Freezing operators in representation theory of quantum loop algebras

Abstract: We prove the Hernandez conjecture on the simple $(q,t)$-characters (an analog of the Kazhdan--Lusztig conjecture) for untwisted quantum loop algebras of classical type. This result is new in type $\mathrm{C}$. We also prove that the folding homomorphism, introduced by Hernandez, gives a dimension-preserving bijective correspondence between the finite-dimensional simple representations of untwisted quantum loop algebras of classical simply-laced type and those of the corresponding doubly-twisted quantum loop algebras. This result is new in type $\mathrm{D}$. In our approach, we develop a bootstrapping method for $q$ and $(q,t)$-characters, based on the freezing operator previously introduced in the context of cluster algebras by the second named author. This method allows us to reduce statements for general simple representations in all classical types to corresponding results on core subcategories in a uniform manner.

• The paper proves the Hernandez conjecture for all classical types by leveraging a novel freezing operator to equate (q,t)-characters with q-characters.
• The freezing operator provides a uniform bootstrapping method that bypasses geometric methods, ensuring correct specialization and folding homomorphisms.
• The results enable algorithmic character computation, bijections between untwisted and twisted algebras, and categorical isomorphisms of Grothendieck rings.

Freezing Operators and $(q,t)$-Characters in Quantum Loop Algebra Representation Theory

Introduction and Context

This paper establishes new algebraic and categorical links in the finite-dimensional representation theory of quantum loop algebras, resolving previously open structural and computational conjectures. Focusing on quantum loop algebras $U_q(\mathcal{L}\mathfrak{g})$, the authors introduce and utilize the freezing operator as a bootstrapping method on $q$- and $(q,t)$-characters. This enables the reduction of Kazhdan–Lusztig-type statements for arbitrary classical types to manageable core subcategories. Notably, the paper provides a proof of the Hernandez conjecture for all classical types, completing the picture for type $\mathrm{C}$, and establishes a new, uniform transfer theorem connecting simple representations of untwisted and doubly-twisted quantum loop algebras in classical simply-laced types, covering the previously unresolved case of type $\mathrm{D}$ (2601.00687).

$(q,t)$-Characters and the Hernandez Conjecture

A fundamental object in finite-dimensional representations of $U_q(\mathcal{L}\mathfrak{g})$ is the $q$-character, which encodes the quantum Cartan data of a module and enables algorithmic calculations. The introduction of the $(q,t)$-character by Nakajima, analogous to the Kazhdan–Lusztig theory for simple Lie algebras, circumvented earlier computational barriers but was restricted to simply-laced types via geometric realization using quiver varieties. The Hernandez conjecture posited that, for general types, the specialization at $t=1$ of a combinatorially defined $(q,t)$-character equals the $q$-character of the same simple module:

$\chi_{q,t}(L)|_{t=1} = \chi_q(L).$

While prior work had resolved this conjecture for certain types (notably $\mathrm{A}$, and, via geometric or categorical means, type $\mathrm{B}$), the complete classical series remained unresolved, particularly for type $\mathrm{C}$.

The present work establishes this conjecture for all classical types ($\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$, and $\mathrm{D}$) by constructing a uniform transfer via the freezing operator. Notably, the resolution for type $\mathrm{C}$ is new and, for types $\mathrm{A}$, $\mathrm{B}$, and $\mathrm{D}$, the argument presents an alternative to existing geometric approaches, relying solely on the algebraic category and quantum cluster algebra tools.

The Freezing Operator and Bootstrapping Method

The freezing operator is an algebraic operation first introduced in the context of cluster algebras and adapted herein to modules over quantum loop algebras. The operator $f^{\tilde{\mathfrak{g}}}_{\mathfrak{g}}$ acts on $q$- and $(q,t)$-characters to extract the leading constituent in the restriction from a larger quantum loop algebra $U_q(\mathcal{L}\tilde{\mathfrak{g}})$ to a smaller subalgebra $U_q(\mathcal{L}\mathfrak{g})$ (with the Dynkin diagram of $\mathfrak{g}$ a subdiagram of $\tilde{\mathfrak{g}}$). The key features are:

• It relates the representation-theoretic characters between different ranks, with an explicit control over the highest weight constituents.
• Most importantly, it commutes with the specialization $t=1$ (as per the $(q,t)$-character construction).
• It reduces categorical and character-theoretic results for arbitrary modules to the case of "core subcategories," for which the conjecture is established (via cluster algebra or quiver Hecke techniques).

This bootstrapping technique is essential in proving the main theorems. Through careful embedding and induction on the rank, the authors show that any simple module $L$ for a classical $U_q(\mathcal{L}\mathfrak{g})$ arises as a leading restriction from a (large enough) higher-rank core module. The commutation properties of the freezing operator with both evaluation and folding homomorphisms are then leveraged to transfer identities for $(q,t)$-characters reliably.

Strong Theoretical Results

Resolution of the Hernandez Conjecture for All Classical Types

The first principal result is the confirmation, for all classical types, that the algebraically constructed $(q,t)$-character specializes at $t=1$ to the $q$-character for simple modules:

• The method is independent of quiver variety geometry, enabling a uniform treatment for types $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$, and $\mathrm{D}$ [(2601.00687), Thm H-conj-C].
• For type $\mathrm{C}$, this establishes the conjecture for the first time.

Folding and Bijection Results for Twisted Quantum Loop Algebras

The authors further apply the freezing method to twisted quantum loop algebras (those corresponding to twisted affine types) using the folding homomorphism. The key result is a bijection between simple modules of the untwisted algebra $U_q(\mathcal{L}\mathfrak{g})$ and the twisted algebra $U_q(\mathcal{L}\mathfrak{g}^\sigma)$:

• The folding homomorphism is shown to be dimension-preserving and bijective for types $\mathrm{A}$ and $\mathrm{D}$ with order-2 automorphism.
• The proof does not rely on geometric or categorification technology but follows strictly from the freezing and combinatorial framework.
• This answers an open question for type $\mathrm{D}$, filling a gap in the twisting theory [(2601.00687), Thm tw-main].

Numerical and Categorical Implications

The theoretical statements have strong categorical consequences:

• The Grothendieck rings of the module categories for $U_q(\mathcal{L}\mathfrak{g})$ and for the twisted algebras are shown to be isomorphic in a way that preserves the full character data and dimensions.
• The commutativity of the freezing operator with specialization and folding is essential for ensuring these bijections, as well as for confirming that all subquotients and tensor decomposition rules are preserved.

Broader Implications and Future Directions

The algebraic strategy presented here has substantial implications:

• Algorithmic Character Computation: With the Hernandez conjecture holding uniformly for classical types, explicit (algorithmic) calculation of $q$-characters for simple modules in quantum loop algebras is now available, supporting advanced computational approaches in combinatorics and categorification.
• Cluster Algebras and Canonical Bases: The freezing operator, as adapted from cluster algebra theory, further cements the connection between quantum loop algebra representation categories, quantum cluster structures, and dual canonical bases, suggesting additional generalizations for other quantum algebras or further relations with categorified cluster structures.
• Exceptional Types: The authors remark that their bootstrapping approach does not extend directly to exceptional types, since there is no infinite family of increasing rank algebras to serve as ambient spaces for freezing. This delineates a natural boundary for the present methods and points to potential requirements for new structures or tools (possibly geometric) for exceptional type cases.
• Extensions to Twisted Types: The extension to twisted quantum loop algebras, especially in type $\mathrm{D}$, enhances the understanding of the Langlands duality relations and may have applications in integrable systems, geometric representation theory, and categorical equivalences.

Conclusion

This work completely resolves the structure and combinatorial character formula conjecture for simple modules over quantum loop algebras of classical type, using an algebraic bootstrapping method grounded in the freezing operator and quantum Grothendieck rings. The uniform strategy not only clarifies the combinatorics of $(q,t)$-characters but also establishes precise correspondences between representations of untwisted and twisted quantum loop algebras. These results provide new, robust foundations for further theoretical developments in quantum affine algebra representation theory and its relationships to cluster algebras and categorification (2601.00687).