Pith. sign in

REVIEW 2 major objections 3 minor 42 references

Affine highest weight structures on module categories over quiver Hecke algebras

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Module categories of quiver Hecke algebras are stratified in arbitrary type and over any field, with standard modules built from determinantial modules.

desk verdict A genuinely important theorem with a clean R-matrix approach, but the proper costandard modules are not defined as written because the duality D is only set up for finite-dimensional modules. read the letter →

arxiv 2412.12903 v3 pith:C6CTWZOL submitted 2024-12-17 math.RT math.QA

Haruto Murata This is my paper · ORCID
classification math.RTmath.QA MSC 17B3717B6716G99
keywords quiverHeckealgebraaffinehighestweightcategorystratifieddeterminantialmoduleR-matrixcategorificationquantumunipotentsubgroupgradedmodules
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that, for every symmetrizable root datum and every base field, the category of finitely generated graded modules over a quiver Hecke algebra admits a stratification: the simple modules can be ordered so that each projective cover has a filtration by explicit standard modules. The standard modules are realized as convolution products of affinizations of determinantial modules, which the paper identifies as the categorical counterpart of the PBW basis. A truncation of this result shows that the full subcategory attached to any Weyl group element—the categorical analogue of a quantum unipotent subgroup—is an affine highest weight category with polynomial endomorphism rings and finite global dimension. If correct, this unifies and extends earlier special cases covering finite and symmetric affine types, and it does so without case-by-case computation. The argument is carried by the injectivity of renormalized R-matrices and by a short exact sequence that relates three determinantial modules.

What carries the argument

The central objects are determinantial modules $M(w\Lambda,v\Lambda)$ and their affinizations $\hat M(w\Lambda,v\Lambda)$: graded modules over the quiver Hecke algebra that categorify unipotent quantum minors. The mechanism that carries the argument is the renormalized R-matrix: Proposition 3.20 asserts that renormalized R-matrices of affinized determinantial modules are injective, and Theorem 4.11 uses this injectivity to construct the short exact sequence relating $\hat M(w\Lambda_i,\Lambda_i)$, $\hat M(w s_i\Lambda_i,w\Lambda_i)$, and $\hat M(w s_i\Lambda_i,\Lambda_i)$. This converts the braider structure of the convolution product into homological information: Theorem 4.16 shows that $\hat M(w s_i\Lambda_i,w\Lambda_i)$ is the projective cover of its head, which together with the BGG-type formula gives the stratification.

What would settle it

Take the affinization $\hat M(w\Lambda_i,\Lambda_i)$ constructed after Proposition 3.51 and compute the action of $p_{i,\beta}$ on it: Lemma 3.12 requires this action to be a nonzero power of $z$. If there exists some $w$ and field where $p_{i,\beta}$ acts by zero, then the injectivity of renormalized R-matrices (Proposition 3.20) fails at that step, and the short exact sequence of Theorem 4.11 would need another proof. Since Proposition 3.20 is used to prove that $\hat M_2$ is a projective cover (Theorem 4.16), checking this spectral condition in unexplored types or characteristics would directly settle whether the stratification extends.

Watch

Extended reading notes

Core claim

The central claim is Theorem 5.18: with respect to the map $\rho\colon\Sigma(\beta)\to P_{\preceq}(\beta)$ and the partial order $\le$, the category $R(\beta)$-gMod is a stratified category. Its standard modules are $\Delta(\lambda,s)=\Delta(s)\circ\Delta(\lambda)$, where $\Delta(\lambda)$ is a convolution of affinized determinantial modules and $\Delta(s)$ is a projective cover in the $R_{\ast,w}$ subcategory; the proper costandard modules are the duals $\nabla(\lambda,s)=D(\Delta(\lambda,s))$. Restricting to the subcategory $R_{w,\ast}(\beta)$-gMod yields Theorem 5.21: it is an affine highest weight category with standard modules $\Delta(\lambda)$ and endomorphism rings isomorphic to polynomial algebras. The proof identifies standard modules with convolutions of affinized determinantial modules, shows $\operatorname{Ext}^1$-vanishing through a foundational short exact sequence of the form $0\to q^{(\alpha_i,\alpha_i)+(\gamma_1,\gamma_2)}\hat M_1\circ\hat M_2\to\hat M_2\circ\hat M_1\to\hat M_3\to 0$, and verifies the BGG-type reciprocity formula for projective modules using the coincidence of the Ext bilinear form with Kashiwara's bilinear form under categorification.

Load-bearing premise

The proof rests on the injectivity of renormalized R-matrices for affinized determinantial modules, which in turn assumes the central elements $p_{i,\beta}$ act on the affinizations as nonzero powers of the degree-shift endomorphism $z$; if that spectral condition fails for any determinantial affinization used in Section 3.7, the projectivity of the standard modules and hence the stratification would not be established.

Editorial extensions

If this is right

  • For arbitrary symmetrizable type and any base field, every finitely generated graded module category $R(\beta)$-gMod is stratified, so every module has a well-behaved filtration by the explicit standard modules and BGG-style reciprocity holds.
  • For each Weyl group element $w$, the subcategory $R_{w,\ast}(\beta)$-gMod is an affine highest weight category, which implies it has finite global dimension and its standard modules have polynomial endomorphism rings.
  • The Grothendieck group of $R_{w,\ast}$-gproj becomes an algebra isomorphic to the quantum unipotent subgroup $U_q(n_-\cap wn_+)$, giving explicit modules that categorify the PBW and dual PBW bases.
  • In affine types the stratification can be refined for any convex order, and the standard module corresponding to the minimal imaginary root vector categorifies the imaginary root vector of the affine PBW basis.
  • The $\operatorname{Ext}^k$-vanishing for affinized determinantial modules obtained in the paper holds for all $k\ge 1$, strengthening earlier extension-vanishing results that were only known in special types.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same R-matrix injectivity criterion could be tested on other monoidal categorifications, for example higher-level or exotic analogues, to see whether explicit determinantial standard modules exist there as well.
  • Because the standard modules are now explicitly described, one can attempt concrete computations of $\operatorname{Ext}$-algebras between standard modules in types and characteristics where nothing was known before, which would give the first homological data for non-semisimple quiver Hecke categories outside finite and symmetric affine types.
  • The paper leaves open which simple modules admit an affinization satisfying the spectral condition on the central elements $p_{i,\beta}$; a counterexample in an unexplored type would isolate exactly where the stratification machinery would need to be modified.
  • In affine type $A_{2l}^{(2)}$, the exceptional case handled separately in Section 6 offers a concrete test: verifying the two short exact sequences of Theorem 6.22 in low-rank examples would confirm the pattern predicted by the general stratification.
  • The explicit nature of the standard modules may allow the stratification to be sheared into finer stratifications indexed by other convex orders, connecting the resulting homological invariants to cluster structures on quantum coordinate rings.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper develops a new algebraic approach to proving that, for arbitrary symmetrizable root data and arbitrary base fields, the category R(β)-gMod of finitely generated graded modules over a quiver Hecke algebra admits a Kleshchev stratification, and that the full subcategory R_{w,*} associated with a quantum unipotent subgroup is affine highest weight. The standard modules are realized as convolutions of affinized determinantial modules, and the proofs rely on R-matrix techniques, including a foundational short exact sequence for affinizations and an Ext-vanishing theorem. The paper also treats affine Lie types and identifies certain standard modules with imaginary root vectors.

Significance. If the main theorems are established, this is a substantial advance: it extends prior results of Kato, Brundan-Kleshchev-McNamara, McNamara, and Kleshchev-Muth from finite or symmetric affine types to arbitrary symmetrizable types and arbitrary characteristic, with explicit algebraic descriptions of standard and proper costandard modules. The paper is well-structured, gives many detailed proofs, and introduces a promising use of R-matrices for homological questions. However, the current manuscript contains a load-bearing definitional gap involving the duality functor, which prevents the main theorem from being proven as written. The central ideas appear plausible and the gap is likely fixable, but a revision is necessary.

major comments (2)
  1. [§5.4, Definition 5.14 (and §5.3, Definition 5.12)] The duality functor D is introduced in Section 3.1 only on the category R-gmod of finite-dimensional graded modules. Definition 5.14 sets ∇(λ,s) = D(∆(λ,s)), where ∆(λ,s) = ∆(s)∘∆(λ) is built from affinizations ˆL(β_k) that are free of finite rank over k[z] (Lemma 3.14) and hence are infinite-dimensional over k. Similarly, Definition 5.12 sets ∇(λ) = D(∆(λ)) for the truncated standard modules. Thus D is not defined on these objects. If one interprets D as the restricted dual, ∇(λ,s) is not finite-dimensional and does not belong to R-gmod, contradicting Definition 5.4, which explicitly requires proper costandard modules to be finite-dimensional. This invalidates the claimed existence of proper costandard modules in Theorem 5.17 and undermines the proofs of Lemma 5.15, Proposition 5.16, and the main stratification Theorem 5.18, since the criterion Proposition 5.7 requires finite-dimensional proper costandard modules.
  2. [§4.3, Proof of Theorem 4.11] The proof of the foundational short exact sequence relies on the exactness of the functor F constructed from a type-A₂ quiver Hecke algebra. The paper asserts that F is exact because the global dimension of R_{A₂}(α)-gMod is finite, citing [KKK18, Proposition 3.7] and [KKOP24, Proposition 7.6], but then states that for the needed cases α ∈ {0, α₁, α₂, α₃} one can directly verify finiteness without providing that verification. Since the exactness of F is essential for producing the short exact sequence for arbitrary w and i, this step needs a complete argument rather than a promise.
minor comments (3)
  1. [Throughout] The word 'Definintion' appears in place of 'Definition' in several places (e.g., Definitions 2.2, 2.3, 3.1, 3.8, 3.15, 3.23, 3.29, 3.35, 3.41, 3.54, 4.21, 5.4, 6.17).
  2. [§3.7 and §5.4] The notation ∆(λ,s) is used with two different meanings: in Section 3.7 it denotes L(s)∘L(λ_lβ_l)∘⋯∘L(λ₁β₁), while in Definition 5.14 it denotes ∆(s)∘∆(λ). Although the latter is claimed to coincide with the standard module, the reuse of the same symbol for a priori different objects is confusing and should be clarified.
  3. [§2.2, Lemma 2.5] The proof of Lemma 2.5(3) uses the notation M^{n'} and then concludes ext^k_H(M^{n'},N)=0, but the transition from the construction of the projective resolution of M^{n'} to the vanishing is terse; a few more details would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stratification proof reduces to external categorification, determinantial-module, and R-matrix results, not to its own conclusions.

full rationale

The claimed derivation chain is self-contained in the relevant sense: affinized determinantial modules are analyzed through injectivity of renormalized R-matrices (Proposition 3.20), the key short exact sequence (Theorem 4.11), and projectivity/Ext^1-vanishing (Theorem 4.16); these feed Lemma 5.15, then Proposition 5.16, and finally Theorem 5.18 via the independent Kleshchev criterion Proposition 5.7. The BGG-type identity in Proposition 5.16 is proved by expanding [P(λ,s)] in the basis [∆(µ,t)] and using the derived Ext-pairing Lemma 5.15, not by assuming the stratification or the standard-module property. No fitted parameters are introduced, and no load-bearing result is justified only by a self-citation; the cited categorification, R-matrix, and determinantial-module theorems are external support, and the paper even corrects an argument in [KKOP18, Proposition 4.6], so the dependence is not an uncritical citation chain. The flagged issue that the duality D of Section 3.1 is defined only on finite-dimensional R-gmod while Definition 5.14 applies it to infinite-dimensional ∆(λ,s) is a potential correctness gap in the construction of the proper costandard modules, but it is not circularity: the stratification being proved is not used as an input to define ∇, and the failure mode would be an invalid or missing argument, not a reduction of the target to its own assumptions. Therefore no circular step is exhibited, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted numerical constants appear in the paper. The only choices are structural input data: the root datum, the polynomials Q_{i,j}, reduced expressions, and convex orders. The paper introduces weakly convex preorders as a definitional generalization, but this is not an invented entity with independent evidence, it is a mathematical definition used inside the proof framework.

assumptions (5)
  • domain assumption KLR categorification isomorphisms: K⊕(R-gproj) ≃ U_q^-_{Z[q,q^-1]} and K(R-gmod) ≃ U_q^-up_{Z[q,q^-1]}, including the qdim pairing.
    Invoked throughout, for example in Section 5.4 Proposition 5.16 to convert the BGG reciprocity identity into a Grothendieck group statement. The theorem is cited from KL09, KL11 and used without reproof.
  • domain assumption Existence and affinization of determinantial modules M(wΛ,vΛ), including that (M(wΛ,vΛ), z) is an affinization.
    These modules are the building blocks of the standard modules in Definition 5.12 and Proposition 5.11. The paper cites Lemma 3.48 and Proposition 3.51 from KKOP18 and KKOP21 rather than proving affinization from scratch.
  • domain assumption R-matrix framework: universal R-matrices, renormalized R-matrices, the invariants Λ(M,N), and injectivity of renormalized R-matrices.
    Used in Proposition 3.20, Theorem 4.11, and Theorem 4.16 to establish the fundamental short exact sequences and the Ext^1 vanishing that underlie the stratification.
  • domain assumption Cuspidal decomposition theorem for convex and weakly convex preorders.
    Theorem 3.32 from Tingley-Webster is cited and generalized in Proposition 3.43 to weakly convex preorders. This is used in Section 3.7 to classify self-dual simple modules and to define the partial order on Σ(β).
  • domain assumption Finiteness of the global dimension of finite type A2 quiver Hecke algebras.
    In Step 2 of Theorem 4.11, the exactness of the Schur-Weyl functor F is obtained from KKOP24 Proposition 7.6 together with BKM14 Theorem 4.7. This exactness is needed to transport the A2 short exact sequence to the target quiver Hecke algebra.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Affine highest weight structures on module categories over quiver Hecke algebras." pith.science (2026). https://pith.science/paper/C6CTWZOL

@misc{pith2026241212903,
  author       = {Pith},
  title        = {Pith review of: Affine highest weight structures on module categories over quiver Hecke algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/C6CTWZOL}},
  note         = {Machine review of arXiv:2412.12903}
}
read the original abstract

We prove that the category of finitely generated graded modules over the quiver Hecke algebra of arbitrary type admits numerous stratifications in the sense of Kleshchev. A direct consequence is that the full subcategory corresponding to the quantum unipotent subgroup associated with any Weyl group element is an affine highest weight category. Our results significantly generalize earlier works by Kato, Brundan, Kleshchev, McNamara and Muth. The key ingredient is a realization of standard modules via determinantial modules. We utilize the technique of R-matrices to study these standard modules.

Figures

Figures reproduced from arXiv: 2412.12903 by the authors.

Figure 1
Figure 1. Affine rank 2 positive roots F ∩Φ min + , as in [KKOP18, Lemma 1.20]. Making them even finer, we obtain convex orders + and − on Φmin + [KKOP18, Proposition 1.21]: they satisfy the second condition of Lemma 6.13 and α ′ − ≺ + δ ≺ + α ′ +, α′ + ≺ − δ ≺ − α ′ −. The coarse type of + is w and that of − is wsi . Note that the convex order introduced in [McN17, Example 3.6] is an example of +. Lemma 6.14. Consider c… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

42 extracted references · 25 canonical work pages

  1. [1]

    Susumu Ariki, On the decomposition numbers of the H ecke algebra of G(m,1,n) , J. Math. Kyoto Univ. 36 (1996), no. 4, 789--808. 1443748

    work page 1996

  2. [2]

    (N.S.) 20 (2014), no

    Matthew Bennett, Arkady Berenstein, Vyjayanthi Chari, Anton Khoroshkin, and Sergey Loktev, Macdonald polynomials and BGG reciprocity for current algebras , Selecta Math. (N.S.) 20 (2014), no. 2, 585--607. 3177927

    work page 2014

  3. [3]

    Jonathan Beck, Vyjayanthi Chari, and Andrew Pressley, An algebraic characterization of the affine canonical basis, Duke Math. J. 99 (1999), no. 3, 455--487. 1712630

    1999

  4. [4]

    Jonathan Brundan and Alexander Kleshchev, Blocks of cyclotomic H ecke algebras and K hovanov- L auda algebras , Invent. Math. 178 (2009), no. 3, 451--484. 2551762

    2009

  5. [5]

    McNamara, Homological properties of finite-type K hovanov- L auda- R ouquier algebras , Duke Math

    Jonathan Brundan, Alexander Kleshchev, and Peter J. McNamara, Homological properties of finite-type K hovanov- L auda- R ouquier algebras , Duke Math. J. 163 (2014), no. 7, 1353--1404. 3205728

    work page 2014

  6. [6]

    Jonathan Beck and Hiraku Nakajima, Crystal bases and two-sided cells of quantum affine algebras, Duke Math. J. 123 (2004), no. 2, 335--402. 2066942

    2004

  7. [7]

    Vyjayanthi Chari and Bogdan Ion, B GG reciprocity for current algebras , Compos. Math. 151 (2015), no. 7, 1265--1287. 3371494

    work page 2015

  8. [8]

    Cline, B

    E. Cline, B. Parshall, and L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine Angew. Math. 391 (1988), 85--99. 961165

    work page 1988

Show all 42 references
  1. [9]

    Evgeny Feigin, Anton Khoroshkin, Ievgen Makedonskyi, and Daniel Orr, P eter- W eyl theorem for I wahori groups and highest weight categories , 2023, arXiv:2307.02124

    2023 arXiv

  2. [10]

    C. Gei , B. Leclerc, and J. Schr\" o er, Cluster structures on quantum coordinate rings, Selecta Math. (N.S.) 19 (2013), no. 2, 337--397. 3090232

    2013

  3. [11]

    Kac, Infinite-dimensional L ie algebras , third ed., Cambridge University Press, Cambridge, 1990

    Victor G. Kac, Infinite-dimensional L ie algebras , third ed., Cambridge University Press, Cambridge, 1990. 1104219

    1990

  4. [12]

    Kashiwara, On crystal bases of the Q -analogue of universal enveloping algebras , Duke Math

    M. Kashiwara, On crystal bases of the Q -analogue of universal enveloping algebras , Duke Math. J. 63 (1991), no. 2, 465--516. 1115118

    1991

  5. [13]

    Masaki Kashiwara, Global crystal bases of quantum groups, Duke Math. J. 69 (1993), no. 2, 455--485. 1203234

    1993

  6. [14]

    Syu Kato, Poincar\'e- B irkhoff- W itt bases and K hovanov- L auda- R ouquier algebras , Duke Math. J. 163 (2014), no. 3, 619--663. 3165425

    2014

  7. [15]

    Yoshiyuki Kimura, Quantum unipotent subgroup and dual canonical basis, Kyoto J. Math. 52 (2012), no. 2, 277--331. 2914878

    2012

  8. [16]

    , Remarks on quantum unipotent subgroups and the dual canonical basis, Pacific J. Math. 286 (2017), no. 1, 125--151. 3582403

    2017

  9. [17]

    Seok-Jin Kang and Masaki Kashiwara, Categorification of highest weight modules via K hovanov- L auda- R ouquier algebras , Invent. Math. 190 (2012), no. 3, 699--742. 2995184

    2012

  10. [18]

    Seok-Jin Kang, Masaki Kashiwara, and Myungho Kim, Symmetric quiver H ecke algebras and R -matrices of quantum affine algebras , Invent. Math. 211 (2018), no. 2, 591--685. 3748315

    2018

  11. [19]

    Seok-Jin Kang, Masaki Kashiwara, Myungho Kim, and Se-jin Oh, Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), no. 2, 349--426. 3758148

    2018

  12. [20]

    Seok-Jin Kang, Masaki Kashiwara, and Se-Jin Oh, Categorification of highest weight modules over quantum generalized K ac- M oody algebras , Mosc. Math. J. 13 (2013), no. 2, 315--343, 364. 3134909

    2013

  13. [21]

    Masaki Kashiwara, Myungho Kim, Se-jin Oh, and Euiyong Park, Monoidal categories associated with strata of flag manifolds, Adv. Math. 328 (2018), 959--1009. 3771147

    2018

  14. [22]

    Masaki Kashiwara, Myungho Kim, Se-Jin Oh, and Euiyong Park, Localizations for quiver H ecke algebras , Pure Appl. Math. Q. 17 (2021), no. 4, 1465--1548. 4359265

    2021

  15. [23]

    Masaki Kashiwara, Myungho Kim, Se-jin Oh, and Euiyong Park, Affinizations, R -matrices and reflection functors , Adv. Math. 443 (2024), Paper No. 109598, 83. 4717658

    2024

  16. [24]

    Lauda, A diagrammatic approach to categorification of quantum groups

    Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups. I , Represent. Theory 13 (2009), 309--347. 2525917

    2009

  17. [25]

    , A diagrammatic approach to categorification of quantum groups II , Trans. Amer. Math. Soc. 363 (2011), no. 5, 2685--2700. 2763732

    2011

  18. [26]

    Kleshchev, Affine highest weight categories and affine quasihereditary algebras, Proc

    Alexander S. Kleshchev, Affine highest weight categories and affine quasihereditary algebras, Proc. Lond. Math. Soc. (3) 110 (2015), no. 4, 841--882. 3335289

    2015

  19. [27]

    Notes Ser

    , Representation theory and cohomology of K hovanov- L auda- R ouquier algebras , Modular representation theory of finite and p -adic groups, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 30, World Sci. Publ., Hackensack, NJ, 2015, pp. 109--163. 3495746

    2015

  20. [28]

    Lauda, Marco Mackaay, and Marko Sto s i\' c , Extended graphical calculus for categorified quantum sl (2) , Mem

    Mikhail Khovanov, Aaron D. Lauda, Marco Mackaay, and Marko Sto s i\' c , Extended graphical calculus for categorified quantum sl (2) , Mem. Amer. Math. Soc. 219 (2012), no. 1029, vi+87. 2963085

    2012

  21. [29]

    Algebra 475 (2017), 133--170

    Alexander Kleshchev and Robert Muth, Stratifying KLR algebras of affine ADE types , J. Algebra 475 (2017), 133--170. 3612467

    2017

  22. [30]

    Masaki Kashiwara and Euiyong Park, Affinizations and R -matrices for quiver H ecke algebras , J. Eur. Math. Soc. (JEMS) 20 (2018), no. 5, 1161--1193. 3790066

    2018

  23. [31]

    Kleshchev and David J

    Alexander S. Kleshchev and David J. Steinberg, Homomorphisms between standard modules over finite-type KLR algebras , Compos. Math. 153 (2017), no. 3, 621--646. 3705237

    2017

  24. [32]

    Alain Lascoux, Bernard Leclerc, and Jean-Yves Thibon, Hecke algebras at roots of unity and crystal bases of quantum affine algebras, Comm. Math. Phys. 181 (1996), no. 1, 205--263. 1410572

    1996

  25. [33]

    Math., vol

    Bernard Leclerc, Maxim Nazarov, and Jean-Yves Thibon, Induced representations of affine H ecke algebras and canonical bases of quantum groups , Studies in memory of I ssai S chur ( C hevaleret/ R ehovot, 2000), Progr. Math., vol. 210, Birkh\"auser Boston, Boston, MA, 2003, pp....

    2000

  26. [34]

    auser Classics, Birkh\

    George Lusztig, Introduction to quantum groups, Modern Birkh\"auser Classics, Birkh\"auser/Springer, New York, 2010, Reprint of the 1994 edition. 2759715

    2010

  27. [35]

    Lauda and Monica Vazirani, Crystals from categorified quantum groups, Adv

    Aaron D. Lauda and Monica Vazirani, Crystals from categorified quantum groups, Adv. Math. 228 (2011), no. 2, 803--861. 2822211

    2011

  28. [36]

    McNamara, Finite dimensional representations of K hovanov- L auda- R ouquier algebras I : F inite type , J

    Peter J. McNamara, Finite dimensional representations of K hovanov- L auda- R ouquier algebras I : F inite type , J. Reine Angew. Math. 707 (2015), 103--124. 3403455

    2015

  29. [37]

    , Representations of K hovanov- L auda- R ouquier algebras III : symmetric affine type , Math. Z. 287 (2017), no. 1-2, 243--286. 3694676

    2017

  30. [38]

    McNamara, C luster M onomials are D ual C anonical , 2021, arXiv:2112.04109

    Peter J. McNamara, C luster M onomials are D ual C anonical , 2021, arXiv:2112.04109

    2021 arXiv

  31. [39]

    (N.S.) 24 (2018), no

    Dinakar Muthiah and Peter Tingley, Affine PBW bases and affine MV polytopes , Selecta Math. (N.S.) 24 (2018), no. 5, 4781--4810. 3874704

    2018

  32. [40]

    1836, Springer-Verlag, Berlin, 2004

    Constantin N a st a sescu and Freddy Van Oystaeyen, Methods of graded rings, Lecture Notes in Mathematics, vol. 1836, Springer-Verlag, Berlin, 2004. 2046303

    2004

  33. [41]

    Raphael Rouquier, 2- K ac- M oody algebras , 2008, arXiv:0812.5023

    2008 arXiv

  34. [42]

    Peter Tingley and Ben Webster, Mirkovi\'c- V ilonen polytopes and K hovanov- L auda- R ouquier algebras , Compos. Math. 152 (2016), no. 8, 1648--1696. 3542489

    2016

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.