\mathbb G_a^M degeneration of flag varieties

2010/07/05 by Feigin, Evgeny
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1007.0646

Abstract

Let \Fl_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 \Fl_λ^a, which is a \mathbb G^M_a variety. Moreover, there exists a larger group G^a acting on \Fl_λ^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 embedded into the product of Grassmanians and thus to the product of projective spaces. The defining ideal of \Fl^a_λ is generated by the set of degenerate Pl" ucker relations. We prove that the coordinate ring of \Fl_λ^a is isomorphic to a direct sum of dual PBW-graded \g-modules. We also prove that there exist bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogues of semistandard tableux.

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