Extremal monomial property of q-characters and polynomiality of the X-series (2504.00260v1)

Abstract: The character of every irreducible finite-dimensional representation of a simple Lie algebra has the highest weight property. The invariance of the character under the action of the Weyl group W implies that there is a similar "extremal weight property" for every weight obtained by applying an element of W to the highest weight. In this paper we conjecture an analogous "extremal monomial property" of the q-characters of simple finite-dimensional modules over the quantum affine algebras, using the braid group action on q-characters defined by Chari. In the case of the identity element of W, this is the highest monomial property of q-characters proved in arXiv:math/9911112. Here we prove it for simple reflections. Somewhat surprisingly, the extremal monomial property for each w in W turns out to be equivalent to polynomiality of the "X-series" corresponding to w, which we introduce in this paper. We show that these X-series are equal to certain limits of the generalized Baxter operators for all w in W. Thus, we find a new bridge between q-characters and the spectra of XXZ-type quantum integrable models associated to quantum affine algebras. This leads us to conjecture polynomiality of all generalized Baxter operators, extending the results of arXiv:1308.3444.

Extremal Monomial Property of q-Characters and Polynomiality of X-Series

The paper addresses the intricate interplay between q-characters of finite-dimensional modules over quantum affine algebras and quantum integrable models of the XXZ type. The authors, Frenkel and Hernandez, propose extensions of key concepts from the theory of Lie algebras to quantum affine settings, primarily through the notion of "extremal monomial property" of q-characters.

Main Contributions:

1. Extremal Monomial Property:
   • Extending the classical theory, which links characters of Lie algebras with the highest weight property, the authors conjecture an "extremal monomial property" for q-characters. This property generalizes the highest monomial property and applies to every element of the Weyl group $W$.
2. Braid Group Action and Polynomiality:
   • The relationship between q-characters and braid group actions, originally due to Chari, is utilized. This connection facilitates discussions on extremal monomials and introduces the "X-series," which serve as a bridge between q-characters and generalized Baxter operators' spectra. Notably, the authors establish that the extremal monomial condition for each $w \in W$ is equivalent to the polynomiality of these X-series.
3. Generalized Baxter Operators:
   • The paper introduces generalized Baxter operators and conjectures their full polynomiality, extending prior results on known special cases. These operators are significant in analyzing the spectra of quantum integrable models associated with quantum affine algebras.
4. Theoretical and Technical Framework:
   • A detailed technical framework is provided, including invariant pairings and extensions of the Weyl and braid group actions on q-characters. The authors establish W-equivariance for homomorphisms involved and analyze the roles of root monomials.

Numerical and Theoretical Implications:

The equivalence between extremal monomial properties and the polynomiality of the X-series $Xw(wi)(z)$ is non-trivial and points to deeper symmetries within quantum affine algebras. It implies an existing invariant structure tied to the fundamental representations of quantum affine algebras, facilitating a more aesthetic and computationally effective framework for understanding module representations and their characters.

Future Directions:

The authors suggest the extensions of these properties could set the stage for further advancements in quantum integrable systems, especially in evaluating and predicting the spectral characteristics of associated Hamiltonians with unprecedented accuracy. The paper also opens avenues for extending these theoretical constructs to multifaceted applications such as deformed W-algebras and possibly other algebraic structures where quantum integrability is prevalent.

In conclusion, this work enriches the intersection between the representation theory of quantum groups and the mathematical framework underpinning quantum integrable models, specifically through innovative extensions to the concept of polynomiality in q-characters. The conjectures and theorems presented, especially concerning simple reflections and the longest element in the Weyl group, provide a robust platform for deeper explorations into both algebraic and physical domains.