Nicholas Loehr

Algebraic combinatorics exploits the connections between algebraic structures (group representations, modules, polynomial rings, Lie algebras) and combinatorial objects (tableaux, lattice paths, permutations, partitions) to obtain new information about both fields.

Dr. Loehr's research in this area focuses on the combinatorial properties of symmetric polynomials. One can often gain useful knowledge about a mathematical system by introducing a symmetric polynomial to model the system and then finding a combinatorial formula for the symmetric polynomial. For example, Schur's symmetric polynomials translate fundamental problems in representation theory into enumerative questions about tableaux; the Hall-Littlewood symmetric polynomials encode lattice-theoretic properties of finite Abelian p-groups; and the zonal symmetric polynomials can be used to solve problems in Fourier analysis and multivariate statistics.