Pith. sign in

REVIEW 3 major objections 10 minor 154 references

Coulomb branches, quantized zastavas, Kac polynomials, and shuffle algebras

T0 review · 3 major / 10 minor · reviewed 2026-07-31 · grok-4.5

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

Pith's one-line read Closed embeddings of quantized zastava spaces stabilize to a Borel-type Yangian whose positive part is a refined-wheel shuffle algebra, proving spherical generation after localization.

desk verdict Solid geometric construction of limit quantized zastavas for arbitrary quivers, with a clean isomorphism to refined-wheel shuffles that proves Neguţ’s spherical-generation conjecture. read the letter →

arxiv 2607.24711 v1 pith:MSJHE3IZ submitted 2026-07-27 math.RT math-phmath.AGmath.MPmath.QA

Dinakar Muthiah , Alex Weekes This is my paper · ORCID
classification math.RTmath-phmath.AGmath.MPmath.QA MSC 17B3714M1516G2081R10
keywords CoulombbranchesquantizedzastavaKacpolynomialsshufflealgebrasmonopoleoperatorsYangianssphericalgenerationBFNconstruction
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 builds a limit algebra A from quantized zastava spaces of an arbitrary quiver by showing that natural closed embeddings of those spaces quantize, even though the analogous embeddings of affine Grassmannian slices do not. A plays the role of a Borel Yangian and contains a positive part A^{+} that plays the role of a unipotent Yangian; both have Hilbert series controlled by a limiting form of Hua’s formula for Kac polynomials. The positive part is identified with a shuffle algebra defined by refined wheel conditions, and the identification extends to an isomorphism of an enlarged shuffle algebra with the full limit A. Because the finite-level zastava algebras are spherically generated after a mild localization, the same holds for A^{+} and therefore for the shuffle algebra, confirming a conjecture of Neguţ.

What carries the argument

The FFT homomorphism from the big shuffle algebra into localized difference operators, restricted to the refined-wheel subalgebra S; it is shown to be a filtered graded surjection onto each positive quantized zastava A^{+}(v) and to become an isomorphism in the limit S ≅ A^{+}.

What would settle it

Exhibit a concrete quiver with loops and a homogeneous element of the refined-wheel shuffle algebra whose image under the FFT map fails to lie in the corresponding positive quantized zastava, or whose image cannot be written as a product of fundamental monopole operators of restricted dressing.

Watch

Extended reading notes

Core claim

For any quiver the quantized zastava algebras form a directed system under quantized closed embeddings induced by fundamental monopole operators; their inverse limit A is generated by those operators, decomposes as Λ ⊗ A^{+}, has Hilbert series given by the limiting monopole formula (equivalently Hua’s generating function for Kac polynomials), and is isomorphic to an extended shuffle algebra whose positive part is the refined-wheel shuffle algebra. Localization at an explicit multiplicative set makes A^{+} spherically generated, proving Neguţ’s conjecture.

Load-bearing premise

The refined wheel conditions are precisely the divisibility conditions that force the image of the FFT map to land inside every positive quantized zastava, and the inductive column-by-column lifting argument that proves this continues to work when the quiver has edge loops.

Editorial extensions

If this is right

  • The limit A supplies an intrinsic Borel Yangian for arbitrary quivers, including those with loops and multiple edges, without inverting the deformation parameter.
  • Hilbert series of A and A^{+} are completely determined by the Kac polynomials of the quiver via Hua’s identity.
  • After localization at the multiplicative set generated by closed-path linear forms, the shuffle algebra is generated by its degree-zero and simple-root components.
  • The same circle of ideas is expected to produce an analogous trigonometric (K-theoretic) limit zastava and an integral form of the shuffle algebra defined by explicit generators.

Reading between the lines

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

  • The isomorphism with the loop-nilpotent cohomological Hall algebra announced in concurrent work is now unconditional once both sides are identified with A⁺.
  • The same quantized-embedding technique should produce a limit object for Coulomb branches of quivers with symmetrizers once a substitute for generation by fundamental monopole operators is found.
  • Spherical generation after localization suggests that the associated graded of A^{+} admits a PBW-type basis indexed by multipartitions with restricted dressings, computable directly from the shuffle product.
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

3 major / 10 minor

Summary. The paper constructs, for an arbitrary quiver Q (loops and multiple edges allowed), a "limit quantized zastava" algebra A as the graded inverse limit of the quantized zastava algebras A(v) arising from the BFN construction, using new surjections Φ_{v,v₁} : A(v) ↠ A(v₁) that quantize the "adding defect" closed embeddings of the authors' previous work [MW24]. A positive part A⁺ is defined via FMOs with restricted dressing, giving a triangular decomposition A = Λ ⊗ A⁺. The limit of the monopole formula is shown to coincide, via Hua's identity, with a generating function of Kac polynomials (Cor. 4.12). A shuffle algebra S ⊂ bigS is defined by refined wheel conditions (Def. 5.14) tailored to the GKLO/FFT difference-operator realization, and the main technical theorem (Thm. 5.32, 5.55, 5.66) proves Φ⁺ : S ≅ A⁺ and Φ : extS ≅ A, where extS = S₀ # S adjoins a Cartan part. Finally, a Wee19-style Nullstellensatz argument after localization at an explicit multiplicative set M (generated by loop parameters along closed paths) yields spherical generation of A[M⁻¹] and A⁺[M⁻¹], hence of S_loc and S¹_loc — Neguţ's Conjecture 2.12.

Significance. If correct — and I found no error — this is a substantial contribution. It gives the first intrinsic construction of a Borel/unipotent Yangian attached to an arbitrary quiver with loops, identifies its Hilbert series with Kac polynomials (consistent with, and clarifying, the CoHA picture), proves the shuffle-algebra realization with explicit refined wheel conditions, and settles Neguţ's Conjecture 2.12. Strengths worth naming: the results are proved in detail rather than asserted; the Hilbert-series computation provides an independent, falsifiable consistency check on the wheel conditions (both sides are governed by Hua's identity); the integral form S_Z is treated honestly (§5.7.1); and the relation to the independent ongoing work of Jindal–Neguţ ([JN26a, JN26b]) and Botta–Tamagni is disclosed, with the combined isomorphism (1.19) stated explicitly.

major comments (3)
  1. [§5.6, proof of Theorem 5.32] Proof of Theorem 5.32: after writing h|λ = r_h · (diagonal factors) via Lemma 5.38, the proof applies the refined wheel conditions directly to r_h. But Definition 5.14 gives divisibility of h|λ, not of r_h. The transfer requires that the diagonal linear forms (w_{i,r}−w_{i,s}+ph) of (5.39) are coprime in R[...] to the edge factors (w_{j,s}−w_{i,r}+θ_e±½h) of (5.12). For loops e:i→i with r≠s this holds only because θ_e is an independent indeterminate, and for r=s one needs Lemma A.6 with m=n. This coprimality argument is not stated; since the regularity Φ⁺_v(S)⊂A⁺(v) — hence Theorems 5.55, 5.66 and 6.1 — rests on it, a short lemma should be added. One should also justify that r_h∈Λ^λ_res(v): this uses that h|λ involves only w_{i,r} with r≤|λ_i| and that the diagonal quotient is S_{λ,res}-invariant (exponents vanish within stabilizer blocks).
  2. [§2.4, Theorem 2.62 (proof deferred to §5.7.2)] Theorem 2.62 (the computation gr A⁺(v)=⊕_λ Λ^λ_res(v)·r_λ, hence freeness and good base change of A⁺(v)) has one inclusion deferred to §5.7.2, where it is proved using the surjectivity of Theorem 5.32. Meanwhile the second formula of Theorem 4.4 for J⁺_v(z,q) — and therefore the Kac-polynomial Hilbert series of A⁺ in Corollary 4.12 — already uses Theorem 2.62 to count bases. On my reading there is no actual circle (the proof of Theorem 5.32 uses only the GKLO formula (2.44) and Theorem 5.29, never §4 or Theorem 2.62), but the manuscript should state this explicitly where the proof is deferred, since the dependency chain §2.4 → §4 → §5.7.2 is otherwise easy to misread as circular.
  3. [§6, proof of Theorem 6.1] The deduction of Theorem 6.1 (spherical generation of S¹_loc) from Theorem 6.17 (spherical generation of A⁺[M⁻¹]) passes through Theorem 5.21(b), which identifies S with S¹ only after localization at the set (5.22) of loop parameters θ_e+(n+½)h. The bridging fact — Remark 6.10(4), that M contains (5.22) up to sign — is stated four pages earlier and never cited at this step. Since Theorem 6.1 is the advertised application (Neguţ's Conjecture 2.12), the chain S¹[M⁻¹]=S[M⁻¹]≅A⁺[M⁻¹] should be written out explicitly in the proof of Theorem 6.1.
minor comments (10)
  1. [Theorem 2.84] Statement: "the kernel of the ring homomorphism Φ_{v,v₁} : A(v) → A(v)" should read A(v) → A(v₁).
  2. [§4.1, Remark 4.3] The notation (q)_∞ is defined as ∏_{i∈I}∏_{r≥1}(1−q^r), i.e. the |I|-th power of the standard (q)_∞. Since the formulas in §4 will be compared with the literature on Hua's identity, a warning remark (or a symbol such as (q)_∞^I) would prevent confusion. Similarly (q)_m in (4.2) is a product over I.
  3. [Eq. (5.45)] The symbol "sign" in (−1)^sign is not defined; presumably it is the sum over i and r<s of the relevant exponents from Lemma 5.42. Please make it explicit.
  4. [§2.5, Lemma 2.81, Lemma 5.23] The parameters θ_e are occasionally written ν_a (e.g. in the sentence after (2.17), in Lemma 2.81, and in the proof of Lemma 5.23). Please unify the notation.
  5. [Lemma 5.23] "well-known to experts" — a precise reference (e.g. [Neg23] or [FT22]) would be helpful here, since this lemma is what compares the refined and 3-variable wheel conditions.
  6. [§2.1.4] The defining sentence of I-colored multisets ("Consider tuples of multisets of the form A=(A_i)...") appears twice in succession.
  7. [Definition 5.35] Definition 5.35 introduces both Supp Φ⁺_v(f) and its dominance closure with nearly identical notation, and the closure is subsequently typeset as "Suppp" (Lemma 5.38, proof of Theorem 5.32). Distinct symbols would improve readability of the induction.
  8. [§5.7.1] Remark 5.59 honestly notes the lack of a membership criterion for S_Z; it would be worth adding a sentence comparing with the explicit integral wheel conditions of [JN26b, Definition 2.3], which the introduction already cites.
  9. [Abstract and passim] Typographical: "there is variant of the BFN construction" → "there is a variant"; "a version for Hua's formula" → "a version of"; "R_{G,N} carries an action of G_O⋊C*×F, were G_O is" → "where"; repeated "Consider tuples of mutisets"; "lsomorphism" in the heading of §5.7.
  10. [§6.1, proof of Theorem 6.11] The localization statement uses the Nullstellensatz over C after reducing to k=ℚ; one line justifying descent of the unit ideal from C to ℚ (faithful flatness) would make the step self-contained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: limit algebras, shuffle isomorphism, and spherical generation are derived from BFN geometry, external FFT/Hua inputs, and re-proved generator arguments rather than forced by definition or self-citation.

full rationale

The derivation chain is self-contained against external benchmarks. Quantized closed embeddings Φ_{v,v′} are constructed explicitly on FMOs via GKLO (Theorem 2.78) and form a directed system whose limit A, A⁺ is generated by those FMOs with kernels likewise FMO-generated (Theorem 3.11)—not a renaming of a prior object. The Hilbert series is the v→∞ limit of the BFN monopole formula, then rewritten by Hua’s external identity (Corollary 4.12); no parameter is fitted to Kac data. The refined wheel conditions (Def. 5.14) are chosen so FFT lands in A⁺(v), which is definitional engineering of a subalgebra, not a circular prediction: surjectivity is proved by support induction plus column FMO lifts (Thm. 5.32), injectivity by the GKLO embedding on each degree, and the limit iso Φ⁺: S ≅ A⁺ follows (Thm. 5.55), extended to extS ≅ A by adjoining Cartan series. Spherical generation of A[M⁻¹] generalizes Wee19 with a full Nullstellensatz argument for loops/flavour (Thm. 6.11), then passes to A⁺_loc by the triangular decomposition and word-reordering (Thm. 6.17), proving Neguţ’s conjecture. Self-citations (MW24, Wee19) supply prior embeddings/generator techniques that are re-established or extended here; they do not make the target equal the input by construction. No fitted-input-as-prediction, uniqueness-import, or ansatz-smuggling step reduces a central claim to its own premise.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The work sits on the BFN equivariant-homology construction of Coulomb/zastava algebras, standard facts about affine Grassmannians and partially symmetric polynomials, Hua’s generating-function identity, and the second author’s earlier spherical-generation theorem for Coulomb branches. No numerical free parameters appear. The refined wheel conditions and the limit algebras A, A⁺ are the main new objects; independent evidence for them is internal (isomorphism, Hilbert series, recovery of finite-type Yangians) plus the disclosed match with concurrent CoHA work.

assumptions (5)
  • domain assumption BFN quantized Coulomb/zastava algebras Ap(v) = H^{G_O ⋊ ℂˣ ˆ F}_•(R⁺_{G,N}) are associative, free over Λ(v) and over k, and generated by dressed fundamental monopole operators.
    Taken from BFN18/BFN19 and Wee19; used throughout §§2–3 as the starting algebras of the directed system.
  • standard math Hua’s identity equates the multipartition generating function with the plethystic/Kac-polynomial product (Thm 4.9 / Cor 4.10).
    Cited from Hua00; applied verbatim to convert the limit monopole series into Kac polynomials.
  • domain assumption After localizing at the multiplicative set M generated by closed-path linear forms y_p in the doubled quiver, each Ap(v) is spherically generated by Cartan series Q_i(z) and simple FMO series F_i(z).
    Generalization of Wee19 Thm 3.7 proved in Thm 6.11 by a Nullstellensatz argument on the associated-graded; load-bearing for the transfer to shuffle algebras.
  • domain assumption Base ring k is commutative; for shuffle statements one assumes ℚ ⊆ k so that factorials in the FFT map are invertible.
    Stated at the opening of §5; integral form treated separately in §5.7.1 by rescaling generators.
  • domain assumption The GKLO embedding Ap(v) ↪ rAp(v)^{S_v}_loc is injective and realizes the multipartition filtration by support of difference-operator coefficients.
    BFN18 §5; used to define filtrations algebraically and to compare with shuffle specializations.
invented entities (3)
  • Limit quantized zastava A and positive part A⁺ independent evidence
    purpose: Provide Borel/unipotent Yangian analogues for arbitrary quivers as inverse limits of quantized zastavas.
    Defined in Def 3.9 via directed systems of FMO-respecting surjections; freeness, triangular decomposition, and generation by FMOs proved in Thm 3.11.
  • Refined wheel conditions and shuffle algebra S independent evidence
    purpose: Cut out the precise subalgebra of the big shuffle algebra that maps into every A⁺(v) under FFT.
    Def 5.14–5.17; shown to be a subring and to map isomorphically onto A⁺ (Thm 5.55). Compared with classical 3-variable wheels in Thm 5.21.
  • Extended shuffle algebra extS = S_0 # S independent evidence
    purpose: Incorporate Cartan generators so that the isomorphism upgrades from A⁺ to all of A.
    Def 5.62–5.65; smash-product relation (5.63) matched to GKLO under Φ (Thm 5.66).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Coulomb branches, quantized zastavas, Kac polynomials, and shuffle algebras." pith.science (2026). https://pith.science/paper/MSJHE3IZ

@misc{pith2026260724711,
  author       = {Pith},
  title        = {Pith review of: Coulomb branches, quantized zastavas, Kac polynomials, and shuffle algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MSJHE3IZ}},
  note         = {Machine review of arXiv:2607.24711}
}
abstract

We previously constructed closed embeddings of Kac-Moody affine Grassmannian slices using fundamental monopole operators. These spaces are defined via the Braverman-Finkelberg-Nakajima construction of Coulomb branches for quiver gauge theories, and the embeddings do not quantize in general. However, there is variant of the BFN construction that produces zastava spaces, and we show that the closed embeddings do quantize in that case. This allows us to take a limit and construct the limit quantized zastava $\mathcal{A}$ for an arbitrary quiver. This algebra plays the role of the Borel Yangian. We also construct a positive part $\mathcal{A}^+$, which plays the role of the unipotent Yangian. By taking the limit of the monopole formula, we show that both $\mathcal{A}$ and $\mathcal{A}^+$ have Hilbert series given by a version for Hua's formula for Kac polynomials. We also show that $\mathcal{A}^+$ is isomorphic to a certain shuffle algebra. Finally, using these results we obtain a proof of Negut's conjecture on the spherical generation of localized shuffle algebras via the second author's work on generators of Coulomb branches.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

154 extracted references · 2 canonical work pages

  1. [1]

    Arnaudon and A

    D. Arnaudon and A. Molev and E. Ragoucy , title=. Annales Henri Poincar\'e , volume=7, year=2006, pages=

    2006

  2. [2]

    Beauville, Arnaud , TITLE =. Invent. Math. , FJOURNAL =. 2000 , NUMBER =

    2000

  3. [3]

    Bellamy, Gwyn , title=

  4. [4]

    Pacific Journal of Mathematics , year=

    On deformation quantizations of hypertoric varieties , author=. Pacific Journal of Mathematics , year=

  5. [5]

    Bellamy, Gwyn and Schedler, Travis , title =

  6. [6]

    Bezrukavnikov and D

    R. Bezrukavnikov and D. Kaledin , title=. Moscow Mathematical Journal , volume=4, year=2004, number=3, pages=

    2004

  7. [7]

    Equivariant (

    Bezrukavnikov, Roman and Finkelberg, Michael and Mirkovi. Equivariant (. Compos. Math. , volume = 141, year = 2005, number = 3, pages =

    2005

  8. [8]

    Botta, Tommaso Maria and Davison, Ben , title =

Show all 154 references
  1. [9]

    Botta, Tommaso Maria and Tamagni, Spencer , title =

  2. [10]

    Bourget, Antoine and Hanany, Amihay and Miketa, Dominik , TITLE =. J. High Energy Phys. , FJOURNAL =. 2021 , NUMBER =

    2021

  3. [11]

    and Hanany, Amihay and Sperling, Marcus and Zhong, Zhenghao , title =

    Bourget, Antoine and Grimminger, Julius F. and Hanany, Amihay and Sperling, Marcus and Zhong, Zhenghao , title =

  4. [12]

    and Hanany, Amihay and Kalveks, Rudolph and Sperling, Marcus and Zhong, Zhenghao , TITLE =

    Bourget, Antoine and Grimminger, Julius F. and Hanany, Amihay and Kalveks, Rudolph and Sperling, Marcus and Zhong, Zhenghao , TITLE =. J. High Energy Phys. , FJOURNAL =. 2020 , NUMBER =

    2020

  5. [13]

    2012 , note=

    Braden, Tom and Proudfoot, Nicholas and Webster, Ben , title=. 2012 , note=

    2012

  6. [14]

    Braden, Tom and Licata, Anthony and Proudfoot, Nicholas and Webster, Ben , TITLE =. Ast\'. 2016 , PAGES =

    2016

  7. [15]

    Advances in Mathematics , volume=300, year=2016, pages=

    Braverman, Alexander and Dobrovolska, Galyna and Finkelberg, Michael , title=. Advances in Mathematics , volume=300, year=2016, pages=

    2016

  8. [16]

    Duke Mathematical Journal , volume=152, year=2010, pages=

    Braverman, Alexander and Finkelberg, Michael , title=. Duke Mathematical Journal , volume=152, year=2010, pages=

    2010

  9. [17]

    Geometric representation theory and gauge theory , publisher=

    Braverman, Alexander and Finkelberg, Michael , title=. Geometric representation theory and gauge theory , publisher=

  10. [18]

    Braverman, Alexander and Finkelberg, Michael , TITLE =. J. Amer. Math. Soc. , FJOURNAL =. 2014 , NUMBER =

    2014

  11. [19]

    Braverman, Alexander and Finkelberg, Michael , TITLE =. J. Lond. Math. Soc. (2) , FJOURNAL =. 2017 , NUMBER =

    2017

  12. [20]

    The unity of mathematics , SERIES =

    Braverman, Alexander and Finkelberg, Michael and Gaitsgory, Dennis , TITLE =. The unity of mathematics , SERIES =

  13. [21]

    Braverman, Alexander and Finkelberg, Michael and Nakajima, Hiraku , TITLE =. Adv. Theor. Math. Phys. , FJOURNAL =. 2018 , NUMBER =

    2018

  14. [22]

    Braverman, Alexander and Finkelberg, Michael and Nakajima, Hiraku , TITLE =. Adv. Theor. Math. Phys. , FJOURNAL =. 2019 , NUMBER =

    2019

  15. [23]

    Braverman, Alexander and Finkelberg, Michael and Nakajima, Hiraku , TITLE =

  16. [24]

    Representation Theories and Algebraic Geometry , series=

    Brion, Michel , title=. Representation Theories and Algebraic Geometry , series=

  17. [25]

    Michigan Mathematical Journal , year=2000, volume=48, pages=

    Brion, Michel , title=. Michigan Mathematical Journal , year=2000, volume=48, pages=

    2000

  18. [26]

    Brown, Kenneth , title=. Comment. Math. Helvetici , volume=50, year=1975, pages=

    1975

  19. [27]

    Transform

    Brown, Jonathan , TITLE =. Transform. Groups , FJOURNAL =. 2009 , NUMBER =

    2009

  20. [28]

    Brown, Jonathan , title =

  21. [29]

    Bullimore, Mathew and Dimofte, Tudor and Gaiotto, Davide , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2017 , NUMBER =

    2017

  22. [30]

    Bullimore, Mathew and Dimofte, Tudor and Gaiotto, Davide and Hilburn, Justin , TITLE =. J. High Energy Phys. , FJOURNAL =. 2016 , NUMBER =

    2016

  23. [31]

    Derived equivalences for symmetric groups and

    Chuang, Joseph and Rouquier, Rapha\". Derived equivalences for symmetric groups and. Ann. of Math. (2) , FJOURNAL =. 2008 , NUMBER =

    2008

  24. [32]

    Brown, Kenneth and Gordon, Iain , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2003 , PAGES =

    2003

  25. [33]

    Communications in Mathematical Physics , volume=254, year=2005, number=1, pages=

    Brundan, Jonathan and Kleshchev, Alexander , title=. Communications in Mathematical Physics , volume=254, year=2005, number=1, pages=

    2005

  26. [34]

    Memoirs of the American Mathematical Society , volume=196, year=2008, number=918, pages=

    Brundan, Jonathan and Kleshchev, Alexander , title=. Memoirs of the American Mathematical Society , volume=196, year=2008, number=918, pages=

    2008

  27. [35]

    Advances in Mathematics , volume=200, year=2006, pages=

    Brundan, Jonathan and Kleshchev, Alexander , title=. Advances in Mathematics , volume=200, year=2006, pages=

    2006

  28. [36]

    Cabrera, Santiago and Hanany, Amihay and Zhong, Zhenghao , TITLE =. J. High Energy Phys. , FJOURNAL =. 2017 , NUMBER =

    2017

  29. [37]

    Cremonesi, Stefano and Hanany, Amihay and Zaffaroni, Alberto , title =. J. High Energy Phys. , FJOURNAL =. 2014 , NUMBER =

    2014

  30. [38]

    Chari, Vyjayanthi and Pressley, Andrew , title=

  31. [39]

    Journal f\"ur die reine und angewandte Mathematik , volume=417, year=1991, pages=

    Chari, Vyjayanthi and Pressley, Andrew , title=. Journal f\"ur die reine und angewandte Mathematik , volume=417, year=1991, pages=

    1991

  32. [40]

    Chriss, Neil and Ginzburg, Victor , title=

  33. [41]

    Davison, Ben , TITLE =. Math. Res. Lett. , FJOURNAL =. 2018 , NUMBER =

    2018

  34. [42]

    Demazure, Michel , TITLE =. Invent. Math. , FJOURNAL =. 1973 , PAGES =

    1973

  35. [43]

    Dlab and C.M

    V. Dlab and C.M. Ringel , title=. Representation Theory II, Lecture Notes in Math , volume=832, year=1980, pages=

    1980

  36. [44]

    Dong, Zhijie , TITLE =. Adv. Math. , FJOURNAL =. 2026 , PAGES =

    2026

  37. [45]

    Doklady Akademii Nauk SSSR (in Russian) , volume=

    Drinfeld, Vladimir , title=. Doklady Akademii Nauk SSSR (in Russian) , volume=. 1985 , number=

    1985

  38. [46]

    Doklady Akademii Nauk SSSR (in Russian) , volume=

    Drinfeld, Vladimir , title=. Doklady Akademii Nauk SSSR (in Russian) , volume=. 1987 , number=

    1987

  39. [47]

    Grothendieck, Alexandre , TITLE =. Inst. Hautes \'. 1965 , PAGES =

    1965

  40. [48]

    Journal of the European Mathematical Society , volume=5, year=2003, number=1, pages=

    Faltings, Gerd , title=. Journal of the European Mathematical Society , volume=5, year=2003, number=1, pages=

    2003

  41. [49]

    Feigin, Boris and Feigin, Evgeny and Jimbo, Michio and Miwa, Tetsuji and Mukhin, Evgeny , TITLE =. Lett. Math. Phys. , FJOURNAL =. 2009 , NUMBER =

    2009

  42. [50]

    A commutative algebra on degenerate

    Feigin, Boris and Hashizume, Kenta and Hoshino, Ayumu and Shiraishi,. A commutative algebra on degenerate. J. Math. Phys. , FJOURNAL =. 2009 , NUMBER =

    2009

  43. [51]

    Integrable structures of exactly solvable two-dimensional models of quantum field theory (

    Feigin, Boris and Odesskii, Alexandre , TITLE =. Integrable structures of exactly solvable two-dimensional models of quantum field theory (. 2001 , ISBN =

    2001

  44. [52]

    Proceedings of the

    Finkelberg, Michael , TITLE =. Proceedings of the

  45. [53]

    Funktsional

    Finkelberg, Michael and Goncharov, Evgeny , TITLE =. Funktsional. Anal. i Prilozhen. , FJOURNAL =. 2019 , NUMBER =

    2019

  46. [54]

    Finkelberg, Michael and Kamnitzer, Joel and Pham, Khoa and Rybnikov, Leonid and Weekes, Alex , title =. Adv. Math. , FJOURNAL =. 2018 , Pages =

    2018

  47. [55]

    A note on a symplectic structure on the space of

    Finkelberg, Michael and Kuznetsov, Alexander and Markarian, Nikita and Mirkovi\'. A note on a symplectic structure on the space of. Commun. Math. Phys. , fjournal =

  48. [56]

    Semi-infinite flags

    Finkelberg, Michael and Mirkovi\'. Semi-infinite flags. Differential topology, infinite-dimensional

  49. [57]

    Finkelberg, Michael and Rybnikov, Leonid , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2014 , NUMBER =

    2014

  50. [58]

    Representations and nilpotent Orbits of

    Finkelberg, Michael and Tsymbaliuk, Alexander , title =. Representations and nilpotent Orbits of

  51. [59]

    Arnold Math

    Finkelberg, Michael and Tsymbaliuk, Alexander , TITLE =. Arnold Math. J. , FJOURNAL =. 2019 , NUMBER =

    2019

  52. [60]

    American Journal of Mathematics , volume=95, year=1973, number=1, pages=

    Fogarty, John , title=. American Journal of Mathematics , volume=95, year=1973, number=1, pages=

    1973

  53. [61]

    Flenner, Hubert , TITLE =. Invent. Math. , FJOURNAL =. 1988 , NUMBER =

    1988

  54. [62]

    Frassek, Rouven and Tsymbaliuk, Alexander , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2022 , NUMBER =

    2022

  55. [63]

    Fu, Baohua , title=. Ann. math. Blaise Pascal , volume =13, year=2006, number=2, pages=

    2006

  56. [64]

    Gaiotto, Davide and Witten, Edward , TITLE =. Adv. Theor. Math. Phys. , FJOURNAL =. 2009 , NUMBER =

    2009

  57. [65]

    Selecta Mathematica , year=

    Gautam, Sachin and Toledano Laredo, Valerio , title=. Selecta Mathematica , year=

  58. [66]

    Journal of the AMS , year=

    Gautam, Sachin and Toledano Laredo, Valerio , title=. Journal of the AMS , year=

  59. [67]

    Rigid modules and

    Gei , Christof and Leclerc, Bernard and Schr\". Rigid modules and. arXiv e-prints , keywords =. 2018

    2018

  60. [68]

    Communications in Mathematical Physics , volume=

    Gerasimov, Anton and Kharchev, Sergei and Lebedev, Dmitry and Oblezin, Sergey , title=. Communications in Mathematical Physics , volume=. 2005 , number=

    2005

  61. [69]

    Ginzburg, Victor , title=

  62. [70]

    Selecta Mathematica , volume=9, year=2003, number=3, pages=

    Ginzburg, Victor , title=. Selecta Mathematica , volume=9, year=2003, number=3, pages=

    2003

  63. [71]

    Goresky and R

    M. Goresky and R. Kottwitz and R. MacPherson , title=. Inventiones Mathematicae , volume=131, year=1998, pages=

    1998

  64. [72]

    G\"ortz and T

    U. G\"ortz and T. Wedhorn , title=

  65. [73]

    and Knuth, Donald E

    Graham, Ronald L. and Knuth, Donald E. and Patashnik, Oren , TITLE =. 1994 , PAGES =

    1994

  66. [74]

    Hanany, Amihay and Sperling, Marcus , TITLE =. J. High Energy Phys. , FJOURNAL =. 2018 , NUMBER =

    2018

  67. [75]

    Nuclear Phys

    Hanany, Amihay and Witten, Edward , TITLE =. Nuclear Phys. B , FJOURNAL =. 1997 , NUMBER =. doi:10.1016/S0550-3213(97)00157-0 , URL =

    1997 doi

  68. [76]

    Harada, Megumi , TITLE =. J. Symplectic Geom. , FJOURNAL =. 2006 , NUMBER =

    2006

  69. [77]

    Grimminger, Julius and Hanany, Amihay , title=. J. High Energy Phys. , FJOURNAL =. 2020 , NUMBER =

    2020

  70. [78]

    Bourget, Antoine and Cabrera, Santiago and Grimminger, Julius and Hanany, Amihay and Sperling, Marcus and Zajac, Anton and Zhong, Zhenghao , Title=. J. High Energy Phys. , FJOURNAL =. 2020 , NUMBER =

    2020

  71. [79]

    Hartshorne , title=

    R. Hartshorne , title=

  72. [80]

    Hausel, Tam\'as and Letellier, Emmanuel and Rodriguez-Villegas, Fernando , TITLE =. Ann. of Math. (2) , FJOURNAL =. 2013 , NUMBER =. doi:10.4007/annals.2013.177.3.8 , URL =

    2013 doi

  73. [81]

    Hua, Jiuzhao , TITLE =. J. Algebra , FJOURNAL =. 2000 , NUMBER =

    2000

  74. [82]

    K-theoretic

    Jindal, Shivang and Negu. K-theoretic

  75. [83]

    The loop-nilpotent cohomological

    Jindal, Shivang and Negu. The loop-nilpotent cohomological

  76. [84]

    Kac, Victor , title=

  77. [85]

    Invariant theory (

    Kac, Victor , TITLE =. Invariant theory (

  78. [86]

    Kaledin, Dmitry , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2006 , PAGES =

    2006

  79. [87]

    S\'eminaire Bourbaki, Ast\'erisque , volume=1072, year=2014, pages=

    Kamnitzer, Joel , title=. S\'eminaire Bourbaki, Ast\'erisque , volume=1072, year=2014, pages=

    2014

  80. [88]

    Algebra and Number Theory , volume=

    Kamnitzer, Joel and Webster, Ben and Weekes, Alex and Yacobi, Oded , title=. Algebra and Number Theory , volume=. 2014 , number=

    2014

  81. [89]

    Kamnitzer, Joel and Tingley, Peter and Webster, Ben and Weekes, Alex and Yacobi, Oded , TITLE =. J. Comb. Algebra , FJOURNAL =. 2019 , NUMBER =

    2019

  82. [90]

    Kamnitzer, Joel and Tingley, Peter and Webster, Ben and Weekes, Alex and Yacobi, Oded , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2019 , NUMBER =

    2019

  83. [91]

    Kamnitzer and D

    J. Kamnitzer and D. Muthiah and A. Weekes , title=. On a reducedness conjecture for spherical. Transform. Groups , FJOURNAL =. 2018 , NUMBER =

    2018

  84. [92]

    Kamnitzer, Joel and Muthiah, Dinakar and Weekes, Alex and Yacobi, Oded , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2018 , NUMBER =

    2018

  85. [93]

    Kamnitzer, Joel and Pham, Khoa and Weekes, Alex , title =

  86. [94]

    Kashiwara , title=

    M. Kashiwara , title=. Combinatorial and geometric representation theory , series=

  87. [95]

    Khoroshkin and V.N

    S.M. Khoroshkin and V.N. Tolstoy , title=. Letters in Mathematical Physics , volume=. 1996 , pages=

    1996

  88. [96]

    King , title=

    A. King , title=. Quarterly Journal of Mathematics , volume=45, year=1994, number=4, pages=

    1994

  89. [97]

    , TITLE =

    Khovanov, Mikhail and Lauda, Aaron D. , TITLE =. Quantum Topol. , FJOURNAL =. 2010 , NUMBER =

    2010

  90. [98]

    Khovanov and A

    M. Khovanov and A. Lauda and M. Mackaay and M. Sto. Extended graphical calculus for categorified quantum

  91. [99]

    Kontsevich, Maxim and Soibelman, Yan , TITLE =. Commun. Number Theory Phys. , FJOURNAL =. 2011 , NUMBER =

    2011

  92. [100]

    Krylov, Vasily and Perunov, Ivan , TITLE =. Adv. Math. , FJOURNAL =. 2021 , PAGES =

    2021

  93. [101]

    Labelle, Antoine , title =

  94. [102]

    Levendorski'i , title=

    S. Levendorski'i , title=. Journal of Geometry and Physics , volume=. 1993 , number=1, pages=

    1993

  95. [103]

    Levendorski'i , title=

    S. Levendorski'i , title=. Letters in Mathematical Physics , volume=. 1993 , number=1, pages=

    1993

  96. [104]

    Losev , title=

    I. Losev , title=. Advances in Mathematics , volume=231, year=2012, number=

    2012

  97. [105]

    Losev and B

    I. Losev and B. Webster , title=. Selecta Mathematica , volume=21, year=2015, number=2, pages=

    2015

  98. [106]

    Lusztig , title=

    G. Lusztig , title=. Annales de l'institut. 2000 , pages=

    2000

  99. [107]

    , title=

    Macdonald, Ian G. , title=

  100. [108]

    Maffei , title=

    A. Maffei , title=. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze , volume=1, year=2002, number=3, pages=

    2002

  101. [109]

    Maulik, Davesh and Okounkov, Andrei , title=

  102. [110]

    Annals of Mathematics , volume=166, year=2007, number=1, pages=

    Mirkovi\'c, Ivan and Vilonen, Kari , title=. Annals of Mathematics , volume=166, year=2007, number=1, pages=

    2007

  103. [111]

    Comptes Rendus de l'Acad\'emie des Sciences, Series I , volume=

    Mirkovi\'c, Ivan and Vybornov, Maxim , title=. Comptes Rendus de l'Acad\'emie des Sciences, Series I , volume=. 2003 , volume=

    2003

  104. [112]

    Molev, Alexander , title=

  105. [113]

    Handbook of Algebra , volume=3, year=2003, pages=

    Molev, Alexander , title=. Handbook of Algebra , volume=3, year=2003, pages=

    2003

  106. [114]

    Mozgovoy, Sergey , TITLE =. J. Algebra , FJOURNAL =. 2007 , NUMBER =

    2007

  107. [115]

    Muthiah, Dinakar and Weekes, Alex , TITLE =. Ann. Sci. \'. 2023 , NUMBER =

    2023

  108. [116]

    Muthiah, Dinakar and Weekes, Alex , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2024 , NUMBER =

    2024

  109. [117]

    Muthiah, Dinakar and Weekes, Alex and Yacobi, Oded , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2021 , PAGES =

    2021

  110. [118]

    Nandakumar, Vinoth and Tingley, Peter , title=. Math. Res. Lett. , volume=25, year=2016, number=1, pages=

    2016

  111. [119]

    Duke Mathematical Journal , volume=91, year=1998, number=3, pages=

    Nakajima, Hiraku , title=. Duke Mathematical Journal , volume=91, year=1998, number=3, pages=

    1998

  112. [120]

    Journal of the American Mathematical Society , volume=14, year=2001, pages=

    Nakajima, Hiraku , title=. Journal of the American Mathematical Society , volume=14, year=2001, pages=

    2001

  113. [121]

    Duke Mathematical Journal , volume=76, year=1994, number=2, pages=

    Nakajima, Hiraku , title=. Duke Mathematical Journal , volume=76, year=1994, number=2, pages=

    1994

  114. [122]

    Inventionaes Mathematicae , volume=146, year=2001, number=2, pages=

    Nakajima, Hiraku , title=. Inventionaes Mathematicae , volume=146, year=2001, number=2, pages=

    2001

  115. [123]

    Nakajima, Hiraku , title=

  116. [124]

    Nakajima, Hiraku , TITLE =. Adv. Theor. Math. Phys. , FJOURNAL =. 2016 , NUMBER =

    2016

  117. [125]

    Nakajima, Hiraku , title=. RIMS

  118. [126]

    Selecta Math

    Nakajima, Hiraku and Takayama, Yuuya , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2017 , NUMBER =

    2017

  119. [127]

    Namikawa, Yoshinori , title=

  120. [128]

    Nakajima, Hiraku and Weekes, Alex , TITLE =. J. Eur. Math. Soc. (JEMS) , FJOURNAL =. 2023 , NUMBER =

    2023

  121. [129]

    Shuffle algebras for quivers and wheel conditions , JOURNAL =

    Negu. Shuffle algebras for quivers and wheel conditions , JOURNAL =. 2023 , PAGES =

    2023

  122. [130]

    An integral form of quantum toroidal

    Negu. An integral form of quantum toroidal. Math. Rep. (Bucur.) , FJOURNAL =. 2024 , NUMBER =

    2024

  123. [131]

    Generators of the preprojective

    Negu. Generators of the preprojective. Bull. Lond. Math. Soc. , FJOURNAL =. 2025 , NUMBER =. doi:10.1112/blms.70093 , URL =

    2025 doi

  124. [132]

    Internat

    Ragoucy, Eric , TITLE =. Internat. J. Modern Phys. A , FJOURNAL =. 2001 , NUMBER =

    2001

  125. [133]

    Rouquier, Rapha\". 2-

  126. [134]

    Schiffmann, Olivier and Vasserot, Eric , TITLE =. Publ. Math. Inst. Hautes \'Etudes Sci. , FJOURNAL =. 2013 , PAGES =

    2013

  127. [135]

    Schiffmann, Olivier and Vasserot, Eric , TITLE =

  128. [136]

    Schiffmann, Olivier and Vasserot, Eric , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2020 , PAGES =

    2020

  129. [137]

    Bozec, Tristan and Schiffmann, Olivier and Vasserot, \'Eric , TITLE =. Ann. Sci. \'Ec. Norm. Sup\'er. (4) , FJOURNAL =. 2020 , NUMBER =

    2020

  130. [138]

    Shan and M

    P. Shan and M. Varagnolo and E. Vasserot , title=

  131. [139]

    Topology , volume=14, year=1975, pages=

    Steinberg, Robert , title=. Topology , volume=14, year=1975, pages=

    1975

  132. [140]

    Selecta Math

    Tsymbaliuk, Alexander , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2021 , NUMBER =

    2021

  133. [141]

    Tsymbaliuk, Alexander , TITLE =. Lett. Math. Phys. , FJOURNAL =. 2023 , NUMBER =

    2023

  134. [142]

    1994 , PAGES =

    Vaisman, Izu , TITLE =. 1994 , PAGES =

    1994

  135. [143]

    Letters in Mathematical Physics , volume=53, year=2000, pages=

    Varagnolo, Michela , title=. Letters in Mathematical Physics , volume=53, year=2000, pages=

    2000

  136. [144]

    Waterhouse, William , title=

  137. [145]

    Webster, Ben , title=

  138. [146]

    Journal of Pure and Applied Algebra , volume=215, year=2011, number=12, pages=

    Webster, Ben , title=. Journal of Pure and Applied Algebra , volume=215, year=2011, number=12, pages=

    2011

  139. [147]

    Webster, Ben , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 2017 , NUMBER =

    2017

  140. [148]

    Webster, Ben and Weekes, Alex and Yacobi, Oded , title=

  141. [149]

    Weekes, Alex , title =

  142. [150]

    Weyman, Jerzy , TITLE =. Invent. Math. , FJOURNAL =. 1989 , NUMBER =

    1989

  143. [151]

    2003 , PAGES =

    Weyman, Jerzy , TITLE =. 2003 , PAGES =

    2003

  144. [152]

    Yang, Yaping and Zhao, Gufang , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2018 , NUMBER =

    2018

  145. [153]

    Zhou, Yehao , title=

  146. [154]

    Zhu, Xinwen , title=

Pith tools

Reviewed July 31, 2026 · model on record in the stance chip above.