Author: Publisher:
ISBN:
Category :
Languages : en
Pages : 20
Book Description
Universal Schubert Polynomials
Author: 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.
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
An Abstract Definition of Schubert Polynomials Extending to the Classical Groups
Author: Sara C. BilleyQuantum Double Schubert Polynomials
Author: 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 Polynomials and the NilCoxeter Algebra
Author: Sergej Vasilʹevič FominPublisher:
ISBN:
Category :
Languages : en
Pages : 11
Book Description
Schubert Polynomials, the Bruhat Order, and the Geometry of Flag Manifolds
Author: Nantel BergeronOn Algebraic and Combinatorial Properties of Schur and Schubert Polynomials
Author: Rudolf WinkelSymmetric Functions and Combinatorial Operators on Polynomials
Author: Alain LascouxPublisher: American Mathematical Soc.
ISBN: 0821828711
Category : Mathematics
Languages : en
Pages : 282
Book Description
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.