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Über dieses Buch

This book gives an introduction to the very active field of combinatorics of affine Schubert calculus, explains the current state of the art, and states the current open problems. Affine Schubert calculus lies at the crossroads of combinatorics, geometry, and representation theory. Its modern development is motivated by two seemingly unrelated directions. One is the introduction of k-Schur functions in the study of Macdonald polynomial positivity, a mostly combinatorial branch of symmetric function theory. The other direction is the study of the Schubert bases of the (co)homology of the affine Grassmannian, an algebro-topological formulation of a problem in enumerative geometry.

This is the first introductory text on this subject. It contains many examples in Sage, a free open source general purpose mathematical software system, to entice the reader to investigate the open problems. This book is written for advanced undergraduate and graduate students, as well as researchers, who want to become familiar with this fascinating new field.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Affine Schubert calculus is a subject that lies at the crossroads of combinatorics, geometry, and representation theory. Its modern development is motivated by two seemingly unrelated directions. One is the introduction of k-Schur functions in the study of Macdonald polynomial positivity, a mostly combinatorial branch of symmetric function theory.
Thomas Lam, Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono, Mike Zabrocki

Chapter 2. Primer on k-Schur Functions

The purpose of this chapter is to outline some of the results and open problems related to k-Schur functions, mostly in the setting of symmetric function theory. This chapter roughly follows the outline of several talks given by Luc Lapointe and Jennifer Morse at a conference titled “Affine Schubert Calculus” held in July of 2010 at the Fields Institute in Toronto.
Thomas Lam, Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono, Mike Zabrocki

Chapter 3. Stanley Symmetric Functions and Peterson Algebras

This purpose of this chapter is to introduce Stanley symmetric functions and affine Stanley symmetric functions from the combinatorial and algebraic point of view. The presentation roughly follows three lectures I gave at a conference titled “Affine Schubert Calculus” held in July of 2010 at the Fields Institute in Toronto.
Thomas Lam, Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono, Mike Zabrocki

Chapter 4. Affine Schubert Calculus

This chapter discusses how k-Schur and dual k-Schur functions can be defined for all types. This is done via some combinatorial problems that come from the geometry of a very large family of generalized flag varieties. They apply to the expansion of products of Schur functions, k-Schur functions and their dual basis, and Schubert polynomials. Despite the geometric origin of these problems, concrete algebraic models will be given for the relevant cohomology rings and their Schubert bases.
Thomas Lam, Luc Lapointe, Jennifer Morse, Anne Schilling, Mark Shimozono, Mike Zabrocki

Backmatter

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