This book is intended for a one-year graduate course on Lie groups and Lie algebras. The book goes beyond the representation theory of compact Lie groups, which is the basis of many texts, and provides a carefully chosen range of material to give the student the bigger picture. The book is organized to allow different paths through the material depending on one's interests. This second edition has substantial new material, including improved discussions of underlying principles, streamlining of some proofs, and many results and topics that were not in the first edition. For compact Lie groups, the book covers the Peter–Weyl theorem, Lie algebra, conjugacy of maximal tori, the Weyl group, roots and weights, Weyl character formula, the fundamental group and more. The book continues with the study of complex analytic groups and general noncompact Lie groups, covering the Bruhat decomposition, Coxeter groups, flag varieties, symmetric spaces, Satake diagrams, embeddings of Lie groups and spin. Other topics that are treated are symmetric function theory, the representation theory of the symmetric group, Frobenius–Schur duality and GL(n) × GL(m) duality with many applications including some in random matrix theory, branching rules, Toeplitz determinants, combinatorics of tableaux, Gelfand pairs, Hecke algebras, the "philosophy of cusp forms" and the cohomology of Grassmannians. An appendix introduces the reader to the use of Sage mathematical software for Lie group computations.
This is a graduate-level text on algebraic geometry that provides a quick and fully self-contained development of the fundamentals, including all commutative algebra which is used. A taste of the deeper theory is given: some topics, such as local algebra and ramification theory, are treated in depth. The book culminates with a selection of topics from the theory of algebraic curves, including the Riemann-Roch theorem, elliptic curves, the zeta function of a curve over a finite field, and the Riemann hypothesis for elliptic curves.
Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang-Baxter equation.
This unique book provides the first introduction to crystal base theory from the combinatorial point of view. Crystal base theory was developed by Kashiwara and Lusztig from the perspective of quantum groups. Its power comes from the fact that it addresses many questions in representation theory and mathematical physics by combinatorial means. This book approaches the subject directly from combinatorics, building crystals through local axioms (based on ideas by Stembridge) and virtual crystals. It also emphasizes parallels between the representation theory of the symmetric and general linear groups and phenomena in combinatorics. The combinatorial approach is linked to representation theory through the analysis of Demazure crystals. The relationship of crystals to tropical geometry is also explained.
Weyl group multiple Dirichlet series are generalizations of the Riemann zeta function. Like the Riemann zeta function, they are Dirichlet series with analytic continuation and functional equations, having applications to analytic number theory. By contrast, these Weyl group multiple Dirichlet series may be functions of several complex variables and their groups of functional equations may be arbitrary finite Weyl groups. Furthermore, their coefficients are multiplicative up to roots of unity, generalizing the notion of Euler products. This book proves foundational results about these series and develops their combinatorics. These interesting functions may be described as Whittaker coefficients of Eisenstein series on metaplectic groups, but this characterization doesn't readily lead to an explicit description of the coefficients. The coefficients may be expressed as sums over Kashiwara crystals, which are combinatorial analogs of characters of irreducible representations of Lie groups. For Cartan Type A, there are two distinguished descriptions, and if these are known to be equal, the analytic properties of the Dirichlet series follow. Proving the equality of the two combinatorial definitions of the Weyl group multiple Dirichlet series requires the comparison of two sums of products of Gauss sums over lattice points in polytopes. Through a series of surprising combinatorial reductions, this is accomplished. The book includes expository material about crystals, deformations of the Weyl character formula, and the Yang-Baxter equation.
Are you looking for a creative and unique blank lined journal that you can write your thoughts, plans and schedule in? Look no further then this awesome and vintage pregnancy journal. A perfect: - Pregnancy Gift - New Mom Journal - Notebook for Pregnant Mothers This notebook comes in a 6x9 size with a matte finish. It's a 108 paged blank lined journal. Check out my other awesome gift journals by clicking my Author Name 'Daniel Timothy.
Be blessed richly by reading how Daniel looked up from the den's bottom and saw the sky disappearing, along with the natural light dimming, as several beams of ghostly light suddenly pierced the cloud of blackness that usually existed when the covering stone was being put into place. And from out of that dimly lit place dozens of hellish eyes all around him all of a sudden were glowing like beacons of incoming death aflame as dozens of lions quickly encircled him. But as they slowly neared, the devilish blood lust within their eyes immediately faded away with the ghostly light's increase that was happening all around Daniel at the very same time. Furthermore, the reddish glare within the eyes of those starving beasts abruptly began glowing a beautiful shade of blue because of the reflection of that intensifying incandescent ghostly light, which was radiating all around Daniel like a blanket of some glorious warmth.Even those lions stopped dead in their tracks, many being only a few feet away from that son of Jerusalem. 'Twas then a delightful moment when that prophet of the King of Heaven unexpectedly felt His anointing fall upon his shoulders, as the sound of a rushing wind supernaturally swept through that cavernous looking den, swiftly causing most of those overgrown cats to back off several feet. So without any warning, Daniel was abruptly being placed under the raging fires of God's blazing love, while his very own heart was immediately inflamed with wonder, once he realized that this den of death was ablaze with life.
This book presents a broad, user-friendly introduction to the Langlands program, that is, the theory of automorphic forms and its connection with the theory of L-functions and other fields of mathematics. Each of the twelve chapters focuses on a particular topic devoted to special cases of the program. The book is suitable for graduate students and researchers.
Things that go Bump is a series of short stories and novels that looks at how the para-beings that we have all heard stories about exist alongside humans. In order to keep their presense secret they have to abide by specific rules. Who can police such powerful Monsters.It will take all to stop the war that is coming. Can the human race survive it?
This book is a collection of 91 fundamental quotes and aphorisms of Jose Marti: "Others go to bed with their mistresses; I with my ideas." "A selfish man is a thief." "Peoples are made of hate and of love, and more of hate than love. But love, like the sun that it is, sets afire and melts everything." "Liberty is the essence of life. Whatever is done without it is imperfect." "Life on earth is a hand-to-hand mortal combat... between the law of love and the law of hate.
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