This project relates combinatorial, algebraic, and topological aspects of affine Lie algebras and loop groups, and consists of three connected investigations. The first part explores the factorization phenomenon, a direct and unexpected link between the representations of finite-dimensional Lie groups and infinite-dimensional loop groups. The second part seeks to relate two important combinatorial tools, the Littelmann and Kyoto path models, to the geometric topology of loop groups as studied by Bott and Mirkovic-Vilonen. The aim is to give a natural geometric framework for these tools, which are originally defined in purely combinatorial terms, in order to better understand their combinatorial structure. The third part seeks to compute explicit bases for the (co)homology and K-theory of loop groups, so-called affine Schubert and Grothendieck polynomials, applying the geometric approach of Bott-Samelson to extend and simplify the recent work of combinatorists.
Loop groups are symmetry groups of infinite-dimensional objects, but are closely analogous to symmetries of finite-dimensional objects such as spheres. Loop group symmetry, at the boundary between tame (finite) and wild (intractable) problems, is crucial in many areas, such as the Langlands program in number theory, conformal field theory in particle physics, soliton theory in integrable systems, and solvable lattice models in statistical physics. Precise combinatorial analysis of loop-group representations makes possible explicit formulas and solutions in all these areas. Furthermore, the resulting combinatorics of periodic permutations and Young tableaux has applications to problems of computer science such as sorting algorithms and network design in cases which are infinite, but periodic.
