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Type $B$ - and $C$ - trigonometric weight functions

This paper introduces and analyzes type $B$ and $C$ trigonometric weight functions arising from integrable lattice models as $q$ -deformations of factorial $K$ -theoretic Schur $P$ - and $Q$ -functions, deriving their algebraic properties, recursive relations, and orthogonality formulas through the Yang-Baxter equation and partition functions of six-vertex models with triangular boundary conditions.

Original authors: Kohei Motegi

Published 2026-10-14

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Kohei Motegi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of modern mathematics, there exists a hidden architecture connecting two seemingly distant worlds: the study of symmetry in shapes and the behavior of particles in quantum systems. On one side, mathematicians study how complex geometric spaces, like those formed by arranging lines and planes in high dimensions, can be classified. These spaces often have special points where the geometry is particularly simple, and understanding the properties of these points helps solve deep problems in algebra and topology. On the other side, physicists and mathematicians study "integrable systems," which are models of particles interacting in a way that is so perfectly balanced that their behavior can be predicted exactly, without needing to rely on approximations. For decades, researchers have discovered that the formulas describing these balanced particle systems often match the formulas describing the geometry of those high-dimensional spaces. This connection is not just a coincidence; it suggests that the laws governing the microscopic world of particles are deeply woven into the fabric of geometric space itself.

A specific family of mathematical objects known as "weight functions" sits at the heart of this connection. Originally developed for a type of symmetry called "type A," these functions act like a universal translator, converting the language of quantum particles into the language of geometric shapes. They have been incredibly useful, allowing mathematicians to calculate properties of geometric spaces by solving problems in physics. However, for many years, a significant gap remained. While the "type A" functions were well understood, their counterparts for other fundamental types of symmetry, specifically "type B" and "type C," were missing. These missing pieces correspond to different geometric shapes, such as those found in orthogonal and symplectic spaces, which appear frequently in physics and geometry but had resisted the same kind of clean mathematical description. Without these functions, the bridge between the quantum world and these specific geometric shapes was incomplete.

In this work, the researcher Kohei Motegi has successfully constructed these missing functions. By building a new mathematical model based on a grid of interacting particles, he has created explicit formulas for the type B and type C weight functions. The construction begins with a six-vertex model, a simplified representation of a physical system where particles move along the edges of a grid. Imagine a grid where lines cross at every intersection, and at each crossing, the particles must follow specific rules about how they enter and exit. The researcher arranged these crossings in a triangular pattern and applied specific boundary conditions, essentially telling the particles how to behave at the edges of the grid. By calculating the total number of ways the particles could arrange themselves to satisfy these rules—a quantity known as a "partition function"—he derived the new weight functions.

The result is a set of formulas that behave exactly like the previously known type A functions but are adapted for the more complex symmetries of types B and C. The paper proves that these new functions possess a remarkable set of properties. They vanish, or become zero, under specific conditions that mirror the geometric structure of the spaces they describe. When they do not vanish, they can be evaluated at specific points to produce exact values that are products of simple terms, much like counting the ways to arrange a deck of cards. Furthermore, the researcher showed that these functions are divisible by certain factors, meaning they can be broken down into smaller, predictable pieces, and that their complexity is bounded in a precise way. These properties are not just mathematical curiosities; they are the defining characteristics that allow these functions to serve as reliable tools for calculation.

A key achievement of the paper is the demonstration of how these new functions relate to the older, well-known type A functions. The researcher derived "branching formulas," which show how to break down a complex type B or C function into a sum of simpler type A functions. This is akin to taking a large, intricate tapestry and showing that it is woven from many smaller, familiar threads. This relationship is powerful because it allows mathematicians to use the vast existing knowledge of type A functions to understand the new, more complex ones. The paper also establishes "orthogonality" relations, which are mathematical rules that ensure these functions are independent of one another in a precise sense, much like how the x, y, and z axes in a room are perpendicular to each other. This independence is crucial for using these functions to solve equations and classify geometric objects.

The work is rigorous and complete, providing not just the formulas but a full proof of their properties using the underlying physics of the grid model. The researcher utilized the "Yang-Baxter equation," a fundamental rule in integrable systems that ensures the consistency of particle interactions, to prove that the new functions behave correctly. By showing that these functions satisfy the same deep algebraic rules as their type A predecessors, the paper confirms that the connection between quantum integrable systems and geometric spaces holds true for these broader classes of symmetry. The findings are presented as exact mathematical proofs, leaving no room for ambiguity. This completion of the mathematical framework means that researchers can now apply these powerful tools to a wider range of geometric problems, potentially unlocking new insights into the structure of orthogonal and symplectic spaces. The paper stands as a definitive step in unifying the algebraic and geometric descriptions of these complex systems, providing a clear and complete set of tools for future exploration.

Technical Summary: Type B- and C- trigonometric weight functions

Problem Statement
The paper addresses the construction and algebraic characterization of trigonometric weight functions for types B and C. These functions are sought as $q$ -deformations of the factorial $K$ -theoretic Schur $P$ - and $Q$ -functions introduced by Ikeda and Naruse, which represent equivariant $K$ -theoretic Schubert classes for orthogonal and symplectic Grassmannians. While type A trigonometric weight functions are well-established as off-shell Bethe wavefunctions and are linked to Grothendieck polynomials and stable envelopes, a parallel algebraic construction for types B and C, including their normalization, diagonal evaluations, and orthogonality properties, was the subject of this work.

Methodology
The author employs the framework of quantum integrable systems, specifically utilizing the six-vertex model with triangular boundary conditions.

Lattice Model Construction: The study begins with the partition function $F_{N,n,r}(u|w|I)$ of a six-vertex model defined by $R$ -matrices (stochastic $R$ -matrices) and triangular $K$ -matrices. The $K$ -matrices are specialized to $K_C(u)$ and $K_B(u)$ to correspond to types C and B, respectively.
Symmetrization and Specialization: Using an explicit symmetrization formula for the partition function, the author defines unnormalized weight functions $W_{C/N,n,r}$ and $W_{B/N,n,r}$ by specific parameter choices ( $\gamma=0, \beta=1$ for type C; $\gamma=-1, \beta=1$ for type B).
Normalization: Normalized weight functions $\widehat{W}_{C/N,n,r}$ and $\widehat{W}_{B/N,n,r}$ are introduced by dividing the unnormalized functions by specific scalar factors $E^C_{N,n,r}$ and $E^B_{N,n,r}$ . This normalization is designed to yield diagonal evaluations expressible as products over roots and to satisfy specific divisibility and degree bounds.
Algebraic Derivation: The properties of these functions are derived using the Yang–Baxter equation, the reflection equation for the $K$ -matrices, and the "railroad" argument (graphical manipulations of the lattice model).
Reduction to Type A: A crucial methodological step involves establishing a branching formula that relates the type B and C weight functions to type A trigonometric weight functions (identified with $K$ -theoretic stable bases for type A flag varieties). This reduction allows the transfer of known type A properties (such as orthogonality and uniqueness characterizations) to the B and C cases.

Key Contributions and Results
The paper establishes the following main results for the normalized weight functions:

Algebraic Characterization (Equal-Cardinality): For specializations where the number of spectral variables equals the cardinality of the indexing subset ( $|M|=|I|=r$ ), the normalized weight functions are uniquely characterized by:
- Triangular Vanishing: The function vanishes unless the subset $M$ dominates the subset $I$ in a specific partial order ( $M \ge I$ ).
- Diagonal Evaluation: At $M=I$ , the function evaluates to a product of terms corresponding to the inversion set and ascent set of the associated strict partition, involving factors like $(1-e^{-\alpha})$ and $(1-qe^{+\alpha})$ .
- Divisibility: The functions satisfy specific divisibility conditions by products of linear factors related to the root system.
- Newton Polytope Bounds: The functions satisfy "smallness" conditions regarding their Newton polytopes (bounds on degrees), analogous to those in type A.
- Uniqueness: These properties, combined with exchange congruences, uniquely determine the equal-cardinality specializations.
Recursive Relations and Stability: The paper derives recursive relations connecting weight functions with different numbers of spectral variables and different cardinalities. It also establishes stability properties, showing that the functions remain invariant under the addition of trivial variables (setting $u_n=1$ ).
Branching Formulas: Explicit branching formulas are derived for specializations of arbitrary cardinality ( $s \ge r$ ). These formulas express the type B/C weight functions as sums over subsets of the type A weight functions, weighted by specific rational factors. This provides a bridge between the different root systems.
Orthogonality and Biorthogonality:
- Equal-Cardinality Orthogonality: The paper proves orthogonality relations for the equal-cardinality specializations, utilizing the inverse of the type A specialization matrix.
- All-Cardinality Biorthogonality: By combining the branching formulas with the orthogonality of type A functions, the author establishes a biorthogonality relation that holds across all cardinalities. This leads to an inversion formula for the branching coefficients.
Laurent Polynomiality and Support: The functions are shown to be Laurent polynomials in the variables $w$ . Support conditions are proven, showing that the functions vanish unless the indexing subsets satisfy specific inclusion criteria related to the underlying strict partitions.

Significance
The paper claims to provide a natural algebraic construction of $q$ -deformations of the factorial $K$ -theoretic Schur $P$ - and $Q$ -functions via integrable lattice models. By realizing these functions as specializations of six-vertex model partition functions, the work connects integrable systems with the Schubert calculus of orthogonal and symplectic Grassmannians. The derived algebraic properties (diagonal values, divisibility, Newton bounds) mirror the geometric properties of stable envelopes in type A, suggesting a deep structural parallel between the integrable model constructions and the geometry of these varieties. The results offer explicit formulas and recursive structures that facilitate the study of these functions, with the author noting that a detailed geometric interpretation involving motivic Chern classes and stable bases is reserved for subsequent work.

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