Grossberg-Karshon扭曲立方体与无基点因子
Grossberg-Karshon twisted cubes and basepoint-free divisors
摘要：
设$G$是复半单单连通线性代数群.本文的主要结果是给出了Grossberg和Karshon提出的扭立方体无扭性的几个等价判据。在表示论的某些情况下，Grossberg和Karshon得到了不可约$G$ -表示的Demazure型特征公式，作为这些扭曲立方体的格点（根据密度函数用符号计数）的总和。当一个扭曲的立方体是一个“真”（即封闭的凸）多面体时，它是解扭曲的;在这种情况下，Grossberg和Karshon的特征公式变成了一个没有多重性的纯正公式，即每个格点在公式中只出现一次，系数为$+1$。我们的一个等价条件为untwedness的是，一定的除数的特殊纤维的环面退化的Bott-Samelson品种，作为建设的Pasquier，是基点免费的。我们还表明，严格的正性的扭曲立方体的一些定义常数，连同凸性（其支持），足以保证untwistedness。最后，在扭曲立方体由外尔群元素的整数权$λ$和选择词分解$\underline{w}$的表示论数据产生的特殊情况下，给出了用$λ$和$\underline{w}$表示的扭曲立方体无扭曲的两个简单必要条件.
英文摘要：
Let $G$ be a complex semisimple simply connected linear algebraic group. The main result of this note is to give several equivalent criteria for the untwistedness of the twisted cubes introduced by Grossberg and Karshon. In certain cases arising from representation theory, Grossberg and Karshon obtained a Demazure-type character formula for irreducible $G$-representations as a sum over lattice points (counted with sign according to a density function) of these twisted cubes. A twisted cube is untwisted when it is a "true" (i.e. closed, convex) polytope; in this case, Grossberg and Karshon's character formula becomes a purely positive formula with no multiplicities, i.e. each lattice point appears precisely once in the formula, with coefficient $+1$. One of our equivalent conditions for untwistedness is that a certain divisor on the special fiber of a toric degeneration of a Bott-Samelson variety, as constructed by Pasquier, is basepoint-free. We also show that the strict positivity of some of the defining constants for the twisted cube, together with convexity (of its support), is enough to guarantee untwistedness. Finally, in the special case when the twisted cube arises from the representation-theoretic data of $λ$ an integral weight and $\underline{w}$ a choice of word decomposition of a Weyl group element, we give two simple necessary conditions for untwistedness which is stated in terms of $λ$ and $\underline{w}$.
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