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The theory of linear algebraic monoids culminates in a coherent blend of algebraic groups, convex geometry, and semigroup theory. The book discusses all the key topics in detail, including classification, orbit structure, representations, universal constructions, and abstract analogues. An explicit cell decomposition is constructed for the wonderful compactification, as is a universal deformation for any semisimple group. A final chapter summarizes important connections with other areas of algebra and geometry. The book will serve as a solid basis for further research. Open problems are discus

Monoids --- Semigroup algebras --- Monoids. --- Semigroup algebras. --- Algebras, Semigroup --- Mathematics. --- Algebra. --- Geometry. --- Combinatorics. --- Linear algebraic groups. --- Algebra --- Algebraic groups, Linear --- Geometry, Algebraic --- Group theory --- Algebraic varieties --- Semigroups --- Linear algebraic groups --- Combinatorics --- Mathematical analysis --- Mathematics --- Euclid's Elements

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Discrete mathematics --- Generating functions. --- Combinatorial analysis. --- Linear algebraic groups. --- Fonctions génératrices --- Analyse combinatoire --- Groupes algébriques linéaires --- Combinatorial analysis --- Generating functions --- Linear algebraic groups --- Algebraic groups, Linear --- Geometry, Algebraic --- Group theory --- Algebraic varieties --- Functions, Generating --- Combinatorics --- Algebra --- Mathematical analysis --- Fonctions génératrices. --- Analyse combinatoire. --- Groupes algébriques linéaires.

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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. --- Geometry, Algebraic.

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The representation theory of symmetric groups is one of the most beautiful, popular and important parts of algebra, with many deep relations to other areas of mathematics such as combinatories, Lie theory and algebraic geometry. Kleshchev describes a new approach to the subject, based on the recent work of Lascoux, Leclerc, Thibon, Ariki, Grojnowski and Brundan, as well as his own. Much of this work has previously appeared only in the research literature. However to make it accessible to graduate students, the theory is developed from scratch, the only prerequisite being a standard course in abstract algebra. For the sake of transparency, Kleshchev concentrates on symmetric and spin-symmetric groups, though methods he develops are quite general and apply to a number of related objects. In sum, this unique book will be welcomed by graduate students and researchers as a modern account of the subject.

Linear algebraic groups. --- Representations of groups. --- Algebras, Linear. --- Geometry, Projective. --- Symmetry groups. --- Groups, Symmetry --- Symmetric groups --- Crystallography, Mathematical --- Quantum theory --- Representations of groups --- Projective geometry --- Geometry, Modern --- Linear algebra --- Algebra, Universal --- Generalized spaces --- Mathematical analysis --- Calculus of operations --- Line geometry --- Topology --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Algebraic groups, Linear --- Geometry, Algebraic --- Algebraic varieties --- Modular representations of groups. --- Hecke algebras. --- Superalgebras. --- Nonassociative algebras --- Algebras, Hecke --- Group algebras

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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 --- Particles (Nuclear physics)