Representation Theory and Combinatorics

所提出的研究问题都在组合表示理论领域，并以组合理解某些刚性代数结构的表示为主要目标，所有这些都来自李论对象，或者是李论对象的推广。提出的研究项目的目标是：(1)构造分区代数的q-类似；(2)将施普林格对应推广到有限复反射群；(3)提供广义Schur函子的单位根的详细分析；(4)找到q-权多重的纯正公式。本研究强烈使用代数几何和组合工具。大多数对称群都很复杂，很难直接研究它们，而表示是提取这些对称信息的一种方式。一般的表示是一个像分子一样的大而复杂的结构，由类似于原子的更小的“不可约成分”组成。组合表示理论（以及本建议）的目标是找到允许（更）容易地确定这些表示的属性的基本模型。这些模型使人们能够获得有关相应表示的精细结构的非常明确的信息，这些表示具有活细胞微观结构的复杂性。

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.