Demazure

Demazure refers to Michel Demazure, a French mathematician known for his foundational contributions to algebraic geometry and the representation theory of algebraic groups. His work helped illuminate the structure of flag varieties, Schubert calculus, and the interaction between geometry and representation theory.

Demazure introduced a family of operators, now called Demazure operators or divided-difference operators, acting on polynomial algebras associated with root systems. These operators underpin the Demazure character formula, which computes the character of certain submodules of highest-weight representations of semisimple Lie algebras.

In this area, the submodules are called Demazure modules. Given a highest-weight module and a Weyl group element, a Demazure module is generated by applying an appropriate Borel action to the weight space, yielding finite-dimensional representations that capture intermediate structure between highest-weight modules and simple modules. The Demazure character formula expresses the graded dimension of these modules.

Demazure also contributed to the study of Schubert varieties through Demazure resolutions, a construction that provides desingularizations of Schubert varieties in flag manifolds by iterated projective bundles tied to a reduced expression of a Weyl group element.

His ideas have had a lasting influence in representation theory, algebraic geometry, and combinatorics, forming standard tools in Schubert calculus and the study of algebraic groups.