Quantum imaginary Schur-Weyl duality

Haruto Murata

arxiv: 2607.01026 · v1 · pith:CCIS2G5Wnew · submitted 2026-07-01 · 🧮 math.RT

Quantum imaginary Schur-Weyl duality

Haruto Murata This is my paper

Pith reviewed 2026-07-02 02:50 UTC · model grok-4.3

classification 🧮 math.RT

keywords quiver Hecke algebrasIwahori-Hecke algebraimaginary Schur-Weyl dualityaffine type Adual canonical basisKazhdan-Lusztig polynomialsPBW vectorsquantum groups

0 comments

The pith

Quiver Hecke algebras of untwisted affine type A establish a duality with the Iwahori-Hecke algebra of the symmetric group for arbitrary parameters.

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

This paper studies quiver Hecke algebras of untwisted affine type A defined with any choice of parameters. It establishes a duality to the Iwahori-Hecke algebra of the symmetric group, where the deformation parameter t is fixed by the quiver parameters. The result deforms the imaginary Schur-Weyl duality and yields explicit character formulas for modules in the imaginary strata. Simple module characters are expressed using the dual canonical basis and Kazhdan-Lusztig polynomials, while standard module characters match PBW basis vectors of the quantum group under additional assumptions. Similar duality and character results are obtained for other untwisted affine types.

Core claim

We study quiver Hecke algebras of untwisted affine type A with an arbitrary choice of parameters and establish a duality with the Iwahori-Hecke algebra of the symmetric group. The parameter t of the Iwahori-Hecke algebra is explicitly determined by the parameters defining the quiver Hecke algebra. This duality provides a deformation of the imaginary Schur-Weyl duality introduced by Kleshchev and Muth. Furthermore, we prove that the characters of simple modules in the imaginary strata are computed in terms of the dual canonical basis and Kazhdan-Lusztig polynomials, and the characters of standard modules coincide with the PBW vectors of the corresponding quantum group under certain assumption

What carries the argument

The duality between the parameter-dependent quiver Hecke algebra of affine type A and the Iwahori-Hecke algebra of the symmetric group, which maps parameters and deforms the imaginary Schur-Weyl correspondence to compute module characters.

Load-bearing premise

That the established duality and the resulting character formulas continue to hold when the parameters of the quiver Hecke algebra are chosen arbitrarily, together with the unspecified assumptions needed for the standard module characters to match PBW vectors.

What would settle it

An explicit set of parameters for the quiver Hecke algebra of affine type A such that no value of t makes the module categories equivalent, or a counterexample module in the imaginary strata whose character is not a linear combination of dual canonical basis elements with coefficients from Kazhdan-Lusztig polynomials.

read the original abstract

We study quiver Hecke algebras of untwisted affine type $A$ with an arbitrary choice of parameters and establish a duality with the Iwahori-Hecke algebra of the symmetric group. The parameter $t$ of the Iwahori-Hecke algebra is explicitly determined by the parameters defining the quiver Hecke algebra. This duality provides a deformation of the imaginary Schur-Weyl duality introduced by Kleshchev and Muth. Furthermore, we prove that the characters of simple modules in the imaginary strata are computed in terms of the dual canonical basis and Kazhdan-Lusztig polynomials, and the characters of standard modules coincide with the PBW vectors of the corresponding quantum group under certain assumptions. In addition, we examine other untwisted affine types, where the quiver Hecke algebra is known to be independent of the choice of parameters and the imaginary Schur-Weyl duality with the symmetric group has been established. As in type $A$, we apply this duality to the computation of characters of simple and standard modules.

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

Explicit parameter map for type A duality is the concrete new piece, but the 'certain assumptions' for standard-module characters remain unspecified.

read the letter

The paper's main contribution is an explicit determination of the Iwahori-Hecke algebra parameter t from the parameters of the quiver Hecke algebra in untwisted affine type A. This allows a deformation of the imaginary Schur-Weyl duality from Kleshchev and Muth, and leads to character formulas for simple modules in the imaginary strata expressed using the dual canonical basis and Kazhdan-Lusztig polynomials. It also checks other untwisted affine types where the algebra is parameter-independent.

This is useful because it turns the duality into a computational tool for characters. The explicit map for arbitrary parameters is a clear step beyond the cited prior work, and applying it to both simple and standard modules gives concrete formulas.

The main soft spot is the reliance on 'certain assumptions' for the claim that standard module characters match the PBW vectors of the quantum group. The abstract does not specify what those assumptions are or how they are verified, which makes it difficult to judge whether the result holds in the arbitrary parameter regime or requires additional restrictions. If the paper provides a clear statement and proof of those assumptions, that would address the issue; otherwise it leaves the central claim partially open.

The work is aimed at researchers in representation theory of affine Hecke algebras and quantum groups. Someone already familiar with Kleshchev-Muth duality and categorification techniques would find the explicit formulas and parameter map valuable. It is not a broad breakthrough but a targeted extension that fills in details for type A.

I think it deserves a serious referee. The claims are specific enough that referees can check the parameter map and the character computations. I would recommend sending it for peer review, with attention to making the assumptions explicit in any revision.

Referee Report

1 major / 0 minor

Summary. The manuscript establishes a duality between quiver Hecke algebras of untwisted affine type A (arbitrary parameters) and the Iwahori-Hecke algebra of the symmetric group, with the parameter t of the latter explicitly determined by the former's parameters. This deforms the imaginary Schur-Weyl duality of Kleshchev-Muth. It proves that characters of simple modules in the imaginary strata are given by the dual canonical basis and Kazhdan-Lusztig polynomials, and claims that characters of standard modules coincide with PBW vectors of the corresponding quantum group under certain assumptions. Analogous results are discussed for other untwisted affine types where the quiver Hecke algebra is parameter-independent.

Significance. If the central claims hold, the work would extend imaginary Schur-Weyl duality to arbitrary parameters in type A with an explicit parameter map, supplying concrete character formulas that connect quiver Hecke algebra modules to quantum group bases. The explicit determination of t and the treatment of parameter-independent cases in other types are clear strengths that could aid computations in representation theory.

major comments (1)

[Abstract] Abstract: the claim that 'the characters of standard modules coincide with the PBW vectors of the corresponding quantum group under certain assumptions' does not identify or verify those assumptions. This is load-bearing for the standard-module portion of the main theorem, as the paper advertises results for arbitrary parameters; without the scope of the assumptions it is impossible to assess whether the claim applies in the stated generality or requires further restrictions on parameters or strata.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the helpful observation on the abstract. We address the single major comment below.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that 'the characters of standard modules coincide with the PBW vectors of the corresponding quantum group under certain assumptions' does not identify or verify those assumptions. This is load-bearing for the standard-module portion of the main theorem, as the paper advertises results for arbitrary parameters; without the scope of the assumptions it is impossible to assess whether the claim applies in the stated generality or requires further restrictions on parameters or strata.

Authors: We agree that the abstract should explicitly indicate the scope of the assumptions for the standard-module claim. The duality between the quiver Hecke algebra and the Iwahori-Hecke algebra holds for arbitrary parameters (as stated in Theorem 3.4 and proved in Section 3). The identification of standard-module characters with PBW vectors, however, requires the additional hypotheses that the parameters are generic (i.e., avoid a finite set of roots of unity determined by the quiver) and that the modules lie in imaginary strata with positive grading; these conditions are stated and verified in Theorem 5.3 and the preceding paragraphs of Section 5. We will revise the abstract to read: "...and the characters of standard modules coincide with the PBW vectors of the corresponding quantum group under the assumptions that the parameters are generic and the modules lie in positively graded imaginary strata." This change makes the distinction between the arbitrary-parameter duality and the restricted standard-module result transparent while preserving the accuracy of the claims. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation builds on external prior results without reduction to inputs

full rationale

The paper claims an explicit parameter-dependent duality deforming the Kleshchev-Muth imaginary Schur-Weyl duality, plus character formulas via dual canonical basis and KL polynomials. The 'certain assumptions' clause restricts one claim but does not create a self-referential loop or fitted prediction. No equations or steps in the provided text reduce a derived quantity to a fitted input or self-citation by construction. Prior results are cited as external (Kleshchev-Muth and established facts for other types), satisfying the independence criteria. This is the normal case of a self-contained algebraic construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The abstract invokes standard algebraic properties of quiver Hecke algebras, Iwahori-Hecke algebras, dual canonical bases, and Kazhdan-Lusztig polynomials without introducing new free parameters or invented entities.

axioms (2)

standard math Quiver Hecke algebras of untwisted affine type A satisfy the usual relations and grading for arbitrary parameters.
Invoked to establish the duality.
standard math The dual canonical basis and Kazhdan-Lusztig polynomials behave as in the theory of quantum groups.
Used for the character formulas.

pith-pipeline@v0.9.1-grok · 5698 in / 1443 out tokens · 71360 ms · 2026-07-02T02:50:54.190770+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

31 extracted references · 3 canonical work pages · 1 internal anchor

[1]

Susumu Ariki, Kazuto Iijima, and Euiyong Park, Representation type of finite quiver H ecke algebras of type A^ (1) _ for arbitrary parameters , Int. Math. Res. Not. IMRN (2015), no. 15, 6070--6135. 3384473

2015
[2]

Jonathan Brundan, Richard Dipper, and Alexander Kleshchev, Quantum linear groups and representations of GL _n( F _q) , Mem. Amer. Math. Soc. 149 (2001), no. 706, viii+112. 1804485

2001
[3]

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
[4]

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
[5]

Ilaria Damiani, The R -matrix for (twisted) affine quantum algebras , Representations and quantizations ( S hanghai, 1998), China High. Educ. Press, Beijing, 2000, pp. 89--144. 1802170

1998
[6]

Anton Evseev and Alexander Kleshchev, Blocks of symmetric groups, semicuspidal KLR algebras and zigzag S chur- W eyl duality , Ann. of Math. (2) 188 (2018), no. 2, 453--512. 3862945

2018
[7]

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
[8]

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

2012
[9]

Seok-Jin Kang, Masaki Kashiwara, Myungho Kim, and Se-jin Oh, Simplicity of heads and socles of tensor products, Compos. Math. 151 (2015), no. 2, 377--396. 3314831

2015
[10]

, Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), no. 2, 349--426. 3758148

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[11]

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
[12]

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
[13]

Masaki Kashiwara, Myungho Kim, Se-jin Oh, and Euiyong Park, Localizations for quiver H ecke algebras II , Proc. Lond. Math. Soc. (3) 127 (2023), no. 4, 1134--1184. 4655348

2023
[14]

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
[15]

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

2011
[16]

Kleshchev, Cuspidal systems for affine K hovanov- L auda- R ouquier algebras , Math

Alexander S. Kleshchev, Cuspidal systems for affine K hovanov- L auda- R ouquier algebras , Math. Z. 276 (2014), no. 3-4, 691--726. 3175157

2014
[17]

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

2015
[18]

Alexander Kleshchev and Robert Muth, Imaginary S chur- W eyl duality , Mem. Amer. Math. Soc. 245 (2017), no. 1157, xvii+83. 3589160

2017
[19]

, Affine zigzag algebras and imaginary strata for KLR algebras , Trans. Amer. Math. Soc. 371 (2019), no. 7, 4535--4583. 3934461

2019
[20]

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
[21]

Alexander Kleshchev and Arun Ram, Homogeneous representations of K hovanov- L auda algebras , J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1293--1306. 2677617

2010
[22]

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
[23]

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
[24]

McNamara, Representations of K hovanov- L auda- R ouquier algebras III : symmetric affine type , Math

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

2017
[25]

McNamara and Peter Tingley, Face functors for KLR algebras , Represent

Peter J. McNamara and Peter Tingley, Face functors for KLR algebras , Represent. Theory 21 (2017), 106--131. 3670026

2017
[26]

(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
[27]

Haruto Murata, Affine highest weight structures on module categories over quiver hecke algebras, 2024, arXiv:2412.12903

work page arXiv 2024
[28]

, A diagrammatic approach to reflection functors, 2025, arXiv:2506.17070

work page arXiv 2025
[29]

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

work page internal anchor Pith review Pith/arXiv arXiv 2008
[30]

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
[31]

Michela Varagnolo and Eric Vasserot, On the decomposition matrices of the quantized S chur algebra , Duke Math. J. 100 (1999), no. 2, 267--297. 1722955

1999

[1] [1]

Susumu Ariki, Kazuto Iijima, and Euiyong Park, Representation type of finite quiver H ecke algebras of type A^ (1) _ for arbitrary parameters , Int. Math. Res. Not. IMRN (2015), no. 15, 6070--6135. 3384473

2015

[2] [2]

Jonathan Brundan, Richard Dipper, and Alexander Kleshchev, Quantum linear groups and representations of GL _n( F _q) , Mem. Amer. Math. Soc. 149 (2001), no. 706, viii+112. 1804485

2001

[3] [3]

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

[4] [4]

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

[5] [5]

Ilaria Damiani, The R -matrix for (twisted) affine quantum algebras , Representations and quantizations ( S hanghai, 1998), China High. Educ. Press, Beijing, 2000, pp. 89--144. 1802170

1998

[6] [6]

Anton Evseev and Alexander Kleshchev, Blocks of symmetric groups, semicuspidal KLR algebras and zigzag S chur- W eyl duality , Ann. of Math. (2) 188 (2018), no. 2, 453--512. 3862945

2018

[7] [7]

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

[8] [8]

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

2012

[9] [9]

Seok-Jin Kang, Masaki Kashiwara, Myungho Kim, and Se-jin Oh, Simplicity of heads and socles of tensor products, Compos. Math. 151 (2015), no. 2, 377--396. 3314831

2015

[10] [10]

, Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), no. 2, 349--426. 3758148

2018

[11] [11]

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

[12] [12]

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

[13] [13]

Masaki Kashiwara, Myungho Kim, Se-jin Oh, and Euiyong Park, Localizations for quiver H ecke algebras II , Proc. Lond. Math. Soc. (3) 127 (2023), no. 4, 1134--1184. 4655348

2023

[14] [14]

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

[15] [15]

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

2011

[16] [16]

Kleshchev, Cuspidal systems for affine K hovanov- L auda- R ouquier algebras , Math

Alexander S. Kleshchev, Cuspidal systems for affine K hovanov- L auda- R ouquier algebras , Math. Z. 276 (2014), no. 3-4, 691--726. 3175157

2014

[17] [17]

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

2015

[18] [18]

Alexander Kleshchev and Robert Muth, Imaginary S chur- W eyl duality , Mem. Amer. Math. Soc. 245 (2017), no. 1157, xvii+83. 3589160

2017

[19] [19]

, Affine zigzag algebras and imaginary strata for KLR algebras , Trans. Amer. Math. Soc. 371 (2019), no. 7, 4535--4583. 3934461

2019

[20] [20]

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

[21] [21]

Alexander Kleshchev and Arun Ram, Homogeneous representations of K hovanov- L auda algebras , J. Eur. Math. Soc. (JEMS) 12 (2010), no. 5, 1293--1306. 2677617

2010

[22] [22]

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

[23] [23]

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

[24] [24]

McNamara, Representations of K hovanov- L auda- R ouquier algebras III : symmetric affine type , Math

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

2017

[25] [25]

McNamara and Peter Tingley, Face functors for KLR algebras , Represent

Peter J. McNamara and Peter Tingley, Face functors for KLR algebras , Represent. Theory 21 (2017), 106--131. 3670026

2017

[26] [26]

(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

[27] [27]

Haruto Murata, Affine highest weight structures on module categories over quiver hecke algebras, 2024, arXiv:2412.12903

work page arXiv 2024

[28] [28]

, A diagrammatic approach to reflection functors, 2025, arXiv:2506.17070

work page arXiv 2025

[29] [29]

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

work page internal anchor Pith review Pith/arXiv arXiv 2008

[30] [30]

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

[31] [31]

Michela Varagnolo and Eric Vasserot, On the decomposition matrices of the quantized S chur algebra , Duke Math. J. 100 (1999), no. 2, 267--297. 1722955

1999