Symmetric Functions, Schubert Polynomials and Degeneracy Loci - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Symmetric Functions, Schubert Polynomials and Degeneracy Loci write by Laurent Manivel. This book was released on 2001. Symmetric Functions, Schubert Polynomials and Degeneracy Loci available in PDF, EPUB and Kindle. 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
Schubert Varieties and Degeneracy Loci - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Schubert Varieties and Degeneracy Loci write by William Fulton. This book was released on 2006-11-13. Schubert Varieties and Degeneracy Loci available in PDF, EPUB and Kindle. 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
Symmetric Functions and Combinatorial Operators on Polynomials - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Symmetric Functions and Combinatorial Operators on Polynomials write by Alain Lascoux. This book was released on 2003. Symmetric Functions and Combinatorial Operators on Polynomials available in PDF, EPUB and Kindle. 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.
Advances in Algebra
Advances in Algebra - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Advances in Algebra write by Jörg Feldvoss. This book was released on 2019-02-27. Advances in Algebra available in PDF, EPUB and Kindle. 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
Schubert Calculus and Its Applications in Combinatorics and Representation Theory - read free eBook in online reader or directly download on the web page. Select files or add your book in reader. Download and read online ebook Schubert Calculus and Its Applications in Combinatorics and Representation Theory write by Jianxun Hu. This book was released on 2020-10-24. Schubert Calculus and Its Applications in Combinatorics and Representation Theory available in PDF, EPUB and Kindle. 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.
