Symmetric Functions Schubert Polynomials And Degeneracy Loci
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Symmetric Functions Schubert Polynomials And Degeneracy Loci
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Author : Laurent Manivel
language : en
Publisher: American Mathematical Soc.
Release Date : 2001
Symmetric Functions Schubert Polynomials And Degeneracy Loci written by Laurent Manivel and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2001 with Computers categories.
This text grew out of an advanced course taught by the author at the Fourier Institute (Grenoble, France). It serves as an introduction to the combinatorics of symmetric functions, more precisely to Schur and Schubert polynomials. Also studied is the geometry of Grassmannians, flag varieties, and especially, their Schubert varieties. This book examines profound connections that unite these two subjects. The book is divided into three chapters. The first is devoted to symmetricfunctions and especially to Schur polynomials. These are polynomials with positive integer coefficients in which each of the monomials correspond to a Young tableau with the property of being ``semistandard''. The second chapter is devoted to Schubert polynomials, which were discovered by A. Lascoux andM.-P. Schutzenberger who deeply probed their combinatorial properties. It is shown, for example, that these polynomials support the subtle connections between problems of enumeration of reduced decompositions of permutations and the Littlewood-Richardson rule, a particularly efficacious version of which may be derived from these connections. The final chapter is geometric. It is devoted to Schubert varieties, subvarieties of Grassmannians, and flag varieties defined by certain incidenceconditions with fixed subspaces. This volume makes accessible a number of results, creating a solid stepping stone for scaling more ambitious heights in the area. The author's intent was to remain elementary: The first two chapters require no prior knowledge, the third chapter uses some rudimentary notionsof topology and algebraic geometry. For this reason, a comprehensive appendix on the topology of algebraic varieties is provided. This book is the English translation of a text previously published in French.
Schubert Varieties And Degeneracy Loci
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Author : William Fulton
language : en
Publisher: Springer
Release Date : 2006-11-13
Schubert Varieties And Degeneracy Loci written by William Fulton and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2006-11-13 with Mathematics categories.
Schubert varieties and degeneracy loci have a long history in mathematics, starting from questions about loci of matrices with given ranks. These notes, from a summer school in Thurnau, aim to give an introduction to these topics, and to describe recent progress on these problems. There are interesting interactions with the algebra of symmetric functions and combinatorics, as well as the geometry of flag manifolds and intersection theory and algebraic geometry.
Symmetric Functions And Combinatorial Operators On Polynomials
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Author : Alain Lascoux
language : en
Publisher: American Mathematical Soc.
Release Date :
Symmetric Functions And Combinatorial Operators On Polynomials written by Alain Lascoux and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on with Science categories.
The theory of symmetric functions is an old topic in mathematics which is used as an algebraic tool in many classical fields. With $\lambda$-rings, one can regard symmetric functions as operators on polynomials and reduce the theory to just a handful of fundamental formulas. One of the main goals of the book is to describe the technique of $\lambda$-rings. The main applications of this technique to the theory of symmetric functions are related to the Euclid algorithm and itsoccurrence in division, continued fractions, Pade approximants, and orthogonal polynomials. Putting the emphasis on the symmetric group instead of symmetric functions, one can extend the theory to non-symmetric polynomials, with Schur functions being replaced by Schubert polynomials. In two independentchapters, the author describes the main properties of these polynomials, following either the approach of Newton and interpolation methods or the method of Cauchy. The last chapter sketches a non-commutative version of symmetric functions, using Young tableaux and the plactic monoid. The book contains numerous exercises clarifying and extending many points of the main text. It will make an excellent supplementary text for a graduate course in combinatorics.
On Combinatorics Of Degeneracy Loci And H G B
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Author : Alexander T. F. Yong
language : en
Publisher:
Release Date : 2003
On Combinatorics Of Degeneracy Loci And H G B written by Alexander T. F. Yong and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2003 with categories.
Advances In Algebra
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Author : Jörg Feldvoss
language : en
Publisher: Springer
Release Date : 2019-02-27
Advances In Algebra written by Jörg Feldvoss and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-02-27 with Mathematics categories.
This proceedings volume covers a range of research topics in algebra from the Southern Regional Algebra Conference (SRAC) that took place in March 2017. Presenting theory as well as computational methods, featured survey articles and research papers focus on ongoing research in algebraic geometry, ring theory, group theory, and associative algebras. Topics include algebraic groups, combinatorial commutative algebra, computational methods for representations of groups and algebras, group theory, Hopf-Galois theory, hypergroups, Lie superalgebras, matrix analysis, spherical and algebraic spaces, and tropical algebraic geometry. Since 1988, SRAC has been an important event for the algebra research community in the Gulf Coast Region and surrounding states, building a strong network of algebraists that fosters collaboration in research and education. This volume is suitable for graduate students and researchers interested in recent findings in computational and theoretical methods in algebra and representation theory.
Schubert Calculus And Its Applications In Combinatorics And Representation Theory
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Author : Jianxun Hu
language : en
Publisher: Springer Nature
Release Date : 2020-10-24
Schubert Calculus And Its Applications In Combinatorics And Representation Theory written by Jianxun Hu and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-10-24 with Mathematics categories.
This book gathers research papers and surveys on the latest advances in Schubert Calculus, presented at the International Festival in Schubert Calculus, held in Guangzhou, China on November 6–10, 2017. With roots in enumerative geometry and Hilbert's 15th problem, modern Schubert Calculus studies classical and quantum intersection rings on spaces with symmetries, such as flag manifolds. The presence of symmetries leads to particularly rich structures, and it connects Schubert Calculus to many branches of mathematics, including algebraic geometry, combinatorics, representation theory, and theoretical physics. For instance, the study of the quantum cohomology ring of a Grassmann manifold combines all these areas in an organic way. The book is useful for researchers and graduate students interested in Schubert Calculus, and more generally in the study of flag manifolds in relation to algebraic geometry, combinatorics, representation theory and mathematical physics.
Representation Theory Of The Symmetric Groups
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Author : Tullio Ceccherini-Silberstein
language : en
Publisher: Cambridge University Press
Release Date : 2010-02-04
Representation Theory Of The Symmetric Groups written by Tullio Ceccherini-Silberstein and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2010-02-04 with Mathematics categories.
The representation theory of the symmetric groups is a classical topic that, since the pioneering work of Frobenius, Schur and Young, has grown into a huge body of theory, with many important connections to other areas of mathematics and physics. This self-contained book provides a detailed introduction to the subject, covering classical topics such as the Littlewood–Richardson rule and the Schur–Weyl duality. Importantly the authors also present many recent advances in the area, including Lassalle's character formulas, the theory of partition algebras, and an exhaustive exposition of the approach developed by A. M. Vershik and A. Okounkov. A wealth of examples and exercises makes this an ideal textbook for graduate students. It will also serve as a useful reference for more experienced researchers across a range of areas, including algebra, computer science, statistical mechanics and theoretical physics.
Recent Trends In Algebraic Combinatorics
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Author : Hélène Barcelo
language : en
Publisher: Springer
Release Date : 2019-01-21
Recent Trends In Algebraic Combinatorics written by Hélène Barcelo and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-01-21 with Mathematics categories.
This edited volume features a curated selection of research in algebraic combinatorics that explores the boundaries of current knowledge in the field. Focusing on topics experiencing broad interest and rapid growth, invited contributors offer survey articles on representation theory, symmetric functions, invariant theory, and the combinatorics of Young tableaux. The volume also addresses subjects at the intersection of algebra, combinatorics, and geometry, including the study of polytopes, lattice points, hyperplane arrangements, crystal graphs, and Grassmannians. All surveys are written at an introductory level that emphasizes recent developments and open problems. An interactive tutorial on Schubert Calculus emphasizes the geometric and topological aspects of the topic and is suitable for combinatorialists as well as geometrically minded researchers seeking to gain familiarity with relevant combinatorial tools. Featured authors include prominent women in the field known for their exceptional writing of deep mathematics in an accessible manner. Each article in this volume was reviewed independently by two referees. The volume is suitable for graduate students and researchers interested in algebraic combinatorics.
Hasse Schmidt Derivations On Grassmann Algebras
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Author : Letterio Gatto
language : en
Publisher: Springer
Release Date : 2016-07-08
Hasse Schmidt Derivations On Grassmann Algebras written by Letterio Gatto and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2016-07-08 with Mathematics categories.
This book provides a comprehensive advanced multi-linear algebra course based on the concept of Hasse-Schmidt derivations on a Grassmann algebra (an analogue of the Taylor expansion for real-valued functions), and shows how this notion provides a natural framework for many ostensibly unrelated subjects: traces of an endomorphism and the Cayley-Hamilton theorem, generic linear ODEs and their Wronskians, the exponential of a matrix with indeterminate entries (Putzer's method revisited), universal decomposition of a polynomial in the product of two monic polynomials of fixed smaller degree, Schubert calculus for Grassmannian varieties, and vertex operators obtained with the help of Schubert calculus tools (Giambelli's formula). Significant emphasis is placed on the characterization of decomposable tensors of an exterior power of a free abelian group of possibly infinite rank, which then leads to the celebrated Hirota bilinear form of the Kadomtsev-Petviashvili (KP) hierarchy describing the Plücker embedding of an infinite-dimensional Grassmannian. By gathering ostensibly disparate issues together under a unified perspective, the book reveals how even the most advanced topics can be discovered at the elementary level.
Combinatorial Commutative Algebra
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Author : Ezra Miller
language : en
Publisher: Springer Science & Business Media
Release Date : 2005-06-21
Combinatorial Commutative Algebra written by Ezra Miller and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005-06-21 with Mathematics categories.
Recent developments are covered Contains over 100 figures and 250 exercises Includes complete proofs