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The theory of Frobenius splittings has made a significant impact in the study of the geometry of flag varieties and representation theory. This work, unique in book literature, systematically develops the theory and covers all its major developments. Key features: * Concise, efficient exposition unfolds from basic introductory material on Frobenius splittings—definitions, properties and examples—to cutting edge research * Studies in detail the geometry of Schubert varieties, their syzygies, equivariant embeddings of reductive groups, Hilbert Schemes, canonical splittings, good filtrations, among other topics * Applies Frobenius splitting methods to algebraic geometry and various problems in representation theory * Many examples, exercises, and open problems suggested throughout * Comprehensive bibliography and index This book will be an excellent resource for mathematicians and graduate students in algebraic geometry and representation theory of algebraic groups.

Algebraic varieties. --- Frobenius algebras. --- Representations of groups. --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Algebras, Frobenius --- Associative algebras --- Varieties, Algebraic --- Geometry, Algebraic --- Linear algebraic groups --- Geometry, algebraic. --- Group theory. --- Algebraic Geometry. --- Group Theory and Generalizations. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Algebraic geometry --- Geometry --- Algebraic geometry.

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This is a textbook for a senior-level undergraduate course for math, physics and chemistry majors. This one course can play two different but complimentary roles: it can serve as a capstone course for students finishing their education, and it can serve as motivating story for future study of mathematics. Some textbooks are like a vigorous regular physical training program, preparing people for a wide range of challenges by honing their basic skills thoroughly. Some are like a series of day hikes. This book is more like an intended trek to a particularly beautiful goal. We’ll take the easiest route to the top, and we’ll stop to appreciate local flora as well as distant peaks worthy of the vigorous training one would need to scale them. Advice to the Student: This book was written with many different readers in mind. Some will be mathematics students interested to see a beautiful and powerful application of a “pure” mathematical subject. Some will be students of physics and chemistry curious about the mathematics behind some tools they use, such as spherical harmonics. Because the readership is so varied, no single reader should be put off by occasional digressions aimed at certain other readers.

Group theory. --- Hydrogen. --- Atoms. --- Linear algebraic groups. --- Symmetry (Physics) --- Representations of groups. --- Quantum theory. --- Chemistry, Physical and theoretical --- Matter --- Stereochemistry --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Constitution --- Quantum dynamics --- Quantum mechanics --- Quantum physics --- Physics --- Mechanics --- Thermodynamics --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Invariance principles (Physics) --- Symmetry (Chemistry) --- Conservation laws (Physics) --- Algebraic groups, Linear --- Geometry, Algebraic --- Algebraic varieties --- Nonmetals --- Mathematical physics. --- Mathematics. --- Mechanics. --- Atomic, Molecular, Optical and Plasma Physics. --- Elementary Particles, Quantum Field Theory. --- Group Theory and Generalizations. --- Mathematical Methods in Physics. --- Applications of Mathematics. --- Classical Mechanics. --- Math --- Science --- Physical mathematics --- Classical mechanics --- Newtonian mechanics --- Dynamics --- Quantum theory --- Mathematics --- Physics. --- Elementary particles (Physics). --- Quantum field theory. --- Applied mathematics. --- Engineering mathematics. --- Engineering --- Engineering analysis --- Mathematical analysis --- Relativistic quantum field theory --- Field theory (Physics) --- Relativity (Physics) --- Elementary particles (Physics) --- High energy physics --- Nuclear particles --- Nucleons --- Nuclear physics --- Natural philosophy --- Philosophy, Natural --- Physical sciences