Newton–Okounkov bodies for categories of modules over quiver Hecke algebras
Annales de l'Institut Fourier, Volume 72 (2022) no. 5, pp. 1773-1818.

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.

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.

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DOI: 10.5802/aif.3492
Classification: 05E10,  16G10
Keywords: Newton–Okounkov bodies, quiver Hecke algebras, cluster algebras
Casbi, Elie 1

1 Université de Paris, Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR 7586, CNRS F-75013 Paris (FRANCE)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Casbi, Elie. Newton–Okounkov bodies for categories of modules over quiver Hecke algebras. Annales de l'Institut Fourier, Volume 72 (2022) no. 5, pp. 1773-1818. doi : 10.5802/aif.3492. https://aif.centre-mersenne.org/articles/10.5802/aif.3492/

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