# ALGEBRAIC COMBINATORICS

On generalized Steinberg theory for type AIII
Algebraic Combinatorics, Volume 6 (2023) no. 1, pp. 165-195.

The multiple flag variety $𝔛=\mathrm{Gr}\left({ℂ}^{p+q},r\right)×\left(\mathrm{Fl}\left({ℂ}^{p}\right)×\mathrm{Fl}\left({ℂ}^{q}\right)\right)$ can be considered as a double flag variety associated to the symmetric pair $\left(G,K\right)=\left({\mathrm{GL}}_{p+q}\left(ℂ\right),$ ${\mathrm{GL}}_{p}\left(ℂ\right)×{\mathrm{GL}}_{q}\left(ℂ\right)\right)$ 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.

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
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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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