no. 5
Newton–Okounkov bodies for categories of modules over quiver Hecke algebras
[Corps de Newton–Okounkov pour des catégories de modules sur les algèbres de Hecke carquois]
Casbi, Elie1
1 Université de Paris, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS F-75013 Paris (FRANCE)
Annales de l'Institut Fourier, Tome 72 (2022) no. 5, pp. 1773-1818.

Nous montrons que pour toute algèbre de Lie 𝔤 de type fini, la théorie des représentations des algèbres de Hecke carquois fournit un cadre naturel pour la construction de corps de Newton–Okounkov associés aux anneaux de coordonnée quantiques 𝒜 q (𝔫(w)). Lorsque 𝔤 est simplement lacée, nous utilisons les catégorifications monoïdales de Kang–Kashiwara–Kim–Oh pour étudier ces corps en lien avec la théorie des algèbres amassées. En particulier, nous construisons des simplexes Δ 𝒮 pour chaque graine 𝒮 de 𝒜 q (𝔫(w)). Nous étudions diverses propriétés de ces simplexes, notamment leurs points rationnels, leurs éventails normaux, ainsi que leurs volumes. Comme application, nous établissons une égalité entre fractions rationnelles reliant la formule des équerres de Nakada avec les partitions de racines associées aux variables d’amas, ce qui suggère davantage de liens possibles entre la théorie des algèbres amassées et la combinatoire des éléments complètement commutatifs des groupes de Weyl.

We show that for a finite-type Lie algebra 𝔤, the representation theory of quiver Hecke algebras provides a natural framework for the construction of Newton–Okounkov bodies associated to the quantum coordinate rings 𝒜 q (𝔫(w)). When 𝔤 is simply-laced, we use Kang–Kashiwara–Kim–Oh’s monoidal categorification to investigate the cluster theory of these bodies. In particular, our construction yields a simplex Δ 𝒮 for every seed 𝒮 of 𝒜 q (𝔫(w)). We exhibit various properties of these simplices by characterizing their rational points, normal fans, and volumes. As an application, we prove an equality of rational functions relating Nakada’s hook formula with the root partitions associated to cluster variables, suggesting further connections between cluster theory and the combinatorics of fully-commutative elements of Weyl groups.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3492
Classification : 05E10, 16G10
Keywords: Newton–Okounkov bodies, quiver Hecke algebras, cluster algebras
Mot clés : Corps de Newton–Okounkov, algèbres de Hecke carquois, algèbres amassées

Casbi, Elie 1

1 Université de Paris, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS F-75013 Paris (FRANCE)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@article{AIF_2022__72_5_1773_0,
     author = {Casbi, Elie},
     title = {Newton{\textendash}Okounkov bodies for categories of modules over quiver {Hecke} algebras},
     journal = {Annales de l'Institut Fourier},
     pages = {1773--1818},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {72},
     number = {5},
     year = {2022},
     doi = {10.5802/aif.3492},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3492/}
}
TY  - JOUR
AU  - Casbi, Elie
TI  - Newton–Okounkov bodies for categories of modules over quiver Hecke algebras
JO  - Annales de l'Institut Fourier
PY  - 2022
SP  - 1773
EP  - 1818
VL  - 72
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3492/
DO  - 10.5802/aif.3492
LA  - en
ID  - AIF_2022__72_5_1773_0
ER  - 
%0 Journal Article
%A Casbi, Elie
%T Newton–Okounkov bodies for categories of modules over quiver Hecke algebras
%J Annales de l'Institut Fourier
%D 2022
%P 1773-1818
%V 72
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3492/
%R 10.5802/aif.3492
%G en
%F AIF_2022__72_5_1773_0
Casbi, Elie. Newton–Okounkov bodies for categories of modules over quiver Hecke algebras. Annales de l'Institut Fourier, Tome 72 (2022) no. 5, pp. 1773-1818. doi : 10.5802/aif.3492. https://aif.centre-mersenne.org/articles/10.5802/aif.3492/

[1] Baumann, Pierre; Kamnitzer, Joel; Knutson, Allen The Mirkovic–Vilonen basis and Duistermaat–Heckman measures (2020) (https://arxiv.org/abs/1905.08460, with an appendix by Anne Dranowski and Joel Kamnitzer and Calder Morton-Ferguson, to appear in Acta Mathematica)

[2] Bossinger, Lara Toric degenerations: a bridge between representation theory, tropical geometry, and cluster algebras (2018) (https://arxiv.org/abs/1806.02090v1, preliminary version to the submitted Phd Thesis from the author)

[3] Boucksom, Sébastien Corps d’Okounkov (d’après Okounkov, Lazarsfeld–Mustata, et Kaveh–Khovanskii), Séminaire Bourbaki. Volume 2012/2013. Exposés 1059–1073. Avec table par noms d’auteurs de 1948/49 à 2012/13 (Astérisque), Volume 361, Société Mathématique de France, 2014, pp. 1-41 (Exp. No.1059) | Zbl

[4] Boucksom, Sébastien; Chen, Huayi Okounkov bodies of filtered linear series, Compositio. Math., Volume 147 (2011) no. 4, pp. 1205-1229 | DOI | MR | Zbl

[5] Casbi, Elie Dominance order and monoidal categorification of cluster algebras, Pac. J. Math., Volume 305 (2020) no. 2, pp. 473-537 | DOI | MR | Zbl

[6] Cautis, Sabin; Williams, Harold Cluster theory of the coherent Satake category, J. Am. Math. Soc., Volume 32 (2019) no. 3, pp. 709-778 | DOI | MR | Zbl

[7] Derksen, Harm; Weyman, Jerzy; Zelevinsky, Andrei Quivers with potential and their representations II: Applications to cluster algebras, J. Am. Math. Soc., Volume 23 (2010) no. 3, pp. 749-790 | DOI | MR | Zbl

[8] Fang, Xin; Littelmann, Peter From standard monomial theory to semi-toric degenerations via Newton–Okounkov bodies, Trans. Mosc. Math. Soc., Volume 2017 (2017), pp. 275-297 published also in Tr. Mosk. Mat. O.-va, vol. 78, No. 2, 331-356 (2017) | DOI | MR | Zbl

[9] Fock, Vladimir V.; Goncharov, Alexander B. Cluster ensembles, quantization and the dilogarithm, Ann. Sci. Éc. Norm. Supér., Volume 42 (2009) no. 6, pp. 865-930 | DOI | Numdam | MR | Zbl

[10] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. I: Foundations, J. Am. Math. Soc., Volume 15 (2002) no. 2, pp. 497-529 | DOI | MR | Zbl

[11] Fomin, Sergey; Zelevinsky, Andrei Cluster algebras. IV: Coefficients, Compos. Math., Volume 143 (2007) no. 1, pp. 112-164 | DOI | MR | Zbl

[12] Fujita, Naoki; Oya, Hironori A comparison of Newton–Okounkov polytopes of Schubert varieties, J. Lond. Math. Soc., Volume 96 (2017) no. 1, pp. 201-227 | DOI | MR | Zbl

[13] Geiß, Christof; Leclerc, Bernard; Schröer, Jan Cluster structures on quantum coordinate rings, Sel. Math., New Ser., Volume 19 (2013) no. 2, pp. 337-397 | DOI | MR | Zbl

[14] Geiß, Christof; Leclerc, Bernard; Schröer, Jan Factorial cluster algebras, Doc. Math., Volume 18 (2013) no. 1, pp. 249-274 | MR | Zbl

[15] Gross, Mark; Hacking, Paul; Keel, Sean; Kontsevich, Maxim Canonical bases for cluster algebras, J. Am. Math. Soc., Volume 31 (2018) no. 2, pp. 497-608 | DOI | MR | Zbl

[16] Hernandez, David; Leclerc, Bernard Cluster algebras and quantum affine algebras, Duke Math. J., Volume 154 (2010) no. 2, pp. 265-341 | MR | Zbl

[17] Hernandez, David; Leclerc, Bernard Quantum Grothendieck rings and derived Hall algebras, J. Reine Angew. Math., Volume 701 (2015), pp. 77-126 | MR | Zbl

[18] Hernandez, David; Leclerc, Bernard A cluster algebra apporach to q-characters of Kirillov–Reshetikhin modules, J. Eur. Math. Soc., Volume 18 (2016) no. 5, pp. 1113-1159 | DOI | Zbl

[19] Hohlweg, Christophe; Pilaud, Vincent; Stella, Salvatore Polytopal realizations of finite type g-vector fans, Adv. Math., Volume 328 (2018), pp. 713-749 | DOI | MR | Zbl

[20] Kang, Seok-Jin; Kashiwara, Masaki; Kim, Myungho Symmetric quiver Hecke algebras and R-matrices of quantum affine algebras, Invent. Math., Volume 211 (2018) no. 2, pp. 591-685 | DOI | MR | Zbl

[21] Kang, Seok-Jin; Kashiwara, Masaki; Kim, Myungho; Oh, Se-Jin Simplicity of heads and socles of tensor products, Compos. Math., Volume 151 (2015) no. 2, pp. 377-396 | DOI | MR | Zbl

[22] Kang, Seok-Jin; Kashiwara, Masaki; Kim, Myungho; Oh, Se-Jin Monoidal categorification of cluster algebras, J. Am. Math. Soc., Volume 31 (2018) no. 2, pp. 349-426 | DOI | MR | Zbl

[23] Kashiwara, Masaki; Kim, Myungho Laurent phenomenon and simple modules of quiver Hecke algebras, Compos. Math., Volume 155 (2019) no. 12, pp. 2263-2295 | DOI | MR | Zbl

[24] Kashiwara, Masaki; Kim, Myungho; Oh, Se-Jin; Park, Euiyong Monoidal categorification and quantum affine algebras II (2021) (https://arxiv.org/abs/2103.10067v1)

[25] Kaveh, Kiumars Crystal bases and Newton–Okounkov bodies, Duke Math. J., Volume 164 (2015) no. 13, pp. 2461-2506 | MR | Zbl

[26] Kaveh, Kumars; Khovanskiĭ, Askold G. Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. Math., Volume 176 (2012) no. 2, pp. 925-978 | DOI | MR | Zbl

[27] Khovanov, Mikhail; Lauda, Aaron D. A diagrammatic approach to categorification of quantum groups I, Represent. Theory, Volume 13 (2009), pp. 309-347 | DOI | MR | Zbl

[28] Kleshchev, Alexander Cuspidal systems for affine Khovanov–Lauda–Rouquier algebras, Math. Z., Volume 276 (2014) no. 3-4, pp. 691-726 (Read: 1) | DOI | MR | Zbl

[29] Kleshchev, Alexander; Ram, Arun Homogeneous representations of Khovanov–Lauda algebras, J. Eur. Math. Soc., Volume 12 (2010) no. 5, pp. 1293-1306 | DOI | MR | Zbl

[30] Kleshchev, Alexander; Ram, Arun Representations of Khovanov–Lauda–Rouquier algebras and combinatorics of Lyndon words, Math. Ann., Volume 349 (2011) no. 4, pp. 943-975 | DOI | MR | Zbl

[31] Lazarsfeld, Robert; Mustată, Mircea Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér., Volume 42 (2009) no. 5, pp. 783-835 | DOI | Numdam | MR | Zbl

[32] Leclerc, Bernard Dual canonical bases, quantum shuffles and q-characters, Math. Z., Volume 246 (2004) no. 4, pp. 691-732 | DOI | MR | Zbl

[33] McNamara, Peter J. Representations of Khovanov–Lauda–Rouquier algebras III: symmetric affine type (2016) (https://arxiv.org/abs/1407.7304v4)

[34] Nakada, Kento Colored hook formula for a generalized Young diagram, Osaka J. Math., Volume 45 (2008) no. 4, pp. 1085-1120 | MR | Zbl

[35] Nakajima, Hiraku Quiver varieties and cluster algebras, Kyoto J. Math., Volume 51 (2011) no. 1, pp. 71-126 | MR | Zbl

[36] Nakanishi, Tomoki Tropicalization method in cluster algebras, Tropical geometry and integrable systems. A conference on tropical geometry and integrable systems, School of Mathematics and Statistics, Glasgow, UK, July 3–8, 2011 (Contemporary Mathematics), Volume 580, American Mathematical Society, 2012, pp. 95-115 | MR | Zbl

[37] Okounkov, Andrei Brunn–Minkowski inequality for multiplicities, Invent. Math., Volume 125 (1996) no. 3, pp. 405-411 | DOI | MR | Zbl

[38] Proctor, Robert A. Dynkin diagram classification of λ-minuscule Bruhat lattices and of d-complete posets, J. Algebr. Comb., Volume 9 (1999) no. 1, pp. 61-94 | DOI | MR | Zbl

[39] Proctor, Robert A. Minuscule elements of Weyl groups, the numbers game, and d-complete posets, J. Algebra, Volume 213 (1999) no. 1, pp. 272-303 | DOI | MR | Zbl

[40] Qin, Fan Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. J., Volume 166 (2017), pp. 2337-2442 | MR | Zbl

[41] Rietsch, Konstanze; Williams, Lauren K. Newton–Okounkov bodies, cluster duality, and mirror symmetry for Grassmannians, Duke Math. J., Volume 168 (2019) no. 18, pp. 3437-3527 | MR | Zbl

[42] Rouquier, Raphaël Quiver Hecke algebras and 2-Lie algebras, Algebra Colloq., Volume 19 (2012) no. 2, pp. 359-410 | DOI | MR | Zbl

[43] Stembridge, John R. Minuscule elements of Weyl groups, J. Algebra, Volume 235 (2001) no. 2, pp. 722-743 | DOI | MR | Zbl

[44] Varagnolo, Michela; Vasserot, Eric Canonical bases and KLR algebras, J. Reine Angew. Math., Volume 659 (2011), pp. 67-100 | MR | Zbl

Cité par Sources :