FAVOURABLE MODULES: FILTRATIONS, POLYTOPES, NEWTON–OKOUNKOV BODIES AND FLAT DEGENERATIONS
FAVOURABLE MODULES: FILTRATIONS, POLYTOPES, NEWTON–OKOUNKOV BODIES AND FLAT DEGENERATIONS
复制标题
有利的模块：过滤、多面体、牛顿-奥孔科夫体和平面简并
作者：
E. Feigin;G. Fourier;P. Littelmann
我们介绍了一个有利的模的概念，一个复杂的幂幺代数群，其性质是由组合的一个相关联的多面体。我们描述了模的两个滤子，一个由相应李代数的PBW基上的全度给出，另一个由PBW基上的齐次单项式阶给出。在有利的情况下，一个模块的基础是参数化的正常多面体的格点。过滤诱导相应的ag品种退化到其abelianized版本和一个复曲面品种，退化的特殊纤维是投影正常和算术科恩-麦考利。多面体本身可以恢复为牛顿-奥昆科夫体。最后，我们给出了有利的模块类的例子。
We introduce the notion of a favourable module for a complex unipotent algebraic group, whose properties are governed by the combinatorics of an associated polytope. We describe two filtrations of the module, one given by the total degree on the PBW basis of the corresponding Lie algebra, the other by fixing a homogeneous monomial order on the PBW basis. In the favourable case a basis of the module is parametrized by the lattice points of a normal polytope. The filtrations induce at degenerations of the corresponding ag variety to its abelianized version and to a toric variety, the special fibres of the degenerations being projectively normal and arithmetically Cohen-Macaulay. The polytope itself can be recovered as a Newton-Okounkov body. We conclude the paper by giving classes of examples for favourable modules.