Representation Theoretic Combinatorics: Constructive and Integral

利用并发展数环上有限基代数的表示理论，研究非阿贝尔差集、类Williamson Hadamard矩阵、图几何的积分Hecke代数和分圆结合方案。典型的问题包括现在开放的小参数情形的显式构造，经典关联映射的算术不变量(Smith范式)的计算，以及经典Hecke代数内融合方案的计数。通过从域转移到数环，该工作在纯数学(数论，抽象代数)和组合结构(有限几何，代数组合学)之间建立了更广泛的桥梁。这项工作将导致更深入的数学工具，以研究已知的组合对象，并有可能更好地设计和建立组合对象，以满足特定的“光谱”规范。这种组合结构在密码学和数字通信系统设计(如高速调制解调器和数字蜂窝电话)中具有很大的价值。这笔赠款还支持在一个长期举办的地区性代数组合学研讨会上的嘉宾演讲者，该研讨会包括怀俄明大学、科罗拉多大学丹佛分校和科罗拉多州立大学(http://www.math.cudenver.edu/~wcherowi/algcomb.html)的教职员工。

Non Abelian difference sets, Williamson-like Hadamard Matrices, integral Hecke algebras of diagram geometries and cyclotomic association schemes will be studied using and developing a representation theory of finitely based algebras over number rings. Typical problems include explicit constructions for small parameter cases that are now open, computation of the arithmetic invariants (Smith normal form) of classical incidence maps, and enumeration of fusion schemes within a classical Hecke algebra.By moving from fields to number rings this work builds a broader bridge between pure mathematics (number theory, abstract algebra) and combinatorial constructions (finite geometry, algebraic combinatorics). This work will lead to more penetrating mathematical tools to study known combinatorial objects and the possibility to better design and build combinatorial objects to meet specific "spectral" specifications. These kinds of ombinatorial constructions have had great value in cryptography and digital communication system design (e.g. high speed modems and digital cellular telephony). This grant also supports guest speakers at a long running regional Algebraic Combinatorics Seminar that includes and benefits faculty from The University of Wyoming, The University of Colorado at Denver and Colorado State University (http://www.math.cudenver.edu/~wcherowi/algcomb.html)