This research is concerned with monoids of Lie type. The principal investigator will study the relation between the coefficients of the unity of Solomon's Hecke algebra and the Kazhdan-Lusztig polynomials; universal monoids of various types; and finite-regular monoids associated with complex representations of groups of Lie type. The algebra M of nxn matrices over a field is of fundamental importance in many branches of science. As an algebra, M is indecomposable. However, as a multiplicative monoid, M decomposes into n-1 local semigroups. Putting together these and similar local semigroups yields a monoid of Lie type which can be thought of as deformations of M. This research will use these monoids of Lie type to examine M.
