ALGEBRAIC COMBINATORICS

no. 1
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, Volume 6 (2023) no. 1, pp. 165-195.

The multiple flag variety 𝔛=Gr(ℂ p+q ,r)×(Fl(ℂ p )×Fl(ℂ q )) can be considered as a double flag variety associated to the symmetric pair (G,K)=(GL p+q (ℂ), GL p (ℂ)×GL q (ℂ)) 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.

Received:
Revised:
Accepted:
Published online:
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
Fresse, Lucas 1; Nishiyama, Kyo 2

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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Fresse, Lucas; Nishiyama, Kyo. On generalized Steinberg theory for type AIII. Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 165-195. doi : 10.5802/alco.245. https://alco.centre-mersenne.org/articles/10.5802/alco.245/

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