Mathematical Sciences: Presidential Young Investigator Award

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This research overlaps the areas of combinatorics, representation theory and homological algebra. A fundamental aspect is the use of supercomputers. Algorithms are designed that use a mix of algebraic and combinatorial methods. The results of these computations give valuable insight into the solutions of the ongoing research problems described now. The first problem is to compute the Lie algebra cohomology of the tensor product of a finite dimensional complex Lie algebra L with a certain polynomial quotient ring. If L is semisimple, a solution to this problem would imply the Macdonald root-system conjectures. Partial results and strong computer evidence points to a complete solution when L is either semisimple or nilpotent. The second problem is to determine the algebra structure of the symplectic and orthogonal centralizer algebras defined by Richard Brauer. These algebras are fundamental to the representation theory of the symplectic and orthogonal groups and are rich in combinatorial structure. Again, strong computer evidence and partial results suggest that a complete solution to this structure problem will have and elegant combinatorial statement. This research investigates the combinatorial structure of algebraic problems. Many of these questions can be described as determining certain coefficients of certain polynomials which on the one hand are defined in a complicated algebraic way but on the other hand have a simple combinatorial interpretation. Hanlon has in his short career become one of the world's leading authorities on this important subject. His use of the supercomputers to aid this work is very innovative. His research will certainly have the most profound affect.