Representation Theory and Combinatorics

点击翻译按钮获取中文摘要

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.