Representation Theoretic Combinatorics: Constructive and Integral

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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)