PBW-filtration of representations, degenerate flag varieties and polytopes

该项目是一个长期计划的一部分，目的是更好地了解表示的PBW过滤的几何、代数和组合方面。设g是单李代数，n-是反Borel子代数的幂零根。U(n-)的PBW-滤子诱导在有限维不可约表示Vλ上的滤子，Vλ表示相应的分次空间。这种构造自然导致了一个代数群Ga，它可以被看作是具有李代数g的原始群G的退化版本。该群作用于Vλa，并且在嵌入一个标志簇Fλ_P(Vλ)的引导下，用Fλa表示通过P(Vλa)中最高权线的像的Ga轨道的闭包。在E.Feigin，M.Finkelberg，G.Fourier和Pi的一系列论文中，研究了G=SLn和Sp2m的几何、代数和组合方面的问题。这个项目的目的是将结果推广到其他类型，并将新开发的方法和工具与表示理论中的其他已知结构联系起来，如晶基和广义Gelfand-Tsetlin图案。

The project is part of a long term program to better understand the geometric, algebraic and combinatorial aspects of the PBW-filtration of a representation. Let g be a simple Lie algebra and let n- be the nilpotent radical of an opposite Borel subalgebra. The PBW-filtration of U(n-) induces a filtration on a finite dimensional irreducible representation Vλ, denote by Vλa the associated graded space. This construction leads naturally to an algebraic group Ga, which can be viewed as a degenerate version of the original group G with Lie algebra g. The group acts on Vλa, and, guided by the embedding of a flag variety Fλ _ P(Vλ), denote by Fλa the closure of the Ga-orbit through the image of highest weight line in P(Vλa).Geometric, algebraic and combinatorial aspects of this construction have been investigated for G = SLn and Sp2m in a series of papers by E. Feigin, M. Finkelberg, G. Fourier and the PI. The aim of this project is to generalize the results to other types as well as to link the newly developed methods and tools to other known constructions in representation theory like crystal bases and generalized Gelfand-Tsetlin patterns.