Combinatorics of affine Lie algebras and loop groups

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This project relates combinatorial, algebraic, and topological aspects ofaffine Lie algebras and loop groups, and consists of three connectedinvestigations. The first part explores the factorization phenomenon, adirect and unexpected link between the representations offinite-dimensional Lie groups and infinite-dimensional loop groups. Thesecond part seeks to relate two important combinatorial tools, theLittelmann and Kyoto path models, to the geometric topology of loop groupsas studied by Bott and Mirkovic-Vilonen. The aim is to give a naturalgeometric framework for these tools, which are originally defined in purelycombinatorial terms, in order to better understand their combinatorialstructure. The third part seeks to compute explicit bases for the(co)homology and K-theory of loop groups, so-called affine Schubert andGrothendieck polynomials, applying the geometric approach of Bott-Samelsonto extend and simplify the recent work of combinatorists.Loop groups are symmetry groups of infinite-dimensional objects, but areclosely analogous to symmetries of finite-dimensional objects such asspheres. Loop group symmetry, at the boundary between tame (finite) andwild (intractable) problems, is crucial in many areas, such as theLanglands program in number theory, conformal field theory in particlephysics, soliton theory in integrable systems, and solvable lattice modelsin statistical physics. Precise combinatorial analysis of loop-grouprepresentations makes possible explicit formulas and solutions in all theseareas. Furthermore, the resulting combinatorics of periodic permutationsand Young tableaux has applications to problems of computer science such assorting algorithms and network design in cases which are infinite, butperiodic.