Flag kinds are vital geometric items and their research consists of an interaction of geometry, combinatorics, and illustration idea. This booklet is targeted account of this interaction. within the sector of illustration concept, the publication provides a dialogue of advanced semisimple Lie algebras and of semisimple algebraic teams; additionally, the illustration conception of symmetric teams is usually mentioned. within the zone of algebraic geometry, the ebook provides a close account of the Grassmannian types, flag forms, and their Schubert subvarieties. end result of the connections with root structures, some of the geometric effects admit based combinatorial description, a standard instance being the outline of the singular locus of a Schubert style. this is often proven to be a final result of ordinary monomial thought (abbreviated SMT). therefore the e-book contains SMT and a few very important purposes - singular loci of Schubert kinds, toric degenerations of Schubert forms, and the connection among Schubert types and classical invariant idea.

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