Degenerate coordinate rings of flag varieties and Frobenius splitting

Abstract

Recently E. Feigin introduced the G N a -degenerations of semisimple algebraic groups and their associated degenerate flag varieties. It has been shown by Feigin, Finkelberg, and Littelmann 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.

PBW degenerations, quiver Grassmannians, and toric varieties

“…One can show that the number of shape λ semistandard PBWtableaux is equal to dim V λ . Moreover, the following theorem holds [52] (see also [78,60]).…”

Symmetric Dellac configurations and symplectic/orthogonal flag varieties

“…. , V N −1 ) of subspaces of E such that dim V k = k and pr k+1 V k ⊂ V k+1 for all k. Theses varieties are singular, but share many nice properties with their classical analogues, see [F1,F2,FFi,H,LS]. It was shown that the varieties F a N can be realized as Schubert varieties [CL, CLL] and as quiver Grassmannains [CFR1].…”

Torus fixed points in Schubert varieties and normalized median Genocchi numbers

“…The degenerate flag variety is one of the main objects in the framework of PBW filtrations and degenerations on universal enveloping algebras of simple Lie algebras (see for various aspects [FFoL11a,FFoL11b,FFoL13,FFR15,Hag14,Fou14,Fou15,CFR12]). Here, one obtains degenerate flag varieties F l a (λ) as highest weight orbits of PBW degenerate modules.…”