{\mathbb{G}_a^ M} degeneration of flag varieties |
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Authors: | Evgeny Feigin |
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Institution: | 1. Department of Mathematics, National Research University Higher School of Economics, Vavilova str. 7, 117312, Moscow, Russia 2. Tamm Theory Division, Lebedev Physics Institute, Moscow, Russia
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Abstract: | Let ${\mathcal{F}_\lambda}$ 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 ${\mathcal{F}_\lambda^a}$ , which is a ${\mathbb{G}^M_a}$ variety. Moreover, there exists a larger group G a acting on ${\mathcal{F}_\lambda^a}$ , which is a degeneration of the group G. The group G a contains ${\mathbb{G}^M_a}$ 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 ${\mathcal{F}_\lambda}$ is generated by the set of degenerate Plücker relations. We prove that the coordinate ring of ${\mathcal{F}_\lambda^a}$ is isomorphic to a direct sum of dual PBW-graded ${\mathfrak{g}}$ -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. |
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