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Monoidal Jantzen filtrations

Ryo Fujita, David Hernandez

TL;DR

The paper develops a monoidal analogue of Jantzen filtrations inside monoidal abelian categories with generic braidings, producing a deformation of the Grothendieck ring and proposing an associative star product that quantizes the ring and yields KL-type polynomials. It implements the construction in two major settings: finite-dimensional modules over simply-laced quantum loop algebras and finite-dimensional modules over symmetric quiver Hecke algebras, proving associativity in both and linking to Nakajima–Varagnolo–Vasserot geometric realizations of the quantum Grothendieck ring. The authors provide a cohesive PBW-theory framework, define decategorified invariants [M]_t, and establish a canonical basis in the t-deformed setting, with geometric proofs based on perverse sheaves, hard Lefschetz, and hyperbolic localization. These results give a representation-theoretic interpretation of quantum Grothendieck rings, yield new insights into the homological structure of mixed tensor products, and pave the way for KL-type algorithms in non-simply-laced cases via associativity and deformation theory.

Abstract

We introduce a monoidal analogue of Jantzen filtrations in the framework of monoidal abelian categories with generic braidings. It leads to a deformation of the multiplication of the Grothendieck ring. We conjecture, and we prove in many remarkable situations, that this deformation is associative so that our construction yields a quantization of the Grothendieck ring as well as analogs of Kazhdan-Lusztig polynomials. As a first main example, for finite-dimensional representations of simply-laced quantum loop algebras, we prove the associativity and we establish that the resulting quantization coincides with the quantum Grothendieck ring constructed by Nakajima and Varagnolo-Vasserot in a geometric manner. Hence, it yields a unified representation-theoretic interpretation of the quantum Grothendieck ring. As a second main example, we establish an analogous result for a monoidal category of finite-dimensional modules over symmetric quiver Hecke algebras categorifying the coordinate ring of a unipotent group associated with a Weyl group element. We obtain various applications, in particular on the homological structure of representations.

Monoidal Jantzen filtrations

TL;DR

The paper develops a monoidal analogue of Jantzen filtrations inside monoidal abelian categories with generic braidings, producing a deformation of the Grothendieck ring and proposing an associative star product that quantizes the ring and yields KL-type polynomials. It implements the construction in two major settings: finite-dimensional modules over simply-laced quantum loop algebras and finite-dimensional modules over symmetric quiver Hecke algebras, proving associativity in both and linking to Nakajima–Varagnolo–Vasserot geometric realizations of the quantum Grothendieck ring. The authors provide a cohesive PBW-theory framework, define decategorified invariants [M]_t, and establish a canonical basis in the t-deformed setting, with geometric proofs based on perverse sheaves, hard Lefschetz, and hyperbolic localization. These results give a representation-theoretic interpretation of quantum Grothendieck rings, yield new insights into the homological structure of mixed tensor products, and pave the way for KL-type algorithms in non-simply-laced cases via associativity and deformation theory.

Abstract

We introduce a monoidal analogue of Jantzen filtrations in the framework of monoidal abelian categories with generic braidings. It leads to a deformation of the multiplication of the Grothendieck ring. We conjecture, and we prove in many remarkable situations, that this deformation is associative so that our construction yields a quantization of the Grothendieck ring as well as analogs of Kazhdan-Lusztig polynomials. As a first main example, for finite-dimensional representations of simply-laced quantum loop algebras, we prove the associativity and we establish that the resulting quantization coincides with the quantum Grothendieck ring constructed by Nakajima and Varagnolo-Vasserot in a geometric manner. Hence, it yields a unified representation-theoretic interpretation of the quantum Grothendieck ring. As a second main example, we establish an analogous result for a monoidal category of finite-dimensional modules over symmetric quiver Hecke algebras categorifying the coordinate ring of a unipotent group associated with a Weyl group element. We obtain various applications, in particular on the homological structure of representations.
Paper Structure (51 sections, 62 theorems, 355 equations)

This paper contains 51 sections, 62 theorems, 355 equations.

Table of Contents

  1. Introduction
  2. Main construction
  3. Quantum loop algebras
  4. Applications
  5. Strategy of the proof
  6. Symmetric quiver Hecke algebras
  7. Further application
  8. General definitions for monoidal categories of representations
  9. PBW-theory for monoidal categories of representations
  10. Generically commutative deformations of simple modules
  11. Intertwiners arising from $R$-matrices
  12. Monoidal Jantzen filtrations
  13. Specialized $R$-matrices
  14. Decategorification
  15. Bar-involution and normality
  16. ...and 36 more sections

Key Result

Theorem 1.1

When $\mathfrak{g}$ is of simply-laced type, the following properties hold:

Theorems & Definitions (141)

  • Theorem 1.1: Nak04VV
  • Theorem 1.2: FHOOFHOO2
  • Conjecture 1.3: = Conjecture \ref{['qgrconj']}
  • Theorem 1.4: = Theorem \ref{['simply-laced']}
  • Theorem 1.5
  • Theorem 1.6: = Theorem \ref{['Thm:adapted']}
  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Definition 2.4
  • ...and 131 more