Degenerations of algebraic varieties, with applications to combinatorics and representation theory

Knutson的建议利用代数簇的退化来研究代数组合学，特别是组合表示论。表示论中的Littelmann路径模型已经有了这样的解释：Chirivi给出了每个旗流形G/P（加上充足的线丛）的退化为环面簇的简化的、非正规的并集。不幸的是，这使用了Lusztig的正则基的性质，因此是特定于表示论应用的。Knutson希望用Samuel-Rees-Nagata退化来代替这些性质，以获得更一般的变种退化的类似约化结果。一个具体的应用是计算从群G到对称子群K的分支规则，使用G/P的K-等变退化，Knutson与Jiang-Hua Liu，陈述了一个基于此的数学规则。许多有趣的数字和多项式都与不可约代数集相关联，其中“不可约”本质上意味着“不是以非平凡的方式从较小的片段胶合而成”。一个圆x^2 + y^2 - 1 = 0就是这样一个例子，因为它的方程没有因式，人们可以将2次与这个方程联系起来。但是如果我们将其退化为x^2 + y^2 - 0 = 0，那么方程就有因子（对复数），得到两条直线的并集x = +/- iy。这两个方程都是一次方程，从退化几何中我们得到组合结果2 = 1 + 1。克努森使用这种技术来研究更有趣的不可约集，例如向量空间中所有子空间的最大嵌套链的空间;将它们退化为简单片段的高度可约并集，用组合复杂性取代了它们的几何复杂性，并给出了更有趣的度公式（及其推广）。在他提出的一些工作中，他特别计划采用这种技术来理解具有非紧对称性的量子力学系统（例如在狭义相对论变换下，它允许速度达到但不包括光速）如何分解，当人们只考虑它们的紧对称性时（例如在旋转下，被困在一个圆上的角度不能无限大）。这与ATLAS小组最近在研究E_8的表示时所研究的问题是同一类的（尽管所希望的结果将是完全互补的）。

Knutson's proposal makes use of degeneration of algebraic varieties to study algebraic combinatorics, particularly combinatorial representation theory. Littelmann's path model in representation theory already has such an interpretation: Chirivi gave a degeneration of each flag manifold G/P (plus ample line bundle) to a reduced, seminormal union of toric varieties. Unfortunately, this used properties of Lusztig's canonical basis, so was specific to representation theory applications. Knutson hopes to replace these properties with Samuel-Rees-Nagata degenerations, to achieve similar reducedness results for degenerations of much more general varieties.One specific application is in computing branching rules from a group G to a symmetric subgroup K, using a K-equivariant degeneration of G/P, and Knutson, jointly with Jiang-Hua Liu, states a conjectural rule based on this.Many interesting numbers and polynomials are associated to irreducible algebraic sets, where "irreducible" essentially means "not glued from smaller pieces in a nontrivial way". A circle x^2 + y^2 - 1 = 0 is such an example, as its equation does not factor, and one can associate the degree 2 to this equation. But if we degenerate it to x^2 + y^2 - 0 = 0, then the equation does factor (over the complex numbers), giving the union of two lines x = +/- iy. Those two equations are degree 1, and from the degenerative geometry we obtain the combinatorial result 2 = 1 + 1. Knutson uses this technique to study much more interesting irreducible sets, e.g. the space of all maximal nested chains of subspaces in a vector space; degenerating them to highly reducible unions of simple pieces replaces their geometric complexity with combinatorial complexity, and gives much more interesting formulae for their degrees (and generalizations thereof). In some of the work proposed, he specifically plans to pursue this technique to understand how quantum-mechanical systems with noncompact symmetry (e.g. under special relativity transformations, which allow boosts by speeds up to but not including light-speed) decompose when one only considers their compact symmetries (e.g. under rotation, by angles that are trapped on a circle and cannot run off to infinity). This is the same sort of problem (though the hoped-for results would be wholly complementary) as studied recently by the ATLAS team in the study of the representations of E_8.