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arxiv: 2402.13544 · v4 · submitted 2024-02-21 · 🧮 math.RT · math.QA

Monoidal Jantzen filtrations

Ryo Fujita , David Hernandez This is my paper

Pith reviewed 2026-05-24 04:19 UTC · model grok-4.3

classification 🧮 math.RT math.QA
keywords monoidal Jantzen filtrationsGrothendieck ringsquantum loop algebrasquiver Hecke algebrasquantizationKazhdan-Lusztig polynomialsfinite-dimensional representations
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The pith

A monoidal Jantzen filtration deforms the Grothendieck ring multiplication into an associative quantization that matches the geometric version for simply-laced quantum loop algebras.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a monoidal analogue of Jantzen filtrations inside monoidal abelian categories that carry generic braidings. This filtration produces a deformed product on the Grothendieck ring of the category. For finite-dimensional representations of simply-laced quantum loop algebras the deformed product is shown to be associative and the resulting ring coincides with the quantum Grothendieck ring previously obtained by geometric methods. The same conclusion holds for a monoidal category of modules over symmetric quiver Hecke algebras that categorifies the coordinate ring of a unipotent group attached to a Weyl group element. The construction therefore supplies a representation-theoretic route to the quantization and to analogs of Kazhdan-Lusztig polynomials.

Core claim

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 2.5

What carries the argument

The monoidal Jantzen filtration on objects of a monoidal abelian category with generic braiding, which induces a deformed multiplication on the Grothendieck ring.

If this is right

  • The construction yields analogs of Kazhdan-Lusztig polynomials.
  • It supplies a representation-theoretic interpretation of the quantum Grothendieck ring.
  • The same associativity result holds for modules over symmetric quiver Hecke algebras.
  • The approach gives information on the homological structure of representations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same monoidal filtration technique might apply to other braided monoidal categories arising in representation theory.
  • It could offer a uniform way to produce deformations of Grothendieck rings in settings where geometric constructions are unavailable.
  • Explicit computations in small-rank cases could test whether the associativity holds beyond the simply-laced and quiver-Hecke examples already treated.

Load-bearing premise

The monoidal abelian category admits a generic braiding and satisfies the technical conditions needed for the Jantzen filtration to be well-defined and to induce a deformation whose associativity can be checked via explicit computations.

What would settle it

An explicit pair of finite-dimensional representations of a simply-laced quantum loop algebra for which the deformed multiplication fails to be associative or produces a ring different from the Nakajima-Varagnolo-Vasserot quantum Grothendieck ring.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces monoidal Jantzen filtrations on monoidal abelian categories equipped with generic braidings. These filtrations induce a deformation of the multiplication on the Grothendieck ring. The authors conjecture associativity of the deformed multiplication in general and prove it for two main families: finite-dimensional representations of simply-laced quantum loop algebras (where the resulting quantization coincides with the Nakajima–Varagnolo-Vasserot quantum Grothendieck ring) and finite-dimensional modules over symmetric quiver Hecke algebras categorifying the coordinate ring of a unipotent group associated to a Weyl group element. Applications to the homological structure of representations are derived, along with analogs of Kazhdan–Lusztig polynomials.

Significance. If the proofs hold, the work supplies a representation-theoretic construction of the quantum Grothendieck ring that unifies it with the existing geometric approach, while introducing a general deformation technique applicable to other monoidal categories. The explicit verification of associativity and coincidence in the two main examples, together with the resulting homological applications, constitute a substantive advance in the representation theory of quantum loop algebras and quiver Hecke algebras.

minor comments (2)
  1. [Abstract / §1] The abstract states that associativity is proved 'in many remarkable situations' but only details two main examples; a brief indication of the scope of the additional cases (e.g., a sentence in §1 or the introduction) would help readers assess the breadth of the results.
  2. [§3 / §5] Notation for the deformed multiplication (presumably denoted something like * or ⋆) and the associated filtration should be introduced with a single consistent symbol and cross-referenced at first use in each main example section.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript and for recommending acceptance. We are pleased that the work is viewed as providing a substantive advance.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation introduces a monoidal Jantzen filtration on monoidal abelian categories with generic braidings, defines a deformation of the Grothendieck ring multiplication from this filtration, conjectures associativity, and verifies it (plus coincidence with the Nakajima–Varagnolo–Vasserot quantum Grothendieck ring) via explicit representation-theoretic computations in the target categories. No step reduces by the paper's own equations to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claims rest on independent verification outside the defining construction itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on the existence of a monoidal abelian category with generic braiding and on the specific representation-theoretic properties of the two example categories; no free parameters or invented entities are mentioned in the abstract.

axioms (2)
  • domain assumption The monoidal abelian category admits a generic braiding
    Invoked to define the monoidal Jantzen filtration.
  • domain assumption The categories of finite-dimensional representations of simply-laced quantum loop algebras and of symmetric quiver Hecke algebras satisfy the required monoidal and braiding hypotheses
    Needed for the associativity statements and the identification with the geometric quantum Grothendieck ring.

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    MR3612477 (R. Fujita) Research Institute for Mathematical Sciences, Kyoto Unive rsity, Oiw ake-Kitashirakaw a, Sakyo, Kyoto, 606-8502, Japan Email address : rfujita@kurims.kyoto-u.ac.jp (D. Hernandez) Universit´e P aris Cit´e and Sorbonne Universit ´e, CNRS, IMJ-PRG, F-75013, P aris, France Email address : david.hernandez@imj-prg.fr

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