On generalized Steinberg theory for type AIII

Fresse, Lucas¹ ; Nishiyama, Kyo²

¹ Université de Lorraine CNRS Institut Élie Cartan de Lorraine UMR 7502 Vandoeuvre-lès-Nancy F-54506 France
² Department of Mathematics Aoyama Gakuin University Fuchinobe 5-10-1 Chuo-ku Sagamihara 252-5258 Japan

Algebraic Combinatorics, Tome 6 (2023) no. 1, pp. 165-195

Résumé

The multiple flag variety $𝔛=\mathrm{Gr}({ℂ}^{p+q},r)\times (\mathrm{Fl}\left({ℂ}^p\right)\times \mathrm{Fl}\left({ℂ}^q\right))$ can be considered as a double flag variety associated to the symmetric pair $(G,K)=({\mathrm{GL}}_{p+q}\left(ℂ\right),$ ${\mathrm{GL}}_p\left(ℂ\right)\times {\mathrm{GL}}_q\left(ℂ\right))$ of type AIII. We consider the diagonal action of $K$ on $𝔛$. There is a finite number of orbits for this action, and our first result is a description of these orbits: parametrization (by a certain set of graphs), dimensions, closure relations and cover relations.

In [5], we defined two generalized Steinberg maps from the $K$-orbits of $𝔛$ to the nilpotent $K$-orbits in $𝔨$ and those in the Cartan complement of $𝔨$, respectively. The main result in the present paper is a complete, explicit description of these two Steinberg maps by means of a combinatorial algorithm which extends the classical Robinson–Schensted correspondence.

Détail

Reçu le : 2021-10-14
Révisé le : 2021-12-19
Accepté le : 2022-05-01
Publié le : 2023-02-24

DOI : 10.5802/alco.245

Classification : 14M15, 17B08, 53C35, 05A15
Keywords: double flag variety, conormal bundle, exotic moment map, nilpotent orbits, partial permutations, Robinson–Schensted correspondence, Steinberg variety

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits