Abstract
We define a set of PBW-semistandard tableaux that is in a weight-preserving bijection with the set of monomials corresponding to integral points in the Feigin–Fourier–Littelmann–Vinberg polytope for highest weight modules of the symplectic Lie algebra. We then show that these tableaux parametrize bases of the multihomogeneous coordinate rings of the complete symplectic original and PBW degenerate flag varieties. From this construction, we provide explicit degenerate relations that generate the defining ideal of the PBW degenerate variety with respect to the Plücker embedding. These relations consist of type Α degenerate Plücker relations and a set of degenerate linear relations that we obtain from De Concini’s linear relations.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Balla, G. Fourier, K. Kambaso, PBW filtration and monomial bases for Demazure modules in types Α and Β, in preparation (2022).
G. Balla, J. A. Olarte, The tropical symplectic Grassmannian, Internat. Math. Res. Not. (2021), doi:https://doi.org/10.1093/imrn/rnab267.
L. Bossinger, S. Lambogila, K. Mincheva, F. Mohammadi, Computing toric degenerations of flag varieties, in: Combinatorial Algebraic Geometry, Springer, New York, NY, 2017, pp. 247–281.
Cerulli Irelli, G., Fang, X., Feigin, E., Fourier, G., Reineke, M.: Linear degenerations of flag varieties: partial flags, defining equations, and group actions. Math. Z. 296(1), 453–477 (2020)
Chirivì, R., Littelmann, P., Maffei, A.: Equations defining symmetric varieties and affine Grassmannians. Internat. Math. Res. Not. 2009(2), 291–347 (2009)
Chirivì, R., Maffei, A.: Plücker relations and spherical varieties: application to model varieties. Transform. Groups. 19(4), 979–997 (2014)
De Concini, C.: Symplectic standard tableaux. Adv. Math. 34(1), 1–27 (1979)
Fang, X., Feigin, E., Fourier, G., Makhlin, I.: Weighted PBW degenerations and tropical flag varieties. Comm. Contemp. Math. 21(01), 1850016 (2019)
Feigin, E.: 𝔾aM degenerations of flag varieties. Selecta Math. 18(3), 513–537 (2012)
Feigin, E., Finkelberg, M., Littelmann, P.: Symplectic degenerate flag varieties. Canad. J. Math. 66(3), 1250–1286 (2014)
Feigin, E., Fourier, G., Littelmann, P.: PBW filtration and bases for irreducible modules in type Αn. Transform. Groups. 16(1), 71–89 (2011)
Feigin, E., Fourier, G., Littelmann, P.: PBW Filtration and Bases for symplectic Lie Algebras. Internat. Math. Res. Not. 2011, 5760–5784 (2011)
Fulton, W., Tableaux, Y.: With Applications to Representation Theory and Geometry, London Math. Soc. Student Texts, vol. 35. Cambridge University Press, Cambridge (1997)
Fulton, W., Harris, J.: Representation Theory. A First Course, Graduate Texts in Math, vol. 129. Springer-Verlag, New York (1991)
Hamel, A.M., King, R.C.: Bijective proof of a symplectic dual pair identity. SIAM J. Discr. Math. 25(2), 539–560 (2011)
Hodge, W.V.D.: Some enumerative results in the theory of forms. Math. Proc. Cambr. Phil. Soc. 39(1), 22–30 (1943)
Kashiwara, M., Nakashima, T.: Crystal graphs for representations of the q-analogue of classical Lie algebras. J. Algebra. 165(2), 295–345 (1994)
King, R.C.: Weight multiplicities for the classical groups. In: Group Theoretical Methods in Physics, pp. 490–499. Springer, Berlin (1976)
Lakshmibai, V., Musili, C., Seshadri, C.S.: Geometry of G/P. Bulletin AMS (New Ser.). 1(2), 432–435 (1979)
V. Lakshmibai, C. S. Seshadri, Standard monomial theory, in: Proceedings of the Hyderabad Conference on Algebraic Groups (Hyderabad, 1989), Madras, Manoj Prakashan (1991), pp. 279–322.
P. Littelmann, The path model, the quantum Frobenius map and standard monomial theory, in: Algebraic Groups and their Representations, Springer, Dordrecht, 1998, pp. 175–212.
Proctor, R.A.: A Schensted algorithm which models tensor representations of the orthogonal group. Canad. J. Math. 42(1), 28–49 (1990)
E. Vinberg, On some canonical bases of representation spaces of simple Lie algebras, conference talk (Bielefeld, 2005).
Young, A.: On quantitative substitutional analysis. Proc. London Math. Soc. 2(1), 255–292 (1928)
Funding
Open Access funding enabled and organized by Projekt DEAL.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
BALLA, G. SYMPLECTIC PBW DEGENERATE FLAG VARIETIES; PBW TABLEAUX AND DEFINING EQUATIONS. Transformation Groups 28, 505–540 (2023). https://doi.org/10.1007/s00031-022-09725-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00031-022-09725-9