We show that for a finite-type Lie algebra , the representation theory of quiver Hecke algebras provides a natural framework for the construction of Newton–Okounkov bodies associated to the quantum coordinate rings . When 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 for every seed of . 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 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 . Lorsque 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 pour chaque graine de . 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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Keywords: Newton–Okounkov bodies, quiver Hecke algebras, cluster algebras
Mot clés : Corps de Newton–Okounkov, algèbres de Hecke carquois, algèbres amassées
Casbi, Elie 1
@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, 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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