Flag Varieties: An Interplay of Geometry, Combinatorics, and Representation Theory: Second Edition

Flag Varieties: An Interplay of Geometry, Combinatorics, and Representation Theory: Second Edition


Author Biography

Dr. Brown joined Olivet's faculty as a full-time professor in 2009. He currently teaches calculus, linear algebra, abstract algebra, number theory and differential equations.

For the 2010-2011 academic year, he was a Project NExT fellow. This is a professional development program for new or recent Ph.D.s in the mathematical sciences, sponsored by the Mathematical Association of America. It addresses all aspects of an academic career, with an emphasis on the teaching and learning of mathematics, and provides participants with a network of peers and mentors as they assume these responsibilities.

Before teaching at ONU, he was a graduate assistant at Northeastern University. There, he was named Outstanding Teacher of First Year Engineering Students in 2007. He also collaborated on a published textbook for graduate students studying algebraic geometry, which was the focal point of his doctoral research.

While an undergraduate student at Point Loma, he served on a three-week mission trip to Rwanda. He also helped with a weekly feeding program in downtown San Diego. While studying at Northeastern in Boston, he volunteered with a poverty advocacy group, advocating on behalf of poor families facing difficult financial situations.

He enjoys spending time with his wife, Jody, and their sons, Owen, Evan, and Callen. For relaxation, he likes watching football, kayaking, playing guitar and listening to Bob Dylan.

If you're a baseball fan, you might recognize his voice, too. He's the announcer for Tiger baseball at ONU. Text from Olivet.edu


A publication of Hindustan Book Agency

Flag varieties are important geometric objects. Because of their richness in geometry, combinatorics, and representation theory, flag varieties may be described as an interplay of all three of these fields.

This book gives a detailed account of this interplay. In the area of representation theory, the book presents a discussion on the representation theory of complex semisimple Lie algebras as well as the representation theory of semisimple algebraic groups; in addition, the representation theory of symmetric groups is also discussed. In the area of algebraic geometry, the book gives a detailed account of the Grassmannian varieties, flag varieties, and their Schubert subvarieties. Because of the root system connections, many of the geometric results admit elegant combinatorial description, a typical example being the description of the singular locus of a Schubert variety. This discussion is carried out as a consequence of standard monomial theory (abbreviated SMT). Thus, the book includes SMT and some important applications—singular loci of Schubert varieties, toric degenerations of Schubert varieties, and the relationship between Schubert varieties and classical invariant theory.

In the second edition, two recent results on Schubert varieties in the Grassmannian have been added. The first result gives a free resolution of certain Schubert singularities.The second result is about certain Levi subgroup actions on Schubert varieties in the Grassmannian and derives some interesting geometric and representation-theoretic consequences.

Document Type




Publication Date



Hindustan Book Agency


New Delhi, India

Scholarship Domain(s)

Scholarship of Discovery


Algebraic Geometry


Hindustan titles are distributed in the Americas by the American Mathematical Society. Springer sells the ebook in the United States.

Flag Varieties: An Interplay of Geometry, Combinatorics, and Representation Theory: Second Edition