Author: Ian Grant MacdonaldPublisher: Dép. de mathématique et d'informatique, Université du Québec à Montréal
ISBN:
Category : Mathematics
Languages : en
Pages : 138
Book Description
Notes on Schubert Polynomials
Author: Ian Grant MacdonaldPublisher: Dép. de mathématique et d'informatique, Université du Québec à Montréal
ISBN:
Category : Mathematics
Languages : en
Pages : 138
Book Description
Symmetric Functions, Schubert Polynomials and Degeneracy Loci
Author: Laurent ManivelPublisher: American Mathematical Soc.
ISBN: 9780821821541
Category : Computers
Languages : en
Pages : 180
Book Description
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
Author: William FultonPublisher: Springer
ISBN: 3540698043
Category : Mathematics
Languages : en
Pages : 158
Book Description
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.
An Abstract Definition of Schubert Polynomials Extending to the Classical Groups
Author: Sara C. BilleyEnumerative Combinatorics: Volume 2
Author: Richard P. StanleyPublisher: Cambridge University Press
ISBN: 9780521789875
Category : Mathematics
Languages : en
Pages : 600
Book Description
An introduction, suitable for beginning graduate students, showing connections to other areas of mathematics.
Combinatorics of Schubert Polynomials
Author: Avery J St. DizierPublisher:
ISBN:
Category :
Languages : en
Pages : 95
Book Description
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.
Schubert Calculus and Its Applications in Combinatorics and Representation Theory
Author: Jianxun HuPublisher: Springer Nature
ISBN: 9811574510
Category : Mathematics
Languages : en
Pages : 367
Book Description
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.
Universal Schubert Polynomials
Author: The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics
Author: James HaglundPublisher: American Mathematical Soc.
ISBN: 0821844113
Category : Mathematics
Languages : en
Pages : 178
Book Description
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.
Schubert Polynomials and the NilCoxeter Algebra
Author: Sergej Vasilʹevič FominPublisher:
ISBN:
Category :
Languages : en
Pages : 11
Book Description