# Levi Subgroup Actions on Schubert Varieties, Induced Decompositions of their Coordinate Rings, and Sphericity Consequences

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## Abstract

Let *L* _{ w } be the Levi part of the stabilizer *Q* _{ w } in *G* *L* _{ N } (for left multiplication) of a Schubert variety *X*(*w*) in the Grassmannian G_{ d,N }. For the natural action of *L* _{ w } on \(\mathbb {C}[X(w)]\), the homogeneous coordinate ring of *X*(*w*) (for the Plücker embedding), we give a combinatorial description of the decomposition of \(\mathbb {C}[X(w)]\) into irreducible *L* _{ w }-modules; in fact, our description holds more generally for the action of the Levi part *L* of any parabolic subgroup *Q* that is contained in *Q* _{ w }. This decomposition is then used to show that all smooth Schubert varieties, all determinantal Schubert varieties, and all Schubert varieties in G_{2,N } are spherical *L* _{ w }-varieties.

## Keywords

Schubert varieties Levi subgroup Representation theory Coordinate rings Spherical varieties Algebraic groups## Mathematics Subject Classification (2010)

Primary 20G05 Secondary 20C33 14M27 05E10## Preview

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## Notes

### Acknowledgements

The authors thank Guido Pezzini for many helpful discussions during their visit to FAU Erlangen-Nürnberg in June 2015 and for his insights into spherical varieties in particular. The authors would also like to thank Friedrich Knop for some useful discussions on spherical varieties. Additionally, the authors would like to thank FAU Erlangen-Nürnberg for the hospitality extended to them during their visit in June 2015. Finally, the authors thank the referee for many helpful comments and suggestions.

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