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Combinatorial bases in quantum toroidal $\mathfrak{gl}_2$ modules

Michio Jimbo, Evgeny Mukhin

TL;DR

The paper develops a general combinatorial framework for constructing tame modules of the quantum toroidal gl_2 algebra via balanced q-characters and labeled colored plane partitions. It proves a main theorem that a data pair (S,\Psi) under a set of axioms yields a tame irreducible E-module V(S,\Psi), with explicit action defined by residues at pole locations. This framework is instantiated in numerous explicit examples, including vector and Fock representations, Macmahon plane-partition modules, and a hierarchy of evaluation Verma, relaxed Verma, and slanted relaxed Verma modules, each with concrete l-weight data and character formulas. The results unify and extend known combinatorial constructions while enabling new tame modules, highlighting rich connections to plane partitions, Macmahon formulas, and potential extensions to higher rank algebras and integrable-system applications.

Abstract

We show that many tame modules of the quantum toroidal $\mathfrak{gl}_2$ algebra can be explicitly constructed in a purely combinatorial way using the theory of $q$-characters. The examples include families of evaluation modules obtained from analytic continuation and automorphism twists of Verma modules of the quantum affine $\mathfrak{gl}_2$ algebra. The combinatorial bases in the modules are labeled by colored plane partitions with various properties.

Combinatorial bases in quantum toroidal $\mathfrak{gl}_2$ modules

TL;DR

The paper develops a general combinatorial framework for constructing tame modules of the quantum toroidal gl_2 algebra via balanced q-characters and labeled colored plane partitions. It proves a main theorem that a data pair (S,\Psi) under a set of axioms yields a tame irreducible E-module V(S,\Psi), with explicit action defined by residues at pole locations. This framework is instantiated in numerous explicit examples, including vector and Fock representations, Macmahon plane-partition modules, and a hierarchy of evaluation Verma, relaxed Verma, and slanted relaxed Verma modules, each with concrete l-weight data and character formulas. The results unify and extend known combinatorial constructions while enabling new tame modules, highlighting rich connections to plane partitions, Macmahon formulas, and potential extensions to higher rank algebras and integrable-system applications.

Abstract

We show that many tame modules of the quantum toroidal algebra can be explicitly constructed in a purely combinatorial way using the theory of -characters. The examples include families of evaluation modules obtained from analytic continuation and automorphism twists of Verma modules of the quantum affine algebra. The combinatorial bases in the modules are labeled by colored plane partitions with various properties.
Paper Structure (17 sections, 12 theorems, 78 equations, 12 figures)

This paper contains 17 sections, 12 theorems, 78 equations, 12 figures.

Table of Contents

  1. Introduction
  2. The quantum affine algebra associated to $\mathfrak{gl}_2$
  3. Quantum algebra $U_q{\mathfrak{gl}}_2$
  4. Quantum affine algebra $U_q\widehat{\mathfrak{gl}}_2$
  5. Examples of representations.
  6. Quantum toroidal $\mathfrak{gl}_2$ algebra
  7. The perpendicular realization $\mathcal{E}^\perp$
  8. Parallel realization of $\mathcal{E}^\perp$
  9. An explicit construction of tame $\mathcal{E}$-modules
  10. The proof of Theorem \ref{['main thm']}
  11. Examples of combinatorial $\mathcal{E}$-modules.
  12. Vector representations
  13. Fock modules
  14. Macmahon modules.
  15. Evaluation Verma modules
  16. ...and 2 more sections

Key Result

Proposition 3.1

Let $u\in {\mathbb C}^{\times}$. Let $C=q_3$. There exists a surjecive homomorphism of algebras: $ev_u: \mathcal{E}^\perp \to \widetilde{U}_{q,q_3} \widehat{\mathfrak{gl}}_2,$ such that $ev_u\circ v^\perp= id$ and $\deg(ev_u(E_{0,k}^\perp))=(-1,k)$, $\deg(ev_u(F_{0,k}^\perp))=(1,k)$. ∎

Figures (12)

  • Figure 1: Grading in a relaxed $U_q\widehat{\mathfrak{gl}}_2$ Verma module.
  • Figure 2: Grading in a slope 2 relaxed $U_q\widehat{\mathfrak{gl}}_2$ Verma module.
  • Figure 3: A state in the Fock module.
  • Figure 4: A state in the Macmahon module.
  • Figure 5: The prohibited box and the Fock module.
  • ...and 7 more figures

Theorems & Definitions (20)

  • Proposition 3.1: M2,FJM1
  • Proposition 3.2: M1
  • Proposition 3.3
  • Lemma 3.4
  • Theorem 3.5
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • proof
  • Remark 4.3
  • ...and 10 more