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

Knutson的建议是利用代数变异的退化来研究代数组合学，特别是组合表示理论。表示理论中的Littelmann路径模型已经有了这样的解释：Chirivi给出了每个标志流形G/P（加上充足的线束）退化为环变的约化半正规并。不幸的是，这使用了Lusztig规范基的性质，因此是特定于表示理论应用的。Knutson希望用Samuel-Rees-Nagata退化来取代这些性质，以获得更一般品种退化的类似约简结果。一个具体的应用是利用G/P的K-等变退化计算从群G到对称子群K的分支规则，Knutson和刘江华在此基础上提出了一个推测规则。许多有趣的数和多项式都与不可约代数集有关，其中“不可约”本质上意味着“不能以非平凡的方式从较小的块中粘接”。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，我们可以把2次与这个方程联系起来。但如果我们把它简并为x^2 + y^2 - 0 = 0，那么这个方程就可以因式分解（除以复数），得到两条直线的并集x = +/- iy。这两个方程都是1次，由退化几何得到组合结果2 = 1 + 1。Knutson用这种方法研究了更有趣的不可约集，例如向量空间中所有子空间的极大嵌套链的空间；将它们退化为简单块的高度可约并，用组合复杂性取代了它们的几何复杂性，并为它们的度（及其推广）给出了更有趣的公式。在提出的一些工作中，他特别计划采用这种技术来理解非紧致对称的量子力学系统（例如，在狭义相对论变换下，它允许加速到但不包括光速）如何在只考虑它们的紧致对称时分解（例如，在旋转下，被困在一个圆圈上的角度，不能跑到无限远）。这是最近ATLAS团队在研究E_8表示时所研究的相同类型的问题（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.