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Degenerate coordinate rings of flag varieties and Frobenius splitting. (English) Zbl 1328.14081
Summary: Recently, E. Feigin [Sel. Math., New Ser. 18, No. 3, 513–537 (2012; Zbl 1267.14064)] introduced the $$\mathbb G _a^N$$-degenerations of semisimple algebraic groups and their associated degenerate flag varieties. It has been shown by Feigin, Finkelberg, and Littelmann [E. Feigin et al., Can. J. Math. 66, No. 6, 1250–1286 (2014; Zbl 1316.14095)] that the degenerate flag varieties in types $$A_n$$ and $$C_n$$ are Frobenius split. In this paper, we construct an associated degeneration of homogeneous coordinate rings of classical flag varieties in all types and show that these rings are Frobenius split in most types. It follows that the degenerate flag varieties of types $$A_n, C_n$$, and $$G_2$$ are Frobenius split. In particular, we obtain an alternate proof of splitting in types $$A_n$$ and $$C_n$$; the case $$G_2$$ was not previously known. We also give a representation-theoretic condition on PBW-graded versions of Weyl modules which is equivalent to the existence of a Frobenius splitting of the classical flag variety that maximally compatibly splits the identity.

##### MSC:
 14M15 Grassmannians, Schubert varieties, flag manifolds 13A35 Characteristic $$p$$ methods (Frobenius endomorphism) and reduction to characteristic $$p$$; tight closure 14A15 Schemes and morphisms 20G05 Representation theory for linear algebraic groups 20G15 Linear algebraic groups over arbitrary fields
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