## Abstract

Let F _{λ} be a generalized flag variety of a simple Lie group G embedded into the projectivization of an irreducible G-module V _{λ}. We define a flat degeneration F _{λ} ^{a} variety. Moreover, there exists a larger group G ^{a} acting on F _{λ} ^{a}, which is a degeneration of the group G. The group G ^{a} contains G _{a} ^{M} as a normal subgroup. If G is of type A, then the degenerate flag varieties can be embedde'd into the product of Grassmannians and thus to the product of projective spaces. The defining ideal of F _{λ} ^{a} is generated by the set of degenerate Plücker relations. We prove that the coordinate ring of F _{λ} ^{a} is isomorphic to a direct sum of dual PBW-graded g-modules. We also prove that there exists bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogs of semistandard tableaux.

Original language | English |
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Pages (from-to) | 513-537 |

Number of pages | 25 |

Journal | Selecta Mathematica, New Series |

Volume | 18 |

Issue number | 3 |

DOIs | |

State | Published - Aug 2012 |

Externally published | Yes |

### Funding

Funders | Funder number |
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Russian Foundation for Basic Research | 07-02-00799, 09-01-00058, NSh-3472.2008.2 |

## Keywords

- Degeneration
- Flag varieties
- Lie groups
- Plücker relations

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