Abstract
Let L be any standard Levi subgroup which acts by left multiplication on a Schubert variety X(w) in the Grassmannian. We give a complete classification of the pairs L and X(w), where X(w) is a spherical variety for the action of L.
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References
Borel A, Linear algebraic groups, volume 126 of Graduate Texts in Mathematics, second edition (1991) (New York: Springer-Verlag)
Bravi P and Cupit-Foutou S, Classification of strict wonderful varieties, Ann. Inst. Fourier (Grenoble) 60(2) (2010) 641–681
Bravi P and Pezzini G, Wonderful varieties of type \(d\), Represent. Theory 9 (2005) 578–637
Bravi P and Pezzini G, Wonderful subgroups of reductive groups and spherical systems, J. Algebra 409 (2014) 101–147
Bravi P and Pezzini G, Primitive wonderful varieties, Math. Z. 282(3–4) (2016) 1067–1096
Bravi P, Wonderful varieties of type \(e\), Represent. Theory 11 (2007) 174–191
Brion M, On orbit closures of spherical subgroups in flag varieties, Comment. Math. Helv. 76(2) (2001) 263–299
Camus R, Variétés sphériques affines lisses, PhD thesis (2001) (Grenoble: Institut Fourier)
Can M B, Hodges R and Lakshmibai V, Toroidal Schubert varieties, Algebras and Representation Theory August 2019
Escobar L and Mészáros K, Toric matrix Schubert varieties and their polytopes, Proc. Amer. Math. Soc. 144(12) (2016) 5081–5096
Fulton W and Harris J, Representation theory, volume 129 of Graduate Texts in Mathematics (1991) (New York: Springer-Verlag) A first course, Readings in Mathematics
Fulton W, Young tableaux, volume 35 of London Mathematical Society Student Texts (1997) (Cambridge: Cambridge University Press), Wwith Applications to Representation Theory and Geometry
Gao Y, Hodges R and Yong A, Classification of Levi-spherical Schubert varieties, ar**v e-prints, page ar**v:2104.10101, April 2021
Gutschwager C, On multiplicity-free skew characters and the Schubert calculus, Ann. Comb. 14(3) (2010) 339–353
Hodges R and Lakshmibai V, Levi subgroup actions on schubert varieties, induced decompositions of their coordinate rings, and sphericity consequences, Algebras and Representation Theory (2018) https://doi.org/10.1007/s10468-017-9744-6
Knop F, The Luna–Vust theory of spherical embeddings, in Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989) pp 225–249 (1991) (Madras: Manoj Prakashan)
Lakshmibai V and Brown J, The Grassmannian variety, volume 42 of Developments in Mathematics (2015) (New York: Springer) Geometric and representation-theoretic aspects
Losev V Ivan, Uniqueness property for spherical homogeneous spaces, Duke Math. J. 147(2) (2009) 315–343
Luna D, Variétés sphériques de type \(A\), Publ. Math. Inst. Hautes Études Sci. (94) (2011) 161–226
Luna D and Vust Th, Plongements d’espaces homogènes, Comment. Math. Helv. 58(2) (1983) 186–245
Matsuki T and Ōshima T, Embeddings of discrete series into principal series, in: The Orbit Method in Representation Theory (Copenhagen, 1988), volume 82 of Progr. Math., pp. 147–175 (1990) (Boston, MA: Birkhäuser Boston)
Magyar P, Weyman J and Zelevinsky A, Multiple flag varieties of finite type, Adv. Math. 141(1) (1999) 97–118
Perrin N, On the geometry of spherical varieties, Transform. Groups 19(1) (2014) 171–223
Stanley R P, Enumerative Combinatorics, Vol. 2, volume 62 of Cambridge Studies in Advanced Mathematics (1999) (Cambridge: Cambridge University Press), With a foreword by Gian-Carlo Rota and Appendix 1 by Sergey Fomin
Thomas H and Yong A, Multiplicity-free Schubert calculus, Canad. Math. Bull. 53(1) (2010) 171–186
Wyser B J, Schubert calculus of Richardson varieties stable under spherical Levi subgroups, J. Algebraic Combin. 38(4) (2013) 829–850
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Communicating Editor: V Balaji
This article is part of the “Special Issue in Memory of Professor C S Seshadri”.
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Hodges, R., Lakshmibai, V. A classification of spherical Schubert varieties in the Grassmannian. Proc Math Sci 132, 68 (2022). https://doi.org/10.1007/s12044-022-00713-3
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DOI: https://doi.org/10.1007/s12044-022-00713-3