A homological study  of Green polynomials*

Kato, Syu

doi:10.24033/asens.2265

A homological study of Green polynomials*
[Une étude homologique des polynômes de Green]

Kato, Syu

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 5, pp. 1035-1074.

Résumé
Abstract

La relation d'orthogonalité des polynômes de Kostka émanant des groupes de réflexions complexes ([51, 52] et [35]) est interprétée en termes d'algèbre homologique. Ceci nous conduit à la notion de système Kostka, qui peut être considérée comme une contrepartie catégorique des polynômes de Kostka. Puis, nous démontrons que chaque correspondance de Springer généralisée ([34]) dans une bonne caractéristique engendre un système de Kostka. Nous pouvons ainsi observer la propriété de génération du premier terme de l'homologie (tordue) des fibres de Springer généralisées, ainsi que la formule de transition de polynômes de Kostka entre deux correspondances de Springer généralisées de type $𝖡𝖢$ . Cette dernière fournit un algorithme inductif de calcul des polynômes de Kostka par la mise à niveau de [16] §3 à sa version graduée. Dans les annexes, nous apportons les preuves algébriques que les systèmes de Kostka existent pour les cas de type $𝖠$ et de type $𝖡𝖢$ asymptotique. Aussi, il est possible d'omettre de lire les sections géométriques 3 à 5 et pour entrevoir les idées-clés et parcourir des exemples/techniques de base.

We interpret the orthogonality relation of Kostka polynomials arising from complex reflection groups ([51, 52] and [35]) in terms of homological algebra. This leads us to the notion of Kostka system, which can be seen as a categorical counterpart of Kostka polynomials. Then, we show that every generalized Springer correspondence ([34]) in a good characteristic gives rise to a Kostka system. This enables us to see the top-term generation property of the (twisted) homology of generalized Springer fibers, and the transition formula of Kostka polynomials between two generalized Springer correspondences of type $𝖡𝖢$ . The latter provides an inductive algorithm to compute Kostka polynomials by upgrading [16] §3 to its graded version. In the appendices, we present purely algebraic proofs that Kostka systems exist for type $𝖠$ and asymptotic type $𝖡𝖢$ cases, and therefore one can skip geometric sections §3–5 to see the key ideas and basic examples/techniques.

Publié le : 2015-09-24

MR Zbl

DOI : 10.24033/asens.2265

Classification : 20G99, 33D52.
Keywords: Generalized Springer correspondences, Kostka polynomials, the Lusztig-Shoji algorithm, $\mathrm {Ext}$-orthogonal collections, Kostka systems.
Mot clés : Correspondances de Springer généralisées, polynômes de Kostka, l'algorithme Lusztig-Shoji, ensembles $\mathrm {Ext}$-orthogonales, systèmes de Kostka.

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     title = {A homological study  of {Green} polynomials*},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1035--1074},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
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Kato, Syu. A homological study  of Green polynomials*. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 5, pp. 1035-1074. doi : 10.24033/asens.2265. http://www.numdam.org/articles/10.24033/asens.2265/

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