix and Twisting: The authors utilize the universal R-matrix of $U_q(\mathfrak{sl}_2)$ and apply a universal Drinfeld twister $F(\lambda)$ to generate a dynamical R-matrix $R(z, \lambda)$ that satisfies the dynamical Yang-Baxter equation.
2. Intertwining Relations: They establish an intertwining relationship between the dynamical R-matrix and its non-dynamic limit (obtained by taking a specific limit of the dynamical parameter). This is achieved by analyzing the action of the R-matrix on tensor products of representations (auxiliary and bulk spaces) and utilizing a "ground state" symmetry to convert the R-matrix into a stochastic transition matrix.
3. Algebraic Construction of Duality: By defining a specific operator $B(z, \lambda)$ constructed from the R-matrix and taking limits of the auxiliary space dimension, the authors derive a duality function $D$. The proof relies on transposition formulas and ground-state transformations to relate the stochastic transfer matrix of the dynamic model to the transpose of the stochastic transfer matrix of a non-dynamic model.
4. Verification of Fusion: The paper verifies that the weights derived from this algebraic construction coincide with the fused weights of the dynamic stochastic higher spin vertex models previously derived by Aggarwal using fusion procedures.

Key Contributions and Results
The primary results of the paper are the explicit construction and proof of duality for dynamic stochastic higher spin vertex models:

• General Duality Theorem: The authors prove that the dynamic stochastic higher spin vertex model (with asymmetry to the right) is dual to a non-dynamic stochastic higher spin vertex model (with asymmetry to the left). The duality functions are explicitly given as products of $3\phi_2$ basic hypergeometric functions.
• Orthogonal Polynomial Dualities: A significant contribution is the identification of these duality functions as orthogonal polynomials. Specifically, in a degeneration limit where a parameter $c \to 0$ (or $qc \to 0$), the duality functions become dual q-Krawtchouk polynomials.
• Connection to Existing Work: The paper demonstrates that the orthogonal duality between dynamic ASEP and ASEP constructed by Groenevelt and Wagenaar (using dual q-Krawtchouk polynomials) arises as a specific degeneration of the general duality functions derived in this work.
• New Duality Functions: The method yields a new duality function for the non-dynamic stochastic higher spin vertex model (obtained by taking the limit $\lambda \to -i\infty$), which is expressed in terms of $2\phi_1$ functions.
• Triangular Duality: In a further limit, the duality reduces to a "triangular" duality (where the function is non-zero only if $\mu_i \ge \xi_i$), recovering known results from previous literature.

Significance and Claims
The paper claims that this work represents the first probabilistic application of the universal Drinfeld twister and the universal R-matrix (rather than fusion) to construct duality functions for dynamic models.

• Algebraic Generality: The authors assert that because the algebraic construction is formulated in a general setting involving quasi-triangular quasi-Hopf algebras, the method is expected to produce duality functions for many other dynamic integrable models beyond the specific $U_q(\mathfrak{sl}_2)$ case studied here.
• Orthogonality Advantage: The paper highlights that obtaining orthogonal polynomial dualities is an improvement over "triangular" dualities. Orthogonal dualities possess a priori applications in probability theory, such as in the derivation of Boltzmann-Gibbs principles and correlation formulas, which are not immediately obvious for triangular dualities.
• Unification: The work unifies the understanding of duality in dynamic models by showing that the specific results for dynamic ASEP are special cases of a broader class of dynamic stochastic higher spin models, all derivable from the same algebraic machinery.

The authors maintain a modest tone regarding future implications, stating that extending the perspective of duality functions as vertex weights to other orthogonal polynomial dualities will require new algebraic insights, which they discuss but do not fully resolve. The paper focuses on establishing the existence and explicit form of these dualities for the specific class of models introduced.