Representation Theory and Combinatorics

所提出的研究问题都存在于组合表示理论领域，并且以对某些刚性代数结构的表示的组合理解为主要目标，所有这些刚性代数结构都来自于或推广到李理论对象。所提出的研究项目的目标是(1)构造划分代数的Q-模拟，(2)将Springer对应推广到有限复反射群，(3)详细分析广义Schur函数的单位根，以及(4)找到q-重量乘数的一个纯正公式。这项研究强烈地使用了代数几何和组合工具。大多数对称群足够复杂，以至于很难直接研究它们，而表示是一种提取关于这些对称的信息的方式。一般表示是一个像分子一样的大而复杂的结构，它由类似于原子的较小的不可约分量组成。组合表示理论(和本建议)的目的是找到允许我们(更容易)确定这些表示的性质的基本模型。这些模型使人们能够获得关于相应表示的精细结构的非常明确的信息，它们在活细胞的微观结构上具有复杂性。

The proposed research problems all lie in the field of combinatorialrepresentation theory and have, as primary goals, combinatorial understandingof representations of certain rigid algebraic structures,all of which come from, or are generalizations of, Lie theoretic objects.The goals of the proposed research projectsare to (1) construct a q-analogue of partition algebras,(2) generalize the Springer correspondence to finitecomplex reflection groups,(3) provide detailed analysis of generalized Schur functorsat roots of unity, and (4) find a purely positive formulafor q-weight multiplicities.This research strongly uses both algebraic geometric and combinatorial tools.Most symmetry groups are complex enough that it isdifficult to study them directly and a representationis a way of extracting information about these symmetries.A general representation is a large and complex structure like a moleculeand is composed of smaller ``irreducible components'' which areanalogous to atoms.The goal of Combinatorial Representation Theory (and this proposal)is to find elementary models which allowus to (more) easily determine properties of these representations.These models enable one to obtain very explicit informationabout the fine structure of the corresponding representations,which have complexities on the order of the microscopic structureof living cells.