Pith. sign in

REVIEW 2 major objections 5 minor 97 references

On weight modules over truncated shifted iYangians

T0 review · 2 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read Integral weight modules over truncated shifted iYangians are equivalent to nilpotent modules over interval oKLRW algebras.

desk verdict First diagrammatic description of integral weight modules for shifted iYangians, with a real but addressable gap in the faithfulness of the parity polynomial representation. read the letter →

arxiv 2607.27596 v1 pith:4R5Q5FNS submitted 2026-07-30 math.RT math-phmath.MPmath.QA

Yaolong Shen , Linliang Song , Rui Xiong This is my paper · ORCID
classification math.RTmath-phmath.MPmath.QA MSC 17B3716G99
keywords truncatedshiftediYangiansoKLRWalgebrasweightmodulesquiverswithinvolutionKLRparitypolynomialrepresentationaffineGrassmannianslicesnilpotent
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 the category of finitely generated integral-weight modules over a truncated shifted iYangian—an algebra expected to quantize components of affine Grassmannian slices fixed by an involution—is equivalent to the category of finitely generated nilpotent modules over an interval orientifold KLRW (oKLRW) algebra. The equivalence passes through a flag version of the iYangian and through a new diagrammatic algebra, the KLR iYangian, whose generators are double-reflective strand diagrams. On the diagrammatic side, the paper constructs faithful polynomial representations and explicit bases, so that integral weight spaces are encoded by configurations of strands in intervals between red strands. If correct, the result gives a combinatorial, diagrammatic description of integral weight modules for these twisted Yangians, analogous to the untwisted setting.

What carries the argument

The central objects are the oKLRW algebras (diagrammatic algebras obtained by combining the mirror of orientifold KLR algebras with Webster's red strands) and the KLR iYangian, defined by double reflective KLR diagrams with two facing mirrors. The load-bearing identity is the parity polynomial representation (Definition 4.5, equation (4.11)): even-indexed red strands act by multiplication in the usual polynomial representation while odd-indexed red strands act by the opposite convention. This representation transfers the shift operators Γi,k and Γτi,k of the flag iYangian into long crossings and long reflections of strands between intervals, and its faithfulness is used to conclude that the

What would settle it

A non-zero element in the kernel of the parity polynomial representation (Definition 4.5) — for example, a non-zero diagram in τ0Rλint whose polynomial action vanishes — would invalidate the well-definedness of the functors Θ and Γ, and hence of Theorem 4.11.

Watch

Extended reading notes

Core claim

Theorem 4.11 asserts that the functors Θ and Γ are mutually inverse equivalences τFYλμ-wtmod ≅ τ0Rλint-modnil. Here τFYλμ is the flag truncated shifted iYangian, Morita equivalent (Theorem 3.12) to the truncated shifted iYangian τYλμ, and τ0Rλint is the interval oKLRW algebra, an idempotent truncation of the oKLRW algebra whose objects are interval configurations of strands matching integral weight spaces. The functor Θ rebuilds a weight module by assigning a summand e(νa)M to each integral weight a, while Γ collects weakly increasing integral generalized weight spaces and equips them with the interval action. Combining this with the Morita equivalence yields a diagrammatic description of in

Load-bearing premise

The parity polynomial representation of the interval oKLRW algebra is faithful, and the paper relies on this to show the explicit operators satisfy the iYangian relations, but the proof is only sketched by reference to a similar corollary.

Editorial extensions

If this is right

  • Integral weight modules over truncated shifted iYangians reduce to finite combinatorial data: interval configurations of strands with nilpotent dot actions.
  • The Morita equivalence extends the result to ordinary (non-flag) truncated shifted iYangians, giving diagrammatic control over their integral weight modules.
  • The explicit functors Θ and Γ allow in principle the computation of weight spaces and intertwiners by manipulating strand diagrams.
  • The parity convention shows how an involution (mirror) must be encoded in polynomial representations, a feature that should transfer to other twisted Yangian settings.
  • For quivers of diagonal type, the equivalence reduces to the untwisted case, so the construction recovers the classical statement in that limit.

Reading between the lines

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

  • The faithfulness of the parity polynomial representation is asserted to follow from 'similar arguments' as Corollary 2.16, but the proof is not written out; a direct proof or a counterexample would settle whether the functors in Theorem 4.11 are well-defined.
  • The paper notes that the iGKLO representation is not known to be full; if fullness fails, the interval oKLRW algebra could be strictly larger than the image of the flag iYangian, refining the equivalence rather than breaking it.
  • The same interval/parity encoding suggests a route to defining category O for shifted iYangians via one-sided quotients of the interval oKLRW algebra, as the authors themselves propose.
  • A computational test for small quivers (e.g., type AIII with small vi) comparing weight-space dimensions computed from both sides would offer a concrete check of the equivalence.
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 / 5 minor

Summary. The paper develops an orientifold analogue of the KTW+19 categorical equivalence. It introduces oKLR W algebras (combining the oKLR mirror formalism with Webster's red strands), KLR iYangians (double reflective KLR diagrams), and interval oKLR W algebras. The main result, Theorem 4.11 (restated as Theorem 1.1), asserts an equivalence between the category of finitely generated integral weight modules over the flag truncated shifted iYangian τFYλμ and the category of finitely generated nilpotent modules over the interval oKLR W algebra τ0Rλint. The proof constructs explicit functors Θ and Γ, and relies on a sequence of intermediate results: faithful polynomial representations for oKLR W algebras and KLR iYangians, an iGKLO embedding of the truncated shifted iYangian into a corner of the KLR iYangian, a Morita equivalence between the flag and non-flag shifted iYangians, and a parity-modified polynomial representation of the interval oKLR W algebra.

Significance. If fully substantiated, the main equivalence is a significant advance: it provides a diagrammatic, combinatorial description of weight modules for shifted iYangians associated to quivers with involution, parallel to and extending the untwisted KTW+19 results. The paper introduces several new algebraic objects (oKLR W algebras, KLR iYangians, interval oKLR W algebras) with explicit generators, relations, and polynomial representations, and it formulates precise functors between module categories. The explicit nature of the construction and the extensive use of diagrammatic methods are strengths. However, the paper does not contain machine-checked proofs or reproducible code; its correctness rests on a chain of technical verifications, several of which are currently asserted rather than demonstrated.

major comments (2)
  1. [Definition 4.5, Eq. (4.11); Propositions 4.8 and 4.10] The faithfulness of the parity polynomial representation P̃par is load-bearing but not proved. Definition 4.5 simply states that faithfulness 'follows from similar arguments as in Corollary 2.16'. The parity modification (4.11) swaps the multiplicative and identity actions of red–black crossings depending on the parity of the red strand. This is not a trivial relabelling: the triangularity argument of Corollary 2.16 uses the explicit polynomial actions (2.19), (2.20), and (2.23), and the new convention changes the coefficients of the leading Bruhat terms for diagrams that include red–black crossings. Since Propositions 4.8 and 4.10 both invoke this faithfulness to conclude that the explicit operators (4.15)–(4.18) satisfy all relations of τFYλμ and τ0Rλint, a failure of faithfulness would make the functors Θ and Γ ill-defined and would invalidate Theorem 4.11. The gap is likely repairabl
  2. [Proposition 3.4] The preservation of the polynomial subalgebra PΣ under the iGKLO action is asserted with a reference to 'the same manner as [KTW+19, Thm. 4.6]' but no proof or even a sketch is given. This step is load-bearing: the truncated shifted iYangian τYλμ is defined as the image of τYμ in End(PΣ), and the subsequent flag algebra τFYλμ is built from τYλμ. If the action only preserved Frac(P) and not PΣ, the entire construction would be ill-founded. Since the formulas (3.10)–(3.11) involve nontrivial rational expressions and shift operators, the pole cancellation that preserves PΣ is a substantive check. The authors should provide a full proof or at least a detailed lemma, rather than relying on an analogy with the untwisted case.
minor comments (5)
  1. [Definition 2.1] The last sentence of Definition 2.1 says 'We refer to λ as the framing weight of τμR', but the definition is for a dominant weight μ. This is a typo: it should refer to μ.
  2. [Definition 4.5] In the definition of PolλΩ, the index set is written as 'i∈Ω' but Ω is a set of interval configurations ν. This should be 'ν∈Ω'.
  3. [Proposition 4.8, Eq. (4.16)] The notation eY^a_{i,r} is introduced directly after the display, but it is used inside the display. Moving the definition before the equation would improve readability.
  4. [Theorem 4.11 proof] The assertion 'The intertwiner relations imply χ^2_{i,k}=1' is stated without proof. For the intertwiners defined in (3.29), this is not immediate from the displayed relations and should be either proved or referenced precisely.
  5. [Remark 4.2] The remark explains the orbit O_{1/4} and its two cosets, but it is somewhat dense. A short example of a non-self-dual orbit would help the reader understand the claimed limitation to integral parameters.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Θ/Γ equivalence is proved by explicit comparison of polynomial actions; the flagged faithfulness gap is a rigor gap, not a reduction.

full rationale

Walking the derivation chain, the central equivalence (Theorem 4.11) is not obtained by fitting or by definition. The functors Θ and Γ (Propositions 4.8 and 4.10) are constructed by explicit formulas and checked against the polynomial actions of the generators of τFYλμ computed in Section 3.4 from the iGKLO comparison (Theorems 3.10 and 3.13). Those polynomial actions come from the KLR iYangian I and the shifted iYangian, not from the interval oKLR W algebra. The interval algebra τ0Rλ_int is independently defined as an idempotent truncation of the oKLR W algebra (Definition 4.3–4.5), and the parity representation is a representation of that already-defined algebra. Remark 4.6 explains why the parity convention was chosen to match framing factors, but this is motivation, not a definitional identification of the two categories. The mutually inverse property of Θ and Γ is proved by direct substitution of the explicit formulas (4.15)–(4.28), which is a computation, not a tautology. The main genuine weakness is that Definition 4.5 asserts faithfulness of P̃par by 'similar arguments as in Corollary 2.16' without supplying the triangularity proof for the parity-modified red–black crossings; this is load-bearing in Propositions 4.8 and 4.10. However, an omitted proof is a rigor/correctness gap, not a circular reduction: the asserted faithfulness does not assume the target equivalence, and the same triangularity method cited for Corollary 2.16 is an independent argument. Citations to [SSX] (with overlapping authors) for the iGKLO action are also backed by non-overlapping references [LWW25a, LWW25b], and the paper explicitly notes (Remark 3.11) that fullness of the iGKLO representation is not established—again a scope limitation, not a circular step. I therefore find no circularity.

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

The paper's central claim depends on the specific choices of parameters R, λ, μ, the integrality restriction, and the borrowed iGKLO action. The new diagrammatic algebras are fully constructed in the paper; they are not pulled from a hat for the sake of the conclusion, but they also lack independent evidence outside the paper. The integrality assumption is the main domain restriction, and the faithfulness of the parity polynomial representation is an in-paper assumption that is not fully proved.

free parameters (3)
  • integral parameter sets R_i = R_i ⊂ 1/4 + Z for i∈Q0^(+), R_{τi} ⊂ 3/4 + Z
    These sets determine the truncated shifted iYangian and the interval oKLR W algebra; the main theorem is stated only for such integral parameters. They are inputs, not fitted to data, but they restrict the domain of the result.
  • framing weight λ = λ = Σ λ_i ϖ_i, dominant
    Defines the truncated shifted iYangian τYλμ and the red-strand data in the diagrammatic algebras.
  • shift μ = τ-invariant weight with τλ ≥ μ
    Part of the shifted iYangian data; determines the truncation.
assumptions (5)
  • domain assumption The involution τ on Q0 has no fixed points.
    Assumed throughout (Section 1.2) and needed for the interval/parity construction; the fixed-vertex case is deferred to future work (Remark 2.17).
  • domain assumption The quiver Q is simply laced and loop-free.
    Standard assumption in the KLR/KLR W literature (Section 2.1).
  • domain assumption Integrality of parameters: R_i ⊂ 1/4+Z, R_{τi} ⊂ 3/4+Z.
    Imposed in Section 4 to select the involution-invariant spectral orbit O_{1/4}; the paper says the general case can be treated by orbits (Remark 4.2).
  • domain assumption The iGKLO action of τYμ on Frac(P) preserves the polynomial subalgebra PΣ.
    Prop 3.4 is cited from [SSX, Thm 4.2] and proved 'in the same manner as [KTW+19, Thm. 4.6]'. This is a load-bearing fact for defining τYλμ as a subalgebra of End(PΣ).
  • standard math Standard background on KLR algebras, KLR W algebras, and Morita equivalence.
    The paper builds on [KL09, Rou08, Web17, VV11, Prz23] without reproving standard facts.
invented entities (3)
  • oKLR W algebra τμRλ
    purpose: Diagrammatic algebra combining orientifold KLR with Webster's red strands; primary object of study.
    New algebra defined in (2.22); its consistency is established in the paper via a basis theorem, but it has no external falsifiable handle outside the paper.
  • KLR iYangian I
    purpose: Diagrammatic algebra on double reflective KLR diagrams; used to embed the truncated shifted iYangian via the iGKLO formulas.
    New algebra introduced in Definition 3.7; its faithfulness on polynomials is proved in the paper (Theorem 3.8), but it is a novel object with no independent corroboration.
  • Interval oKLR W algebra τ0Rλint
    purpose: Idempotent truncation of the oKLR W algebra whose nilpotent modules are claimed equivalent to integral weight modules of the flag iYangian.
    Defined in (4.5) specifically to match the integral weight spaces; its module category is the target of the main equivalence.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On weight modules over truncated shifted iYangians." pith.science (2026). https://pith.science/paper/4R5Q5FNS

@misc{pith2026260727596,
  author       = {Pith},
  title        = {Pith review of: On weight modules over truncated shifted iYangians},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4R5Q5FNS}},
  note         = {Machine review of arXiv:2607.27596}
}
read the original abstract

Truncated shifted iYangians are a family of algebras expected to quantize certain components of affine Grassmannian islices. We introduce orientifold KLRW (oKLRW) algebras associated with quivers with involution and establish their faithful polynomial representations and diagrammatic bases. We also define KLR iYangians using double reflective KLR diagrams and construct diagrammatic realizations of the iGKLO homomorphisms. For integral parameters, we introduce interval oKLRW algebras and prove an equivalence between integral weight modules over truncated shifted iYangians and nilpotent modules over the corresponding interval oKLRW algebras.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

97 extracted references · 24 canonical work pages

  1. [1]

    Ariki, Susumu , TITLE =. J. Math. Kyoto Univ. , FJOURNAL =. 1996 , NUMBER =. doi:10.1215/kjm/1250518452 , URL =

    arXiv 1996

  2. [2]

    and Finkelberg, M

    Braverman, A. and Finkelberg, M. and Nakajima, H. , TITLE =. Adv. Theor. Math. Phys. , FJOURNAL =. 2018 , NUMBER =. doi:10.4310/ATMP.2018.v22.n5.a1 , URL =

    doi:10.4310/atmp.2018.v22.n5.a1 2018

  3. [3]

    and Tingley, P

    Kamnitzer, J. and Tingley, P. and Webster, B. and Weekes, A. and Yacobi, O. , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2019 , NUMBER =. doi:10.1112/plms.12254 , URL =

    work page doi:10.1112/plms.12254 2019

  4. [4]

    Braverman, Alexander and Finkelberg, Michael and Nakajima, Hiraku , TITLE =. Adv. Theor. Math. Phys. , FJOURNAL =. 2019 , NUMBER =. doi:10.4310/ATMP.2019.v23.n1.a3 , URL =

    doi:10.4310/atmp.2019.v23.n1.a3 2019

  5. [5]

    Kodera, Ryosuke and Nakajima, Hiraku , TITLE =. String-. 2018 , ISBN =. doi:10.1090/pspum/098/01720 , URL =

    work page doi:10.1090/pspum/098/01720 2018

  6. [6]

    Wang, Weiqiang and Zhang, Weinan , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2023 , NUMBER =. doi:10.1112/plms.12562 , URL =

    work page doi:10.1112/plms.12562 2023

  7. [7]

    , TITLE =

    Araki, S. , TITLE =. J. Math. Osaka City Univ. , FJOURNAL =. 1962 , PAGES =

    1962

  8. [8]

    On generators and defining relations of

    Levendorski. On generators and defining relations of. J. Geom. Phys. , FJOURNAL =. 1993 , NUMBER =. doi:10.1016/0393-0440(93)90084-R , URL =

    work page doi:10.1016/0393-0440(93)90084-r 1993

Show all 97 references
  1. [9]

    Braverman, Alexander and Finkelberg, Michael and Nakajima, Hiraku , TITLE =. Adv. Theor. Math. Phys. , FJOURNAL =. 2019 , NUMBER =. doi:10.4310/ATMP.2019.v23.n2.a1 , URL =

    2019 doi

  2. [10]

    Geometric representation theory and gauge theory , SERIES =

    Braverman, Alexander and Finkelberg, Michael , TITLE =. Geometric representation theory and gauge theory , SERIES =. 2019 , ISBN =. doi:10.1007/978-3-030-26856-5\_1 , URL =

    2019 doi

  3. [11]

    Brundan, Jonathan and Kleshchev, Alexander , TITLE =. Adv. Math. , FJOURNAL =. 2009 , NUMBER =. doi:10.1016/j.aim.2009.06.018 , URL =

    2009 doi

  4. [12]

    and Kolb, S

    Balagovi\'c, M. and Kolb, S. , TITLE =. J. Reine Angew. Math. , FJOURNAL =. 2019 , PAGES =. doi:10.1515/crelle-2016-0012 , URL =

    2019 doi

  5. [13]

    Ast\'erisque , FJOURNAL =

    Bao, Huanchen and Wang, Weiqiang , TITLE =. Ast\'erisque , FJOURNAL =. 2018 , PAGES =

    2018

  6. [14]

    Bao, Huanchen and Wang, Weiqiang , TITLE =. Invent. Math. , FJOURNAL =. 2018 , NUMBER =. doi:10.1007/s00222-018-0801-5 , URL =

    2018 doi

  7. [15]

    Cherednik, I. V. , TITLE =. Teoret. Mat. Fiz. , FJOURNAL =. 1984 , NUMBER =

    1984

  8. [16]

    Guay, Nicolas and Regelskis, Vidas , TITLE =. Math. Z. , FJOURNAL =. 2016 , NUMBER =. doi:10.1007/s00209-016-1649-2 , URL =

    2016 doi

  9. [17]

    Molev, A. I. and Ragoucy, E. , TITLE =. Rev. Math. Phys. , FJOURNAL =. 2002 , NUMBER =. doi:10.1142/S0129055X02001156 , URL =

    2002 doi

  10. [18]

    Zhang, Weinan , TITLE =. Lett. Math. Phys. , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s11005-022-01583-6 , URL =

    2022 doi

  11. [19]

    Olshanski i, G. I. , TITLE =. Quantum groups (. 1992 , ISBN =. doi:10.1007/BFb0101183 , URL =

    1992 doi

  12. [20]

    Sklyanin, E. K. , TITLE =. J. Phys. A , FJOURNAL =. 1988 , NUMBER =

    1988

  13. [21]

    Bao, Huanchen and Wang, Weiqiang , TITLE =. Compos. Math. , FJOURNAL =. 2021 , NUMBER =. doi:10.1112/S0010437X2100734X , URL =

    2021 doi

  14. [22]

    Bao, Huanchen and Wang, Weiqiang and Watanabe, Hideya , TITLE =. Proc. Amer. Math. Soc. , FJOURNAL =. 2018 , NUMBER =. doi:10.1090/proc/13749 , URL =

    2018 doi

  15. [23]

    and Shan, P

    Bao, H. and Shan, P. and Wang, W. and Webster, B. , TITLE =. Quantum Topol. , FJOURNAL =. 2018 , NUMBER =. doi:10.4171/QT/117 , URL =

    2018 doi

  16. [24]

    , title =

    Brundan, J. , title =. 2017 , ARCHIVEPREFIX =

    2017

  17. [25]

    and Comes, J

    Brundan, J. and Comes, J. and Nash, D. and Reynolds, A. , TITLE =. Quantum Topol. , FJOURNAL =. 2017 , NUMBER =. doi:10.4171/QT/87 , URL =

    2017 doi

  18. [26]

    and Wang, W

    Brundan, J. and Wang, W. and Webster, B. , title =. 2023 , ARCHIVEPREFIX =

    2023

  19. [27]

    and Wang, W

    Brundan, J. and Wang, W. and Webster, B. , title =. 2025 , ARCHIVEPREFIX =

    2025

  20. [28]

    and Kamnitzer, J

    Cautis, S. and Kamnitzer, J. and Morrison, S. , TITLE =. Math. Ann. , FJOURNAL =. 2014 , NUMBER =. doi:10.1007/s00208-013-0984-4 , URL =

    2014 doi

  21. [29]

    and Shen, Y

    Cui, W. and Shen, Y. , TITLE =. Math. Res. Lett. , FJOURNAL =. 2024 , NUMBER =. doi:10.4310/mrl.240904211718 , URL =

    2024 doi

  22. [30]

    , TITLE =

    Deligne, P. , TITLE =. Algebraic groups and homogeneous spaces , SERIES =. 2007 , MRCLASS =

    2007

  23. [31]

    and Stroppel, C

    Ehrig, M. and Stroppel, C. , TITLE =. Math. Z. , FJOURNAL =. 2016 , NUMBER =. doi:10.1007/s00209-016-1669-y , URL =

    2016 doi

  24. [32]

    , TITLE =

    Enomoto, N. , TITLE =. Int. Math. Res. Not. IMRN , FJOURNAL =. 2009 , NUMBER =. doi:10.1093/imrn/rnp014 , URL =

    2009 doi

  25. [33]

    and Kashiwara, M

    Enomoto, N. and Kashiwara, M. , TITLE =. Proc. Japan Acad. Ser. A Math. Sci. , FJOURNAL =. 2006 , NUMBER =

    2006

  26. [34]

    Combinatorial representation theory and related topics , SERIES =

    Enomoto, Naoya and Kashiwara, Masaki , TITLE =. Combinatorial representation theory and related topics , SERIES =. 2008 , MRCLASS =

    2008

  27. [35]

    Enomoto, Naoya and Kashiwara, Masaki , TITLE =. Publ. Res. Inst. Math. Sci. , FJOURNAL =. 2008 , NUMBER =

    2008

  28. [36]

    and Gelaki, S

    Etingof, P. and Gelaki, S. and Nikshych, D. and Ostrik, V. , TITLE =. 2015 , PAGES =. doi:10.1090/surv/205 , URL =

    2015 doi

  29. [37]

    Representations and nilpotent orbits of

    Finkelberg, Michael and Tsymbaliuk, Alexander , TITLE =. Representations and nilpotent orbits of. 2019 , ISBN =. doi:10.1007/978-3-030-23531-4\_6 , URL =

    2019 doi

  30. [38]

    and Rui, H

    Gao, M. and Rui, H. and Song, L. , TITLE =. J. Algebra , FJOURNAL =. 2022 , PAGES =. doi:10.1016/j.jalgebra.2022.07.005 , URL =

    2022 doi

  31. [39]

    , TITLE =

    Gavarini, F. , TITLE =. Pacific J. Math. , FJOURNAL =. 2020 , NUMBER =. doi:10.2140/pjm.2020.306.451 , URL =

    2020 doi

  32. [40]

    and Kharchev, S

    Gerasimov, A. and Kharchev, S. and Lebedev, D. and Oblezin, S. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2005 , NUMBER =. doi:10.1007/s00220-005-1417-3 , URL =

    2005 doi

  33. [41]

    and Webster, B

    Kamnitzer, J. and Webster, B. and Weekes, A. and Yacobi, O. , TITLE =. Algebra Number Theory , FJOURNAL =. 2014 , NUMBER =. doi:10.2140/ant.2014.8.857 , URL =

    2014 doi

  34. [42]

    , TITLE =

    Kolb, S. , TITLE =. Adv. Math. , FJOURNAL =. 2014 , PAGES =. doi:10.1016/j.aim.2014.08.010 , URL =

    2014 doi

  35. [43]

    , TITLE =

    Kolb, S. , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2020 , NUMBER =. doi:10.1112/plms.12303 , URL =

    2020 doi

  36. [44]

    and Yakimov, M

    Kolb, S. and Yakimov, M. , TITLE =. Adv. Math. , FJOURNAL =. 2020 , PAGES =. doi:10.1016/j.aim.2020.107042 , URL =

    2020

  37. [45]

    Lascoux, Alain and Leclerc, Bernard and Thibon, Jean-Yves , TITLE =. Comm. Math. Phys. , FJOURNAL =. 1996 , NUMBER =

    1996

  38. [46]

    and Wang, W

    Lu, K. and Wang, W. and Weekes, A. , title =. 2025 , JOURNAL =

    2025

  39. [47]

    , title =

    Webster, B. , title =. 2016 , JOURNAL =

    2016

  40. [48]

    and Webster, B

    Kamnitzer, J. and Webster, B. and Weekes, A. and Yacobi, O. , TITLE =. Forum Math. Sigma , FJOURNAL =. 2024 , PAGES =. doi:10.1017/fms.2024.3 , URL =

    2024 doi

  41. [49]

    and Davidson, N

    Brundan, J. and Davidson, N. , TITLE =. Categorification and higher representation theory , SERIES =. 2017 , ISBN =. doi:10.1090/conm/683 , URL =

    2017 doi

  42. [50]

    and Wang, W

    Lu, K. and Wang, W. and Weekes, A. , title =. 2026 , JOURNAL =

    2026

  43. [51]

    and Przezdziecki, T

    Li, J. and Przezdziecki, T. , title =. 2026 , JOURNAL =

    2026

  44. [52]

    , title =

    Rouquier, R. , title =. 2008 , JOURNAL =

    2008

  45. [53]

    and Su, C

    Shen, Y. and Su, C. and Xiong, R. , TITLE =. to appear on Comm. Math. Phys. , ARCHIVEPREFIX =

  46. [54]

    and Lauda, A

    Khovanov, M. and Lauda, A. , TITLE =. Represent. Theory , FJOURNAL =. 2009 , PAGES =. doi:10.1090/S1088-4165-09-00346-X , URL =

    2009 doi

  47. [55]

    , TITLE =

    Webster, B. , TITLE =. Compos. Math. , FJOURNAL =. 2015 , NUMBER =. doi:10.1112/S0010437X1400760X , URL =

    2015 doi

  48. [56]

    , TITLE =

    Webster, B. , TITLE =. Mem. Amer. Math. Soc. , FJOURNAL =. 2017 , NUMBER =. doi:10.1090/memo/1191 , URL =

    2017 doi

  49. [57]

    , TITLE =

    Webster, B. , TITLE =. Math. Ann. , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s00208-016-1438-6 , URL =

    2017 doi

  50. [58]

    , TITLE =

    Prze\'zdziecki, T. , TITLE =. Pacific J. Math. , FJOURNAL =. 2023 , NUMBER =. doi:10.2140/pjm.2023.322.407 , URL =

    2023 doi

  51. [59]

    Lehrer, G. I. and Zhang, R. B. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2017 , NUMBER =. doi:10.1007/s00220-016-2731-7 , URL =

    2017 doi

  52. [60]

    , TITLE =

    Letzter, G. , TITLE =. J. Algebra , FJOURNAL =. 1999 , NUMBER =. doi:10.1006/jabr.1999.8015 , URL =

    1999

  53. [61]

    , TITLE =

    Letzter, G. , TITLE =. New directions in. 2002 , MRCLASS =

    2002

  54. [62]

    Lu, Ming and Ruan, Shiquan and Wang, Weiqiang , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2023 , NUMBER =. doi:10.1090/tran/8798 , URL =

    2023 doi

  55. [63]

    Lu, Ming and Wang, Weiqiang , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2022 , NUMBER =. doi:10.1112/plms.12423 , URL =

    2022 doi

  56. [64]

    Lu, Ming and Wang, Weiqiang , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2021 , NUMBER =. doi:10.1007/s00220-021-03965-8 , URL =

    2021 doi

  57. [65]

    Lu, Ming and Wang, Weiqiang , TITLE =. Adv. Math. , FJOURNAL =. 2021 , PAGES =. doi:10.1016/j.aim.2021.108111 , URL =

    2021

  58. [66]

    Lu, Ming and Wang, Weiqiang , TITLE =. Adv. Math. , FJOURNAL =. 2021 , PAGES =. doi:10.1016/j.aim.2021.108071 , URL =

    2021

  59. [67]

    and Wang, W

    Lu, K. and Wang, W. and Zhang, W. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2025 , NUMBER =. doi:10.1007/s00220-025-05263-z , URL =

    2025 doi

  60. [68]

    Lu, Ming and Wang, Weiqiang , TITLE =. Adv. Math. , FJOURNAL =. 2023 , PAGES =. doi:10.1016/j.aim.2023.109215 , URL =

    2023

  61. [69]

    Selecta Math

    Lu, Ming and Wang, Weiqiang , TITLE =. Selecta Math. (N.S.) , FJOURNAL =. 2022 , NUMBER =. doi:10.1007/s00029-022-00800-3 , URL =

    2022 doi

  62. [70]

    and Zhang, W

    Lu, K. and Zhang, W. , journal=. A

  63. [71]

    , journal=

    Wang, Z. , journal=. Quivers with Involutions and Shifted Twisted

  64. [72]

    preprint , note =

    Shifted twisted Yangians and Slodowy slices in classical Lie algebras , author=. preprint , note =

  65. [73]

    2025 , JOURNAL =

    Shifted twisted Yangians and finite W-algebras of classical type , author=. 2025 , JOURNAL =

    2025

  66. [74]

    McNamara, P. J. and Savage, A. , TITLE =. Forum Math. Sigma , FJOURNAL =. 2024 , PAGES =. doi:10.1017/fms.2024.102 , URL =

    2024 doi

  67. [75]

    , TITLE =

    Lusztig, G. , TITLE =. 2010 , PAGES =. doi:10.1007/978-0-8176-4717-9 , URL =

    2010 doi

  68. [76]

    , TITLE =

    Mitsuhashi, H. , TITLE =. Algebr. Represent. Theory , FJOURNAL =. 2006 , NUMBEnR =. doi:10.1007/s10468-006-9014-5 , URL =

    2006 doi

  69. [77]

    Molev, A. I. , TITLE =. Czechoslovak J. Phys. , FJOURNAL =. 2003 , NUMBEnR =. doi:10.1023/B:CJOP.0000010536.64174.8e , URL =

    2003

  70. [78]

    Modern geometry: a celebration of the work of

    Nakajima, Hiraku , TITLE =. Modern geometry: a celebration of the work of. 2018 , ISBN =. doi:10.1090/pspum/099/01741 , URL =

    2018 doi

  71. [79]

    and Sartori, A

    Queffelec, H. and Sartori, A. , TITLE =. J. Pure Appl. Algebra , FJOURNAL =. 2019 , NUMBER =. doi:10.1016/j.jpaa.2018.09.014 , URL =

    2019 doi

  72. [80]

    and Song, L

    Rui, H. and Song, L. , TITLE =. Math. Z. , FJOURNAL =. 2019 , NUMBER =. doi:10.1007/s00209-018-2207-x , URL =

    2019 doi

  73. [81]

    and Savage, A

    Samchuck-Schnarch, S. and Savage, A. , TITLE =. Ann. Represent. Theory , FJOURNAL =. 2024 , NUMBER =. doi:10.5802/art.7 , ARCHIVEPREFIX =

    2024 doi

  74. [82]

    and Tubbenhauer, D

    Sartori, A. and Tubbenhauer, D. , TITLE =. Trans. Amer. Math. Soc. , FJOURNAL =. 2019 , NUMBEnR =. doi:10.1090/tran/7583 , URL =

    2019 doi

  75. [83]

    and Webster, B

    Savage, A. and Webster, B. , TITLE =. J. Aust. Math. Soc. , FJOURNAL =

  76. [84]

    and Savage, A

    Salmasian, H. and Savage, A. and Shen, Y. , TITLE =. Forum Math. Sigma , FJOURNAL =. 2026 , PAGES =. doi:10.1017/fms.2025.10148 , URL =

    2026

  77. [85]

    , TITLE =

    Shen, Y. , TITLE =. J. Algebra , FJOURNAL =. 2025 , PAGES =. doi:10.1016/j.jalgebra.2024.07.035 , URL =

    2025 doi

  78. [86]

    Shen, Yaolong and Wang, Weiqiang , TITLE =. Adv. Math. , FJOURNAL =. 2023 , PAGES =. doi:10.1016/j.aim.2023.109131 , URL =

    2023

  79. [87]

    Shen, Yaolong and Wang, Weiqiang , TITLE =. Comm. Math. Phys. , FJOURNAL =. 2025 , NUMBER =. doi:10.1007/s00220-025-05339-w , URL =

    2025 doi

  80. [88]

    Turaev, V. G. , TITLE =. Izv. Akad. Nauk SSSR Ser. Mat. , FJOURNAL =. 1989 , NUMBER =

    1989

  81. [89]

    and Vasserot, E

    Varagnolo, M. and Vasserot, E. , TITLE =. Invent. Math. , FJOURNAL =. 2011 , NUMBER =. doi:10.1007/s00222-011-0314-y , URL =

    2011 doi

  82. [90]

    , TITLE =

    Wang, W. , TITLE =. I. 2023 , ISBN =

    2023

  83. [91]

    , TITLE =

    Watanabe, H. , TITLE =. J. Algebra , FJOURNAL =. 2021 , PAGES =. doi:10.1016/j.jalgebra.2021.02.023 , URL =

    2021 doi

  84. [92]

    , journal=

    Weekes, A. , journal=. Generators for

  85. [93]

    , TITLE =

    Wenzl, H. , TITLE =. J. Algebra , FJOURNAL =. 2012 , PAGES =. doi:10.1016/j.jalgebra.2012.02.017 , URL =

    2012 doi

  86. [94]

    preprint , note=

    3-dimensional mirror symmetry , author=. preprint , note=

  87. [95]

    , TITLE =

    Yamane, H. , TITLE =. Publ. Res. Inst. Math. Sci. , FJOURNAL =. 1994 , NUMBER =. doi:10.2977/prims/1195166275 , URL =

    1994

  88. [96]

    Represent

    Li, Yiqiang , TITLE =. Represent. Theory , FJOURNAL =. 2019 , PAGES =. doi:10.1090/ert/522 , URL =

    2019 doi

  89. [97]

    , TITLE =

    Yamane, H. , TITLE =. Publ. Res. Inst. Math. Sci. , FJOURNAL =. 1999 , NUMBER =. doi:10.2977/prims/1195143607 , URL =

    1999

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.