pith. sign in

arxiv: 2112.12688 · v2 · submitted 2021-12-23 · 🧮 math.QA · math.RT

Minimal presentations of gln-web categories

Genta Latifi , Daniel Tubbenhauer This is my paper

Pith reviewed 2026-05-24 13:04 UTC · model grok-4.3

classification 🧮 math.QA math.RT
keywords gln-websquantum groupspresentationsrepresentation categoriesgenerators and relationsintegral forms
0
0 comments X

The pith

The category of gln-webs admits a minimal presentation over fields with generic quantum parameters.

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

The paper establishes that the category of gln-webs, which encodes the representation theory of the quantum group Uq(gln), can be defined by a minimal collection of generators and relations when the quantum parameters are generic. This presentation is coefficient-free in an integral form that differs from prior versions in the literature. A sympathetic reader would care because such a presentation reduces the data needed to work with morphisms and representations in these categories. It directly supports computations that previously required larger or less explicit sets of relations.

Core claim

The category of gln-webs over a field with generic quantum parameters admits a minimal presentation. The authors additionally construct an integral presentation that is as coefficient-free as possible and differs from earlier versions in the literature.

What carries the argument

The minimal presentation of generators and relations that defines the gln-web category.

If this is right

  • Morphisms between gln-webs can be computed using only the minimal relations.
  • An integral form exists with fewer coefficients than previous presentations.
  • The presentation applies uniformly to the representation category of Uq(gln) at generic parameters.

Where Pith is reading between the lines

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

  • This minimal form may reduce the complexity of diagrammatic calculations in related quantum categories.
  • The approach could be tested on small values of n to verify that all known web relations follow from the minimal set.

Load-bearing premise

The quantum parameters are generic and the gln-web category is the representation category associated to Uq(gln).

What would settle it

An explicit morphism or relation in the gln-web category that cannot be obtained from the proposed minimal generators and relations, or a demonstration that the proposed set is still redundant.

read the original abstract

In this paper we study categories of gln-webs which describe associated representation categories of the quantum group Uq(gln). We give a minimal presentation of the category of gln-webs over a field with generic quantum parameters. We additionally describe an integral presentation which differs from others in the literature because it is "as coefficient-free as possible".

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 paper studies categories of gl_n-webs, which are diagrammatic categories equivalent to the representation category of the quantum group U_q(gl_n). It claims to give a minimal presentation of the gl_n-web category over a field with generic quantum parameters, together with an integral presentation that is as coefficient-free as possible.

Significance. If the claimed minimal presentation is correct and complete, the result supplies a concrete set of generators and relations for a central object in quantum representation theory and low-dimensional topology. The emphasis on genericity and coefficient-minimality is useful for specialization arguments and integral forms; the work therefore has potential value for computations of morphisms and invariants in web categories.

minor comments (2)
  1. The abstract and title use the notation 'gln' and 'Uq(gln)' without subscripts; standardizing to gl_n and U_q(gl_n) throughout would improve readability and consistency with the literature.
  2. The phrase 'as coefficient-free as possible' in the abstract is informal; a precise statement of which coefficients are retained or eliminated (e.g., in which ring the relations live) should appear in the introduction or the statement of the integral presentation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the recognition of its potential value for computations in web categories, and the recommendation of minor revision. No specific major comments appear in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper states a direct existence result for a minimal presentation of the gl_n-web category as equivalent to Rep(U_q(gl_n)) at generic q. The abstract and provided material describe a standard construction of generators and relations without any reduction of the central claim to fitted parameters, self-definitional loops, or load-bearing self-citations. The generic-parameter hypothesis is the conventional one that excludes roots-of-unity relations and is flagged explicitly rather than smuggled in. No equations or steps in the excerpt equate a derived object to its own input by construction, and the completeness argument (standard for such diagrammatic categories) draws on external representation-theoretic facts rather than internal re-labeling.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on abstract; no free parameters, invented entities, or ad-hoc axioms are mentioned. Relies on standard background in quantum group representation theory.

axioms (1)
  • standard math Standard axioms of monoidal categories and representation theory of quantum groups Uq(gln).
    The paper builds directly on established definitions of gln-web categories as modeling Uq(gln) representations.

pith-pipeline@v0.9.0 · 5567 in / 992 out tokens · 20751 ms · 2026-05-24T13:04:20.275910+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. On Hecke and asymptotic categories for a family of complex reflection groups

    math.RT 2024-09 unverdicted novelty 6.0

    Constructs Hecke algebras and asymptotic versions for G(M,M,N) complex reflection groups by generalizing the dihedral case.

  2. Orthogonal webs and semisimplification

    math.RT 2024-01 unverdicted novelty 4.0

    A diagrammatic category equivalent to tilting representations of the orthogonal group is defined and its semisimplification is described, valid in characteristic not equal to two.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · cited by 2 Pith papers

  1. [1]

    Andersen, P

    H.H. Andersen, P. Polo, and K.X. Wen. Representations of quantum algebras. Invent. Math. , 104(1):1--59, 1991

    work page 1991

  2. [2]

    Andersen, C

    H.H. Andersen, C. Stroppel, and D. Tubbenhauer. Cellular structures using U_q -tilting modules. Pacific J. Math. , 292(1):21--59, 2018

    work page 2018

  3. [3]

    Brown, N.J

    G.C. Brown, N.J. Davidson, and J.R. Kujawa. Quantum webs of type q. 2020

    work page 2020

  4. [4]

    Brundan, I

    J. Brundan, I. Entova-Aizenbud, P. Etingof, and V. Ostrik. Semisimplification of the category of tilting modules for GL_n . Adv. Math. , 375:107331, 37, 2020

    work page 2020

  5. [5]

    Berenstein and S

    A. Berenstein and S. Zwicknagl. Braided symmetric and exterior algebras. Trans. Amer. Math. Soc. , 360(7):3429--3472, 2008

    work page 2008

  6. [6]

    Cautis, J

    S. Cautis, J. Kamnitzer, and S. Morrison. Webs and quantum skew H owe duality. Math. Ann. , 360(1-2):351--390, 2014

    work page 2014

  7. [7]

    B. Elias. Light ladders and clasp conjectures. 2015

    work page 2015

  8. [8]

    Kuperberg

    G. Kuperberg. Spiders for rank 2 L ie algebras. Comm. Math. Phys. , 180(1):109--151, 1996

    work page 1996

  9. [9]

    G. Latifi. Minimal and integral presentations of gl _n -web categories. Ph.D. thesis , 2021. Universit \"a t Z \"u rich

    work page 2021

  10. [10]

    a user Classics. Birkh\

    G. Lusztig. Introduction to quantum groups . Modern Birkh\" a user Classics. Birkh\" a user/Springer, New York, 2010. Reprint of the 1994 edition

    work page 2010

  11. [11]

    Rose and D

    D.E.V. Rose and D. Tubbenhauer. Symmetric webs, J ones-- W enzl recursions, and q - H owe duality. Int. Math. Res. Not. IMRN , (17):5249--5290, 2016

    work page 2016

  12. [12]

    u r die V alenztheorie geeignete B asis der bin \

    G. Rumer, E. Teller, and H. Weyl. Eine f \"u r die V alenztheorie geeignete B asis der bin \"a ren V ektorinvarianten. Nachrichten von der Ges. der Wiss. Zu G \"o ttingen. Math.-Phys. Klasse , pages 498--504, 1932. In German

    work page 1932

  13. [13]

    Robert and E

    L.-H. Robert and E. Wagner. Symmetric K hovanov- R ozansky link homologies. J. \'E c. polytech. Math. , 7:573--651, 2020

    work page 2020

  14. [14]

    Sartori and D

    A. Sartori and D. Tubbenhauer. Webs and q - H owe dualities in types BCD . Trans. Amer. Math. Soc. , 371(10):7387--7431, 2019

    work page 2019

  15. [15]

    V.G. Turaev. The category of oriented tangles and its representations. Funktsional. Anal. i Prilozhen. , 23(3):93--94, 1989

    work page 1989

  16. [16]

    Tubbenhauer, P

    D. Tubbenhauer, P. Vaz, and P. Wedrich. Super q - H owe duality and web categories. Algebr. Geom. Topol. , 17(6):3703--3749, 2017

    work page 2017