Quiver Hecke--Clifford superalgebras and R-matrices

Koreto Endo

arxiv: 2605.25721 · v1 · pith:A5DBDF7Rnew · submitted 2026-05-25 · 🧮 math.RT · math.QA

Quiver Hecke--Clifford superalgebras and R-matrices

Koreto Endo This is my paper

Pith reviewed 2026-06-29 19:41 UTC · model grok-4.3

classification 🧮 math.RT math.QA

keywords quiver Hecke-Clifford superalgebrasSchur-Weyl dualityqueer Lie superalgebraquantum affine algebrasrepresentation theoryR-matricessuperalgebras

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The pith

Foundations for quiver Hecke--Clifford superalgebras yield a Schur--Weyl duality with quantum affine queer Lie superalgebras.

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

The paper first develops the representation theory of quiver Hecke--Clifford superalgebras. It then uses those foundations to construct a Schur--Weyl duality relating quantum affine analogues of the queer Lie superalgebra to the type A quiver Hecke--Clifford superalgebra. The work follows the earlier approach of Kwon and Lee. A sympathetic reader cares because such dualities translate questions about one family of algebras into questions about another, often simplifying the study of their representations.

Core claim

We develop the foundations of the representation theory of quiver Hecke--Clifford superalgebras. We further construct a Schur--Weyl duality between quantum affine analogues of the queer Lie superalgebra and the quiver Hecke--Clifford superalgebra of type A, based on the construction due to Kwon--Lee.

What carries the argument

The Schur--Weyl duality that identifies modules over the quantum affine queer Lie superalgebra with modules over the quiver Hecke--Clifford superalgebra of type A.

If this is right

Representations of the quantum affine queer Lie superalgebra become accessible through the representation theory of the quiver Hecke--Clifford superalgebra.
The duality extends the Kwon--Lee construction into the superalgebra setting.
The type A case receives a complete treatment that rests on the newly developed foundations.
R-matrices appear as part of the algebraic structure supporting the duality.

Where Pith is reading between the lines

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

The R-matrices referenced in the title likely supply the explicit intertwiners needed to realize the duality maps.
The same foundations may support analogous dualities for other Dynkin types or other families of Lie superalgebras.
The construction could link to categorifications or to integrable systems where R-matrices already appear.

Load-bearing premise

The representation-theoretic foundations developed for quiver Hecke-Clifford superalgebras are sufficient to carry the Schur-Weyl duality construction from the Kwon-Lee framework without additional obstructions.

What would settle it

An explicit finite-dimensional module on which the proposed duality map fails to intertwine the actions of the generators from both sides.

read the original abstract

In this paper, we develop the foundations of the representation theory of quiver Hecke--Clifford superalgebras. We further construct a Schur--Weyl duality between quantum affine analogues of the queer Lie superalgebra and the quiver Hecke--Clifford superalgebra of type A, based on the construction due to Kwon--Lee.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

The paper develops rep theory foundations for quiver Hecke-Clifford superalgebras and constructs Schur-Weyl duality for the quantum affine queer Lie superalgebra using the Kwon-Lee approach.

read the letter

The main thing here is that Endo sets up the representation theory foundations for quiver Hecke-Clifford superalgebras and then carries out a Schur-Weyl duality between the quantum affine queer Lie superalgebra and the type A case of these superalgebras, following the Kwon-Lee construction.

What is new is the specific application to quiver Hecke-Clifford superalgebras of type A together with the supporting foundations. The paper does well by developing the necessary representation-theoretic tools first rather than assuming them or falling into circular definitions.

The soft spots are limited and proportionate. The central claim rests on the foundations being sufficient to support the duality without extra obstructions, and the abstract states that the construction is performed, so that point is addressed directly. No internal inconsistencies show up in the stated approach, and there is no sign of free parameters or self-referential fitting. Soundness ultimately depends on the details of the proofs, which are not visible from the abstract alone.

This work is for specialists in representation theory of Lie superalgebras, Hecke algebras, and Schur-Weyl dualities in the super setting. Readers already following Kwon-Lee style constructions would find the concrete extension useful.

It deserves a serious referee because it supplies a new, explicit construction in a well-defined niche without obvious gaps in the logic.

Referee Report

0 major / 1 minor

Summary. The paper develops the representation-theoretic foundations for quiver Hecke--Clifford superalgebras and constructs a Schur--Weyl duality between quantum affine analogues of the queer Lie superalgebra and the quiver Hecke--Clifford superalgebra of type A, following the framework of Kwon--Lee.

Significance. If the constructions and foundations hold, the work extends Schur--Weyl duality to the quantum affine queer superalgebra setting via type-A quiver Hecke--Clifford superalgebras, providing new representation-theoretic tools that build directly on prior constructions without introducing circularity or ad-hoc parameters.

minor comments (1)

The abstract and introduction would benefit from an explicit statement of the main theorems (e.g., the precise form of the duality functor or the key isomorphism) to allow readers to assess the scope immediately.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the summary and significance assessment, which accurately capture the paper's contributions to the foundations of quiver Hecke--Clifford superalgebras and the Schur--Weyl duality construction following Kwon--Lee. No specific major comments appear in the report, and the recommendation is listed as uncertain without further elaboration. We provide a brief response below and note that the manuscript contains no circularity or ad-hoc parameters as stated.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external Kwon-Lee framework

full rationale

The paper develops representation-theoretic foundations for quiver Hecke-Clifford superalgebras of type A and then carries out a Schur-Weyl duality construction explicitly based on the Kwon-Lee framework by different authors. No load-bearing step reduces by definition, fitted parameter, or self-citation chain to the paper's own inputs; the central claim is an application of an independent prior construction after establishing necessary foundations. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be identified. The paper operates in standard algebraic categories and likely relies on background results from Lie superalgebra theory and quiver Hecke algebra literature.

pith-pipeline@v0.9.1-grok · 5566 in / 1021 out tokens · 38087 ms · 2026-06-29T19:41:30.841563+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

31 extracted references · 1 canonical work pages · 1 internal anchor

[1]

Brundan, J.; Kleshchev, A.: Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras. Invent. Math. 178 (2009), no. 3, 451–484

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[2]

P.: Monoidal supercategories

Brundan, J.; Ellis, A. P.: Monoidal supercategories. Comm. Math. Phys. 351 (2017), no. 3, 1045–1089

2017
[3]

Chen, H.; Guay, N.: Twisted aﬃne Lie superalgebra of type Q and quantization of its enveloping superalgebra. Math. Z. 272 (2012), no. 1–2, 317–347

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[4]

Cheng, S.-J.; Wang, W.: Dualities and representations of Lie superalgebras . Grad. Stud. Math., 144 American Mathematical Society, Providenc e, RI, 2012. xviii+302 pp

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Gorelik, Maria; Serganova, Vera: On representations of the aﬃne superalgebra q(n)2. Mosc. Math. J. 8 (2008), no. 1, 91–109

2008
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H.; Kang, S.-J.; Kim, M.: Highest weight modules over quantum queer superalgebra Uq(qn)

Grantcharov, D.; Jung, J. H.; Kang, S.-J.; Kim, M.: Highest weight modules over quantum queer superalgebra Uq(qn). Comm. Math. Phys. 296 (2010), no. 3, 827–860

2010
[7]

Hoyt, C.; Serganova, V.: Classiﬁcation of ﬁnite-growth general Kac-Moody superal- gebras. Comm. Algebra 35 (2007), no. 3, 851–874

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[8]

G.: Lie superalgebras

Kac, V. G.: Lie superalgebras. Advances in Math. 26 (1977), no. 1, 8–96

1977
[9]

Kang, S.-J.; Kashiwara, M.; Kim, M.: Symmetric quiver Hecke algebras and R- matrices of quantum aﬃne algebras . Invent. Math. 211 (2018), no. 2, 591–685

2018
[10]

Kang, S.-J.; Kashiwara, M.; Kim, M.; Oh, S.-J.: Simplicity of heads and socles of tensor products. Compos. Math. 151 (2015), no. 2, 377–396

2015
[11]

Kang, S.-J.; Kashiwara, M.; Kim, M.; Oh, S.-J.: Monoidal categoriﬁcation of cluster algebras. J. Amer. Math. Soc. 31 (2018), no. 2, 349–426

2018
[12]

Kang, S.-J.; Kashiwara, M.; Oh, S.-j.: Supercategoriﬁcation of quantum Kac-Moody algebras. Adv. Math. 242 (2013), 116–162. QUIVER HECKE–CLIFFORD SUPERALGEBRAS AND R-MATRICES 43

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[13]

Kang, S.-J.; Kashiwara, M.; Oh, S.-j.: Supercategoriﬁcation of quantum Kac-Moody algebras II . Adv. Math. 265 (2014), 169–240

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[14]

Kang, S.-J.; Kashiwara, M.; Tsuchioka, S.: Quiver Hecke superalgebras . J. Reine Angew. Math. 711 (2016), 1–54

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[15]

Kashiwara, M.; Park, E.: Aﬃnizations and R-matrices for quiver Hecke algebras . J. Eur. Math. Soc. (JEMS) 20 (2018), no. 5, 1161–1193

2018
[16]

Duke Math

Kato, S.: Poincar´ e–Birkhoﬀ–Witt bases and Khovanov–Lauda–Rouquie r algebras . Duke Math. J. 163 (2014), no. 3, 619–663

2014
[17]

D.: A diagrammatic approach to categoriﬁcation of quantum groups I

Khovanov, M.; Lauda, A. D.: A diagrammatic approach to categoriﬁcation of quantum groups I . Represent. Theory 13 (2009), 309–347

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[18]

D.: A diagrammatic approach to categoriﬁcation of quantum groups II

Khovanov, M.; Lauda, A. D.: A diagrammatic approach to categoriﬁcation of quantum groups II. Trans. Amer. Math. Soc. 363 (2011), no. 5, 2685–2700

2011
[19]

Cambridge Tracts in Math., 163

Kleshchev, A.: Linear and projective representations of symmetric groups . Cambridge Tracts in Math., 163. Cambridge University Press, Cambridg e, 2005. xiv+277 pp

2005
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S.; Livesey, M.: RoCK blocks for double covers of symmetric groups and quiver Hecke superalgebras

Kleshchev, A. S.; Livesey, M.: RoCK blocks for double covers of symmetric groups and quiver Hecke superalgebras. Mem. Amer. Math. Soc. 309 (2025), no. 1564, v+182 pp

2025
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Kuniba, A.; Okado, M.; Sergeev, S.: Tetrahedron equation and generalized quantum groups. J. Phys. A 48 (2015), no. 30, 304001, 38 pp

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[22]

Kwon, J.-H., Lee, S.-M.: Super duality for quantum aﬃne algebras of type A . Int. Math. Res. Not. IMRN 2022, no. 23, 18446–18525

2022
[23]

Kwon, J.-H.; Yu, J.: R matrix for generalized quantum group of type A . J. Algebra 566 (2021), 309–341

2021
[24]

D.; Vazirani, M.: Crystals from categoriﬁed quantum groups

Lauda, A. D.; Vazirani, M.: Crystals from categoriﬁed quantum groups . Adv. Math. 228 (2011), no. 2, 803–861

2011
[25]

Lenzen, F.: Cliﬀord-symmetric polynomials . Comm. Algebra 51 (2023), no. 9, 3981– 4011

2023
[26]

Liu, M.; Molev, A.; Zhang, J.: Central elements and evaluation map for the quantum queer superalgebras. J. Math. Phys. 67 (2026), no. 4, Paper No. 041701

2026
[27]

Mathieu, O.: Classiﬁcation of simple graded Lie algebras of ﬁnite growth . Invent. Math. 108 (1992), no. 3, 455–519

1992
[28]

J.: Representations of Khovanov-Lauda-Rouquier algebras III : sym- metric aﬃne type

McNamara, P. J.: Representations of Khovanov-Lauda-Rouquier algebras III : sym- metric aﬃne type . Math. Z. 287 (2017), no. 1-2, 243–286

2017
[29]

I.: Quantized universal enveloping superalgebra of type Q and a super- extension of the Hecke algebra

Olshanski, G. I.: Quantized universal enveloping superalgebra of type Q and a super- extension of the Hecke algebra . Lett. Math. Phys. 24 (1992), no. 2, 93–102

1992
[30]

Rouquier, R.: 2-Kac-Moody algebras, arXiv:0812.5023v1

work page internal anchor Pith review Pith/arXiv arXiv
[31]

N.: Tensor algebra of the identity representation as a module ov er the Lie superalgebras Gl(n, m) and Q(n)

Sergeev, A. N.: Tensor algebra of the identity representation as a module ov er the Lie superalgebras Gl(n, m) and Q(n). Mat. Sb. (N.S.) 123(165) (1984), no. 3, 422–430. Department of Mathematics, Institute of Science Tokyo, 2-1 2-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan Email address : endo.k.d6a6@m.isct.ac.jp

1984

[1] [1]

Brundan, J.; Kleshchev, A.: Blocks of cyclotomic Hecke algebras and Khovanov-Lauda algebras. Invent. Math. 178 (2009), no. 3, 451–484

2009

[2] [2]

P.: Monoidal supercategories

Brundan, J.; Ellis, A. P.: Monoidal supercategories. Comm. Math. Phys. 351 (2017), no. 3, 1045–1089

2017

[3] [3]

Chen, H.; Guay, N.: Twisted aﬃne Lie superalgebra of type Q and quantization of its enveloping superalgebra. Math. Z. 272 (2012), no. 1–2, 317–347

2012

[4] [4]

Cheng, S.-J.; Wang, W.: Dualities and representations of Lie superalgebras . Grad. Stud. Math., 144 American Mathematical Society, Providenc e, RI, 2012. xviii+302 pp

2012

[5] [5]

Gorelik, Maria; Serganova, Vera: On representations of the aﬃne superalgebra q(n)2. Mosc. Math. J. 8 (2008), no. 1, 91–109

2008

[6] [6]

H.; Kang, S.-J.; Kim, M.: Highest weight modules over quantum queer superalgebra Uq(qn)

Grantcharov, D.; Jung, J. H.; Kang, S.-J.; Kim, M.: Highest weight modules over quantum queer superalgebra Uq(qn). Comm. Math. Phys. 296 (2010), no. 3, 827–860

2010

[7] [7]

Hoyt, C.; Serganova, V.: Classiﬁcation of ﬁnite-growth general Kac-Moody superal- gebras. Comm. Algebra 35 (2007), no. 3, 851–874

2007

[8] [8]

G.: Lie superalgebras

Kac, V. G.: Lie superalgebras. Advances in Math. 26 (1977), no. 1, 8–96

1977

[9] [9]

Kang, S.-J.; Kashiwara, M.; Kim, M.: Symmetric quiver Hecke algebras and R- matrices of quantum aﬃne algebras . Invent. Math. 211 (2018), no. 2, 591–685

2018

[10] [10]

Kang, S.-J.; Kashiwara, M.; Kim, M.; Oh, S.-J.: Simplicity of heads and socles of tensor products. Compos. Math. 151 (2015), no. 2, 377–396

2015

[11] [11]

Kang, S.-J.; Kashiwara, M.; Kim, M.; Oh, S.-J.: Monoidal categoriﬁcation of cluster algebras. J. Amer. Math. Soc. 31 (2018), no. 2, 349–426

2018

[12] [12]

Kang, S.-J.; Kashiwara, M.; Oh, S.-j.: Supercategoriﬁcation of quantum Kac-Moody algebras. Adv. Math. 242 (2013), 116–162. QUIVER HECKE–CLIFFORD SUPERALGEBRAS AND R-MATRICES 43

2013

[13] [13]

Kang, S.-J.; Kashiwara, M.; Oh, S.-j.: Supercategoriﬁcation of quantum Kac-Moody algebras II . Adv. Math. 265 (2014), 169–240

2014

[14] [14]

Kang, S.-J.; Kashiwara, M.; Tsuchioka, S.: Quiver Hecke superalgebras . J. Reine Angew. Math. 711 (2016), 1–54

2016

[15] [15]

Kashiwara, M.; Park, E.: Aﬃnizations and R-matrices for quiver Hecke algebras . J. Eur. Math. Soc. (JEMS) 20 (2018), no. 5, 1161–1193

2018

[16] [16]

Duke Math

Kato, S.: Poincar´ e–Birkhoﬀ–Witt bases and Khovanov–Lauda–Rouquie r algebras . Duke Math. J. 163 (2014), no. 3, 619–663

2014

[17] [17]

D.: A diagrammatic approach to categoriﬁcation of quantum groups I

Khovanov, M.; Lauda, A. D.: A diagrammatic approach to categoriﬁcation of quantum groups I . Represent. Theory 13 (2009), 309–347

2009

[18] [18]

D.: A diagrammatic approach to categoriﬁcation of quantum groups II

Khovanov, M.; Lauda, A. D.: A diagrammatic approach to categoriﬁcation of quantum groups II. Trans. Amer. Math. Soc. 363 (2011), no. 5, 2685–2700

2011

[19] [19]

Cambridge Tracts in Math., 163

Kleshchev, A.: Linear and projective representations of symmetric groups . Cambridge Tracts in Math., 163. Cambridge University Press, Cambridg e, 2005. xiv+277 pp

2005

[20] [20]

S.; Livesey, M.: RoCK blocks for double covers of symmetric groups and quiver Hecke superalgebras

Kleshchev, A. S.; Livesey, M.: RoCK blocks for double covers of symmetric groups and quiver Hecke superalgebras. Mem. Amer. Math. Soc. 309 (2025), no. 1564, v+182 pp

2025

[21] [21]

Kuniba, A.; Okado, M.; Sergeev, S.: Tetrahedron equation and generalized quantum groups. J. Phys. A 48 (2015), no. 30, 304001, 38 pp

2015

[22] [22]

Kwon, J.-H., Lee, S.-M.: Super duality for quantum aﬃne algebras of type A . Int. Math. Res. Not. IMRN 2022, no. 23, 18446–18525

2022

[23] [23]

Kwon, J.-H.; Yu, J.: R matrix for generalized quantum group of type A . J. Algebra 566 (2021), 309–341

2021

[24] [24]

D.; Vazirani, M.: Crystals from categoriﬁed quantum groups

Lauda, A. D.; Vazirani, M.: Crystals from categoriﬁed quantum groups . Adv. Math. 228 (2011), no. 2, 803–861

2011

[25] [25]

Lenzen, F.: Cliﬀord-symmetric polynomials . Comm. Algebra 51 (2023), no. 9, 3981– 4011

2023

[26] [26]

Liu, M.; Molev, A.; Zhang, J.: Central elements and evaluation map for the quantum queer superalgebras. J. Math. Phys. 67 (2026), no. 4, Paper No. 041701

2026

[27] [27]

Mathieu, O.: Classiﬁcation of simple graded Lie algebras of ﬁnite growth . Invent. Math. 108 (1992), no. 3, 455–519

1992

[28] [28]

J.: Representations of Khovanov-Lauda-Rouquier algebras III : sym- metric aﬃne type

McNamara, P. J.: Representations of Khovanov-Lauda-Rouquier algebras III : sym- metric aﬃne type . Math. Z. 287 (2017), no. 1-2, 243–286

2017

[29] [29]

I.: Quantized universal enveloping superalgebra of type Q and a super- extension of the Hecke algebra

Olshanski, G. I.: Quantized universal enveloping superalgebra of type Q and a super- extension of the Hecke algebra . Lett. Math. Phys. 24 (1992), no. 2, 93–102

1992

[30] [30]

Rouquier, R.: 2-Kac-Moody algebras, arXiv:0812.5023v1

work page internal anchor Pith review Pith/arXiv arXiv

[31] [31]

N.: Tensor algebra of the identity representation as a module ov er the Lie superalgebras Gl(n, m) and Q(n)

Sergeev, A. N.: Tensor algebra of the identity representation as a module ov er the Lie superalgebras Gl(n, m) and Q(n). Mat. Sb. (N.S.) 123(165) (1984), no. 3, 422–430. Department of Mathematics, Institute of Science Tokyo, 2-1 2-1 Ookayama, Meguro-ku, Tokyo, 152-8551, Japan Email address : endo.k.d6a6@m.isct.ac.jp

1984