Classification of prime modules of quantum affine algebras corresponding to 2-column tableaux

Jian-Rong Li; Nick Early

arxiv: 2406.16879 · v3 · submitted 2024-04-14 · 🧮 math.QA

Classification of prime modules of quantum affine algebras corresponding to 2-column tableaux

Nick Early , Jian-Rong Li This is my paper

Pith reviewed 2026-05-24 02:11 UTC · model grok-4.3

classification 🧮 math.QA

keywords quantum affine algebrasprime modulessemistandard Young tableauxtype Afinite dimensional modules2-column tableauxrepresentation theoryrectangular shapes

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The pith

Prime modules corresponding to 2-column semistandard Young tableaux are classified up to one conjectural property.

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

Finite dimensional simple modules of quantum affine algebras of type A correspond to semistandard Young tableaux of rectangular shapes. The paper determines exactly which of the 2-column tableaux give prime modules. The result is complete once a single conjectural property holds. A further conjecture supplies a sufficient condition for primeness when more than two columns appear. This identifies the indecomposable building blocks in the tensor category of modules.

Core claim

The paper classifies all prime modules corresponding to 2-column semistandard Young tableaux for quantum affine algebras of type A, up to a conjectural property, and supplies a conjectural sufficient condition for a module corresponding to a tableau with more than two columns to be prime.

What carries the argument

The correspondence between finite dimensional simple modules and semistandard Young tableaux of rectangular shapes, used to identify which tableaux produce prime modules.

If this is right

Every 2-column tableau is decided to be prime or not once the conjecture is verified.
A sufficient condition is proposed that may detect primeness for tableaux having three or more columns.
The prime spectrum of the representation category is now known for all 2-column rectangular shapes.
Tensor factorization of modules can be read off directly from the 2-column tableaux.

Where Pith is reading between the lines

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

The same conjectural property may extend the classification to all rectangular shapes if it behaves uniformly across column numbers.
The classification could be used to compute explicit bases or characters for the prime modules in the 2-column case.
Neighboring questions about the Grothendieck ring or the cluster algebra structure attached to these modules become accessible once the primes are listed.

Load-bearing premise

A single conjectural property about the modules must hold in order to finish the classification for the 2-column case.

What would settle it

An explicit 2-column tableau whose corresponding module is prime (or non-prime) in a way that contradicts the conjectural property used in the classification.

read the original abstract

Finite dimensional simple modules of quantum affine algebras of type A correspond to semistandard Young tableaux of rectangular shapes. In this paper, we classify all prime modules corresponding to 2-column semistandard Young tableaux, up to a conjectural property. Moreover, we give a conjectural sufficient condition for a module corresponding to a tableau with more than two columns to be prime.

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 2-column classification is new but explicitly conditional on an unproven property, with a conjectural condition offered for larger cases.

read the letter

The key point is that the authors classify prime modules corresponding to 2-column semistandard Young tableaux, but only up to a conjectural property, and they add a conjectural sufficient condition for primeness when there are more than two columns. This is the main deliverable from the abstract. The work is new in the sense that it claims a classification for the 2-column case that had not been stated before, building directly on the established bijection between finite-dimensional simple modules of quantum affine algebras of type A and semistandard Young tableaux of rectangular shapes. That correspondence is used to organize the prime ones in the 2-column setting, which is a concrete step in a narrow corner of the representation theory literature. The paper is transparent about the conjecture and does not overclaim an unconditional result. The soft spot is exactly the one flagged in the stress test: the classification is not complete without the conjectural property, and the abstract gives no derivations, small-case verifications, or reductions to known results that would support the property independently. If the full text does not supply those checks, the completeness claim stays tentative rather than settled. This is a limitation but not a fatal one for a specialized paper; it simply means the result is best viewed as conditional progress. The paper is for people already working on quantum affine algebra modules and their combinatorial descriptions via tableaux. A reader in that subfield can extract the 2-column list and the suggested condition for larger tableaux as useful data points, even if they treat the conjecture as open. Outside that group the impact is limited. I would send it to peer review. The attempt to classify is substantive enough to deserve referee time, particularly to assess whether the conjecture can be addressed or whether the 2-column results stand usefully on their own.

Referee Report

2 major / 0 minor

Summary. The paper classifies all prime modules corresponding to 2-column semistandard Young tableaux for quantum affine algebras of type A, up to a conjectural property. It also provides a conjectural sufficient condition for a module corresponding to a tableau with more than two columns to be prime.

Significance. If the conjectural property holds, the result would complete the classification of prime modules in the 2-column case, extending the known correspondence between finite-dimensional simple modules and semistandard Young tableaux. The conjectural condition for multi-column cases could serve as a starting point for further investigations.

major comments (2)

[Abstract] Abstract: The classification of all prime modules for 2-column tableaux is explicitly conditional on an unproven conjectural property, which is used to assert completeness of the list; no proof, reduction to known results, or explicit verification for small tableaux is supplied to support the property.
[§1, §4] §1 and §4: The conjectural property is invoked to finish the classification without independent evidence or partial results confirming it holds in the relevant cases, making the central claim of a full classification load-bearing on this assumption.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript. We respond point by point to the major comments below, emphasizing that the results are presented explicitly as conditional on the stated conjecture.

read point-by-point responses

Referee: [Abstract] Abstract: The classification of all prime modules for 2-column tableaux is explicitly conditional on an unproven conjectural property, which is used to assert completeness of the list; no proof, reduction to known results, or explicit verification for small tableaux is supplied to support the property.

Authors: The abstract states that the classification holds up to the conjectural property, and the paper does not assert an unconditional completeness. The main contribution is the explicit list of candidate prime modules together with the formulation of the conjecture needed for completeness. No proof of the conjecture is supplied because establishing it lies beyond the scope of the present work; the manuscript instead focuses on deriving the classification assuming the property. We note that the conjecture can in principle be verified directly for small tableaux via existing algorithms for quantum affine algebra modules, but the current version does not include such checks. revision: no
Referee: [§1, §4] §1 and §4: The conjectural property is invoked to finish the classification without independent evidence or partial results confirming it holds in the relevant cases, making the central claim of a full classification load-bearing on this assumption.

Authors: Sections 1 and 4 present the classification of prime modules for 2-column tableaux as conditional on the conjectural property, which is stated clearly at the outset. The central claim of the paper is therefore not an unconditional classification but a classification under the explicitly formulated assumption. This structure is standard when a key step remains conjectural; the manuscript provides the reduction to the conjecture rather than independent evidence for it. revision: no

Circularity Check

0 steps flagged

No circularity detected; classification is explicitly conditional on an external conjecture

full rationale

The abstract states the classification of prime modules for 2-column tableaux holds 'up to a conjectural property' and offers a separate 'conjectural sufficient condition' for >2 columns. This is an honest admission of an unproven assumption rather than a derivation that reduces to its own inputs by construction. No self-citation, self-definitional loop, fitted-input-as-prediction, or ansatz-smuggling is quoted or exhibited in the provided text. The central claim does not assert completeness without the conjecture, so the derivation chain does not collapse to the inputs. This matches the default expectation of no significant circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

pith-pipeline@v0.9.0 · 5575 in / 933 out tokens · 30951 ms · 2026-05-24T02:11:54.245766+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We classify all prime modules corresponding to 2-column semistandard Young tableaux, up to a conjectural property... L(M) is prime if and only if T1, T2 are not weakly separated (Theorem 3.9)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Conjecture 3.8... Conjecture 5.1: if for every i≠j, Si,Sj are not weakly separated, then T is prime

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages

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K. Baur, D. Bogdanic, A. Garcia Elsener, J.-R. Li, Rigid indecomposable modules in Grassman- nian cluster categories , arXiv:2011.09227

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Chang, B

W. Chang, B. Duan, C. Fraser, J.-R. Li. Quantum aﬃne algebras a nd Grassmannians, Math. Z. 296 (2020), 1539–1583

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Cheung, P.-P

M.-W. Cheung, P.-P. Dechant, Y.-H. He, E. Heyes, E. Hirst, J.-R. Li, Clustering Cluster Algebras with Clusters , Advances in Theoretical and Mathematical Physics, 27 (3), 797–828

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Chari, A

V. Chari, A. Moura, and C. Young, Prime representations from a homological perspective , Math. Z. 274, 613–645 (2013)

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V. Chari and A. Pressley, Quantum aﬃne algebras , Comm. Math. Phys. 142 (1991), 261–283. Classiﬁcation of prime modules corresponding to 2-column t ableaux 15

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Chari and A

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Chari and A

V. Chari and A. Pressley, Minimal aﬃnizations of representations of quantum groups: the simply- laced case, Journal of Algebra 184 (1996), 1–30

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Chari, A

V. Chari, A. Pressley, Factorization of representations of quantum aﬃne algebras , Modular in- terfaces, (Riverside CA 1995), AMS/IP Stud. Adv. Math., 4 (1997), 33–40

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N. Early, Planarity in generalized scattering amplitudes: PK polyto pe, generalized root systems and worldsheet associahedra, arXiv preprint arXiv:2106.07142 (2021)

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Early and J.-R

N. Early and J.-R. Li, Tropical geometry, quantum aﬃne algebras, and scattering a mplitudes, Journal of Physics A: Mathematical and Theoretical, 57 (49), 495201

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Frenkel and E

E. Frenkel and E. Mukhin, Combinatorics of q-characters of ﬁnite-dimensional representations of quantum aﬃne algebras , Commun. Math. Phys. 216, 23–57 (2001)

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Hernandez and B

D. Hernandez and B. Leclerc, Cluster algebras and quantum aﬃne algebras , Duke Math. J., 154(2) (2010), 265–341

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M. Jimbo, A q-diﬀerence analogue of U (g) and the Yang-Baxter equation , Lett. Math. Phys., 10 (1985), no. 1, 63–69

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Kazhdan, and G

D. Kazhdan, and G. Lusztig, Representations of Coxeter Groups and Hecke Algebras , Inventiones Mathematicae, 53, 165–184 (1979)

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Kashiwara, M

M. Kashiwara, M. Kim, S-J. Oh and E. Park, Categories over quantum aﬃne algebras and monoidal categoriﬁcation, Proc. Japan Acad. Ser. A Math. Sci. 97 (2021), no. 7, 39–44

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Kashiwara, M

M. Kashiwara, M. Kim, S-J. Oh and E. Park, Monoidal categoriﬁcation and quantum aﬃne algebras II , Invent. Math. 236 (2024), no. 2, 837–924

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Leclerc, Imaginary vectors in the dual canonical basis of Uq(n), Transform Groups 8 (2003), no

B. Leclerc, Imaginary vectors in the dual canonical basis of Uq(n), Transform Groups 8 (2003), no. 1, 95–104

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Leclerc and A

B. Leclerc and A. Zelevinsky, Quasicommuting families of quantum Pl¨ ucker coordinates, In Kir- illov’s seminar on representation theory, volume 181 of Amer. Math. Soc. Transl. Ser. 2, pages 85–108. Amer. Math. Soc., Providence, RI, 1998

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Le and E

I. Le and E. Yıldırım, Cluster categoriﬁcation of rank 2 webs , arXiv:2402.14501

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Qin, Triangular bases in quantum cluster algebras and monoidal c ategoriﬁcation conjectures, Duke Math

F. Qin, Triangular bases in quantum cluster algebras and monoidal c ategoriﬁcation conjectures, Duke Math. J. 166 (12) 2337–2442, 2017

work page 2017
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Moura, C

A. Moura, C. Silva, On the primality of totally ordered q-factorization graphs, Canadian Journal of Mathematics, 76 (2), 594–637

work page
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S. Oh, A. Postnikov, and D. E. Speyer, Weak separation and plabic graphs , Proceedings of the London Mathematical Society, 110 (3), 721–754. 16 NICK EARLY AND JIAN-RONG LI

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[31]

L. Ren, M. Spradlin, A. Volovich, Symbol alphabets from tensor diagrams , J. High Energ. Phys. 2021, 79 (2021)

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[32]

M. P. Sch¨ utzenberger, Quelques remarques sur une construction de Schensted , Math. Scand. 12 (1963), 117–128

work page 1963
[33]

M. P. Sch¨ utzenberger,Promotion des morphismes d’ensembles ordonn´ es, Discrete Math. 2 (1972), 73–94

work page 1972
[34]

M. P. Sch¨ utzenberger, La correspondance de Robinson , in Combinatoire et repr´ esentation du groupe sym´ etrique, Lecture Notes in Math., Vol. 579, Springer-Verlag, Berlin-New York, 1977, 59–113

work page 1977
[35]

Scott, Grassmannians and cluster algebras , Proceedings of the London Mathematical Society, 92(3) (2006), 345–380

J. Scott, Grassmannians and cluster algebras , Proceedings of the London Mathematical Society, 92(3) (2006), 345–380

work page 2006
[36]

Santos, C

F. Santos, C. Stump, and V. Welker, Noncrossing sets and a Grassmann associahedron , In Forum of Mathematics, Sigma (Vol. 5), Cambridge University Press, 2017

work page 2017
[37]

R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, Cambridge, 1999. Nick Early, Institute for Advanced Study, Princeton, NJ. Email address : earlnick@ias.edu Jian-Rong Li, F aculty of Mathematics, University of Vienna , Oskar-Morgenstern- Platz 1, 1090 Vienna, Austria. Email address ...

work page 1999

[1] [1]

K. Baur, D. Bogdanic, A. Garcia Elsener, J.-R. Li, Rigid indecomposable modules in Grassman- nian cluster categories , arXiv:2011.09227

work page arXiv 2011

[2] [2]

Brito, V

M. Brito, V. Chari, and A. Moura, Demazure modules of level two and prime representations of quantum aﬃne sln+ 1, Journal of the Institute of Mathematics of Jussieu, 17 (2018), no. 1, 75–105

work page 2018

[3] [3]

Brito and V

M. Brito and V. Chari, Higher order Kirillov-Reshetikhin modules, Imaginary mod ules and Monoidal Categoriﬁcation for Uq(A(1) n ), Journal f¨ ur die reine und angewandte Mathematik (Crelles Journal), 2023 (804), 221–262

work page 2023

[4] [4]

E. A. Bender and D. E. Knuth, Enumeration of plane partitions , J. Combinatorial Theory Ser. A 13 (1972), 40–54

work page 1972

[5] [5]

Chang, B

W. Chang, B. Duan, C. Fraser, J.-R. Li. Quantum aﬃne algebras a nd Grassmannians, Math. Z. 296 (2020), 1539–1583

work page 2020

[6] [6]

Cheung, P.-P

M.-W. Cheung, P.-P. Dechant, Y.-H. He, E. Heyes, E. Hirst, J.-R. Li, Clustering Cluster Algebras with Clusters , Advances in Theoretical and Mathematical Physics, 27 (3), 797–828

work page

[7] [7]

Chari, A

V. Chari, A. Moura, and C. Young, Prime representations from a homological perspective , Math. Z. 274, 613–645 (2013)

work page 2013

[8] [8]

Chari and A

V. Chari and A. Pressley, Quantum aﬃne algebras , Comm. Math. Phys. 142 (1991), 261–283. Classiﬁcation of prime modules corresponding to 2-column t ableaux 15

work page 1991

[9] [9]

Chari and A

V. Chari and A. Pressley, A guide to quantum groups , Cambridge University Press, Cambridge,

work page

[10] [10]

Chari and A

V. Chari and A. Pressley, Quantum aﬃne algebras and their representations , in Representations of groups (Banﬀ, AB, 1994), 16 of CMS Conf. Proc. Amer. Math. Soc., Providence, RI, 1995

work page 1994

[11] [11]

Chari and A

V. Chari and A. Pressley, Minimal aﬃnizations of representations of quantum groups: the simply- laced case, Journal of Algebra 184 (1996), 1–30

work page 1996

[12] [12]

Chari, A

V. Chari, A. Pressley, Factorization of representations of quantum aﬃne algebras , Modular in- terfaces, (Riverside CA 1995), AMS/IP Stud. Adv. Math., 4 (1997), 33–40

work page 1995

[13] [13]

Drinfeld, A new realization of Yangians and of quantum aﬃne algebras , Dokl

V. Drinfeld, A new realization of Yangians and of quantum aﬃne algebras , Dokl. Akad. Nauk SSSR, 296 (1987), no.1, 13–17

work page 1987

[14] [14]

Early, Planarity in generalized scattering amplitudes: PK polyto pe, generalized root systems and worldsheet associahedra, arXiv preprint arXiv:2106.07142 (2021)

N. Early, Planarity in generalized scattering amplitudes: PK polyto pe, generalized root systems and worldsheet associahedra, arXiv preprint arXiv:2106.07142 (2021)

work page arXiv 2021

[15] [15]

Early and J.-R

N. Early and J.-R. Li, Tropical geometry, quantum aﬃne algebras, and scattering a mplitudes, Journal of Physics A: Mathematical and Theoretical, 57 (49), 495201

work page

[16] [16]

Frenkel and E

E. Frenkel and E. Mukhin, Combinatorics of q-characters of ﬁnite-dimensional representations of quantum aﬃne algebras , Commun. Math. Phys. 216, 23–57 (2001)

work page 2001

[17] [17]

Frenkel and N

E. Frenkel and N. Reshetikhin, The q-characters of representations of quantum aﬃne algebras and deformations of W -algebras, Recent developments in quantum aﬃne algebras and related topics (Raleigh, NC, 1998), pp. 163–205. Contemp. Math., vol. 248. American Mathematical Society, Providence (1999)

work page 1998

[18] [18]

E. R. Gansner, On the equality of two plane partition correspondences , Discrete Math. 30 (1980), 121–132

work page 1980

[19] [19]

Hopkins, Cyclic sieving for plane partitions and symmetry , SIGMA Symmetry Integrability Geom

S. Hopkins, Cyclic sieving for plane partitions and symmetry , SIGMA Symmetry Integrability Geom. Methods Appl. 16 (2020), Paper No. 130, 40

work page 2020

[20] [20]

Hernandez and B

D. Hernandez and B. Leclerc, Cluster algebras and quantum aﬃne algebras , Duke Math. J., 154(2) (2010), 265–341

work page 2010

[21] [21]

Jimbo, A q-diﬀerence analogue of U (g) and the Yang-Baxter equation , Lett

M. Jimbo, A q-diﬀerence analogue of U (g) and the Yang-Baxter equation , Lett. Math. Phys., 10 (1985), no. 1, 63–69

work page 1985

[22] [22]

Kazhdan, and G

D. Kazhdan, and G. Lusztig, Representations of Coxeter Groups and Hecke Algebras , Inventiones Mathematicae, 53, 165–184 (1979)

work page 1979

[23] [23]

Kashiwara, M

M. Kashiwara, M. Kim, S-J. Oh and E. Park, Categories over quantum aﬃne algebras and monoidal categoriﬁcation, Proc. Japan Acad. Ser. A Math. Sci. 97 (2021), no. 7, 39–44

work page 2021

[24] [24]

Kashiwara, M

M. Kashiwara, M. Kim, S-J. Oh and E. Park, Monoidal categoriﬁcation and quantum aﬃne algebras II , Invent. Math. 236 (2024), no. 2, 837–924

work page 2024

[25] [25]

Leclerc, Imaginary vectors in the dual canonical basis of Uq(n), Transform Groups 8 (2003), no

B. Leclerc, Imaginary vectors in the dual canonical basis of Uq(n), Transform Groups 8 (2003), no. 1, 95–104

work page 2003

[26] [26]

Leclerc and A

B. Leclerc and A. Zelevinsky, Quasicommuting families of quantum Pl¨ ucker coordinates, In Kir- illov’s seminar on representation theory, volume 181 of Amer. Math. Soc. Transl. Ser. 2, pages 85–108. Amer. Math. Soc., Providence, RI, 1998

work page 1998

[27] [27]

Le and E

I. Le and E. Yıldırım, Cluster categoriﬁcation of rank 2 webs , arXiv:2402.14501

work page arXiv

[28] [28]

Qin, Triangular bases in quantum cluster algebras and monoidal c ategoriﬁcation conjectures, Duke Math

F. Qin, Triangular bases in quantum cluster algebras and monoidal c ategoriﬁcation conjectures, Duke Math. J. 166 (12) 2337–2442, 2017

work page 2017

[29] [29]

Moura, C

A. Moura, C. Silva, On the primality of totally ordered q-factorization graphs, Canadian Journal of Mathematics, 76 (2), 594–637

work page

[30] [30]

S. Oh, A. Postnikov, and D. E. Speyer, Weak separation and plabic graphs , Proceedings of the London Mathematical Society, 110 (3), 721–754. 16 NICK EARLY AND JIAN-RONG LI

work page

[31] [31]

L. Ren, M. Spradlin, A. Volovich, Symbol alphabets from tensor diagrams , J. High Energ. Phys. 2021, 79 (2021)

work page 2021

[32] [32]

M. P. Sch¨ utzenberger, Quelques remarques sur une construction de Schensted , Math. Scand. 12 (1963), 117–128

work page 1963

[33] [33]

M. P. Sch¨ utzenberger,Promotion des morphismes d’ensembles ordonn´ es, Discrete Math. 2 (1972), 73–94

work page 1972

[34] [34]

M. P. Sch¨ utzenberger, La correspondance de Robinson , in Combinatoire et repr´ esentation du groupe sym´ etrique, Lecture Notes in Math., Vol. 579, Springer-Verlag, Berlin-New York, 1977, 59–113

work page 1977

[35] [35]

Scott, Grassmannians and cluster algebras , Proceedings of the London Mathematical Society, 92(3) (2006), 345–380

J. Scott, Grassmannians and cluster algebras , Proceedings of the London Mathematical Society, 92(3) (2006), 345–380

work page 2006

[36] [36]

Santos, C

F. Santos, C. Stump, and V. Welker, Noncrossing sets and a Grassmann associahedron , In Forum of Mathematics, Sigma (Vol. 5), Cambridge University Press, 2017

work page 2017

[37] [37]

R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Studies in Advanced Mathematics 62, Cambridge University Press, Cambridge, 1999. Nick Early, Institute for Advanced Study, Princeton, NJ. Email address : earlnick@ias.edu Jian-Rong Li, F aculty of Mathematics, University of Vienna , Oskar-Morgenstern- Platz 1, 1090 Vienna, Austria. Email address ...

work page 1999