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Notes On Schubert Polynomials
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Author :Ian Grant Macdonald Publisher :Dép. de mathématique et d'informatique, Université du Québec à Montréal ISBN 13 : Total Pages :138 pages Book Rating :4.3/5 (91 download)
Book Synopsis Notes on Schubert Polynomials by : Ian Grant Macdonald
Download or read book Notes on Schubert Polynomials written by Ian Grant Macdonald and published by Dép. de mathématique et d'informatique, Université du Québec à Montréal. This book was released on 1991 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Symmetric Functions, Schubert Polynomials and Degeneracy Loci by : Laurent Manivel
Download or read book Symmetric Functions, Schubert Polynomials and Degeneracy Loci written by Laurent Manivel and published by American Mathematical Soc.. This book was released on 2001 with total page 180 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Schubert Varieties and Degeneracy Loci by : William Fulton
Download or read book Schubert Varieties and Degeneracy Loci written by William Fulton and published by Springer. This book was released on 2006-11-13 with total page 158 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis An Abstract Definition of Schubert Polynomials Extending to the Classical Groups by : Sara C. Billey
Download or read book An Abstract Definition of Schubert Polynomials Extending to the Classical Groups written by Sara C. Billey and published by . This book was released on 1994 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Enumerative Combinatorics: Volume 2 by : Richard P. Stanley
Download or read book Enumerative Combinatorics: Volume 2 written by Richard P. Stanley and published by Cambridge University Press. This book was released on 1997 with total page 600 pages. Available in PDF, EPUB and Kindle. Book excerpt: An introduction, suitable for beginning graduate students, showing connections to other areas of mathematics.
Book Synopsis Combinatorics of Schubert Polynomials by : Avery J St. Dizier
Download or read book Combinatorics of Schubert Polynomials written by Avery J St. Dizier and published by . This book was released on 2020 with total page 95 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this thesis, we study several aspects of the combinatorics of various important families of polynomials, particularly focusing on Schubert polynomials. Schubert polynomials arise as distinguished representatives of cohomology classes in the cohomology ring of the flag variety. As polynomials, they enjoy a rich and well-studied combinatorics. Through joint works with Fink and M\'esz\'aros, we connect the supports of Schubert polynomials to a class of polytopes called generalized permutahedra. Through a realization of Schubert polynomials as characters of flagged Weyl modules, we show that the exponents of a Schubert polynomial are exactly the integer points in a generalized permutahedron. We also prove a combinatorial description of this permutahedron. We then study characters of flagged Weyl modules more generally and give an interesting inequality on their coefficients. We next shift our focus onto the coefficients of Schubert polynomials. We describe a construction due to Magyar called orthodontia. We use orthodontia together with the previous inequality for characters to give a complete description of the Schubert polynomials that have only zero and one as coefficients. Through joint work with Huh, Matherne, and M\'esz\'aros, we next show a discrete log-concavity property of the coefficients of Schubert polynomials. The main tool for this purpose is the Lorentzian property introduced by Br\"and\'en and Huh. We prove that something similar to Schubert polynomials is Lorentzian. We extract from this the discrete log-concavity of Schubert polynomials and the Lorentzian property of Schur polynomials. We finish with various conjectures and partial results regarding other families of polynomials.
Book Synopsis Schubert Calculus and Its Applications in Combinatorics and Representation Theory by : Jianxun Hu
Download or read book Schubert Calculus and Its Applications in Combinatorics and Representation Theory written by Jianxun Hu and published by Springer Nature. This book was released on 2020-10-24 with total page 367 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Universal Schubert Polynomials written by and published by . This book was released on 1997 with total page 20 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics by : James Haglund
Download or read book The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics written by James Haglund and published by American Mathematical Soc.. This book was released on 2008 with total page 178 pages. Available in PDF, EPUB and Kindle. Book excerpt: This work contains detailed descriptions of developments in the combinatorics of the space of diagonal harmonics, a topic at the forefront of current research in algebraic combinatorics. These developments have led in turn to some surprising discoveries in the combinatorics of Macdonald polynomials.
Book Synopsis Quantum Double Schubert Polynomials by :
Download or read book Quantum Double Schubert Polynomials written by and published by . This book was released on 1997 with total page 4 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Schubert Polynomials and the NilCoxeter Algebra by : Sergej Vasilʹevič Fomin
Download or read book Schubert Polynomials and the NilCoxeter Algebra written by Sergej Vasilʹevič Fomin and published by . This book was released on 1992 with total page 11 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis On Algebraic and Combinatorial Properties of Schur and Schubert Polynomials by : Rudolf Winkel
Download or read book On Algebraic and Combinatorial Properties of Schur and Schubert Polynomials written by Rudolf Winkel and published by . This book was released on 2000 with total page 223 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Algebraic Methods and Q-special Functions by : Jan Felipe Van Diejen
Download or read book Algebraic Methods and Q-special Functions written by Jan Felipe Van Diejen and published by American Mathematical Soc.. This book was released on 1999-01-01 with total page 302 pages. Available in PDF, EPUB and Kindle. Book excerpt: There has been revived interest in recent years in the study of special functions. Many of the latest advances in the field were inspired by the works of R. A. Askey and colleagues on basic hypergeometric series and I. G. Macdonald on orthogonal polynomials related to root systems. Significant progress was made by the use of algebraic techniques involving quantum groups, Hecke algebras, and combinatorial methods. The CRM organized a workshop for key researchers in the field to present an overview of current trends. This volume consists of the contributions to that workshop. Topics include basic hypergeometric functions, algebraic and representation-theoretic methods, combinatorics of symmetric functions, root systems, and the connections with integrable systems.
Book Synopsis Combinatorial Commutative Algebra by : Ezra Miller
Download or read book Combinatorial Commutative Algebra written by Ezra Miller and published by Springer Science & Business Media. This book was released on 2005-06-21 with total page 442 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recent developments are covered Contains over 100 figures and 250 exercises Includes complete proofs
Book Synopsis Encyclopaedia of Mathematics, Supplement III by : Michiel Hazewinkel
Download or read book Encyclopaedia of Mathematics, Supplement III written by Michiel Hazewinkel and published by Springer Science & Business Media. This book was released on 2007-11-23 with total page 564 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is the third supplementary volume to Kluwer's highly acclaimed twelve-volume Encyclopaedia of Mathematics. This additional volume contains nearly 500 new entries written by experts and covers developments and topics not included in the previous volumes. These entries are arranged alphabetically throughout and a detailed index is included. This supplementary volume enhances the existing twelve volumes, and together, these thirteen volumes represent the most authoritative, comprehensive and up-to-date Encyclopaedia of Mathematics available.
Book Synopsis Symmetric Functions and Combinatorial Operators on Polynomials by : Alain Lascoux
Download or read book Symmetric Functions and Combinatorial Operators on Polynomials written by Alain Lascoux and published by American Mathematical Soc.. This book was released on 2003 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 its occurrence 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 independent chapters, the author describes the main properties of these polynomials, following either the approach of Newton and interpolation methods, or the method of Cauchy and the diagonalization of a kernel generalizing the resultant. The last chapter sketches a non-commutative version of symmetric functions, with the help of Young tableaux and the plactic monoid. The book also contains numerous exercises clarifying and extending many points of the main text.
Download or read book Young Tableaux written by William Fulton and published by Cambridge University Press. This book was released on 1997 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: Describes combinatorics involving Young tableaux and their uses in representation theory and algebraic geometry.
