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\(k\)-Schur functions and affine Schubert calculus. (English) Zbl 1360.14004
Fields Institute Monographs 33. New York, NY: Springer; Toronto: The Fields Institute for Research in the Mathematical Sciences (ISBN 978-1-4939-0681-9/hbk; 978-1-4939-0682-6/ebook). viii, 219 p. (2014).
This book presents the current state of the art and ongoing work on affine Schubert calculus with an accent on the combinatorics of a family of polynomials called \(k\)-Schur functions. Several generalizations of \(k\)-Schur functions are also discussed.
In [Duke Math. J. 116, No. 1, 103–146 (2003; Zbl 1020.05069)], L. Lapointe et al. found computational evidence of a conjectural property for a family of new bases for a filtration on the symmetric function space: the property is that Macdonald polynomials expand positively in terms of it (see Section 4.11). This gave rise to \(k\)-Schur functions, which then were proven to be connected to a vast set of subjects, see the introduction of the book.
Chapter 2 (which occupies 2/3 of the book) presents basics on \(k\)-Schur functions, emphasizing combinatoric aspects in the symmetric function setting. In particular, \(k\)-Pieri rule for the product of \(k\)-Schur functions is discussed. Also, \(k\)-Schur functions (resp. their duals) generate so called strong (resp., weak) tableaux, which is explained by means of an affine insertion algorithm. A lot of example in Sage is given, the authors hope that this will encourage the reader to generate new data and new conjectures.
Chapter 3 explains the combinatorial connections between Stanley symmetric functions (appeared when Stanley was enumerating reduced words in the symmetric group) and \(k\)-Schur functions, using root systems, nilCoxeter and nilHecke rings. There are exercises in this chapter. Several geometric interpretations of the material are listed at the end of this chapter.
T. Lam showed [Am. J. Math. 128, No. 6, 1553–1586 (2006; Zbl 1107.05095)] that the dual \(k\)-Schur functions are a special case of affine analogs of Stanley symmetric functions. Then, the way how Stanley symmetric functions are related to nilCoxeter algebra [S. Fomin and R. P. Stanley, Adv. Math. 103, No. 2, 196–207 (1994; Zbl 0809.05091)] can be reproduced in the affine setting.
Chapter 4 presents the nilHecke ring in the general Kac-Moody setting, and then this ideology is applied for affine Grassmannians. The nilHecke ring was introduced to study the torus equivariant cohomology of Kac-Moody partial flag varieties, and so this chapter presents this geometric aspect of the story. The algebraic part of correspondence between polynomial representatives for the Schubert classes of the affine Grassmannian, and \(k\)-Schur functions in homology and the dual \(k\)-Schur functions in cohomology is presented.

14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14N15 Classical problems, Schubert calculus
14M17 Homogeneous spaces and generalizations
05E05 Symmetric functions and generalizations
05E10 Combinatorial aspects of representation theory
20G44 Kac-Moody groups
33D52 Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.)
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