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k-Schur Functions and Affine Schubert Calculus pp 1–7Cite as

Introduction

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Part of the book series: Fields Institute Monographs ((FIM,volume 33))

Abstract

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.

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Acknowledgements

This material is based upon work supported by the National Science Foundation under Grant No. DMS-0652641. “Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.” We are grateful to the Fields Institute in Toronto for helping organize and support the summer school and workshop on “Affine Schubert calculus”.

We would like to thank Tom Denton and Karola Mészáros for helpful comments and additions on Chap. 2, and Jason Bandlow, Chris Berg, Nicolas M. Thiéry as well as many other Sage developers for their help with the Sage implementations.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

    Thomas Lam

  2. Instituto de Matemática y Física, Universidad de Talca, Talca, Chile

    Luc Lapointe

  3. Department of Mathematics, Drexel University, Philadelphia, PA, USA

    Jennifer Morse

  4. Department of Mathematics, University of California, Davis, CA, USA

    Anne Schilling

  5. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Mark Shimozono

  6. Mathematics and Statistics, York University, Toronto, ON, Canada

    Mike Zabrocki

Authors
  1. Thomas Lam
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  2. Luc Lapointe
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  3. Jennifer Morse
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  4. Anne Schilling
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  5. Mark Shimozono
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  6. Mike Zabrocki
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Appendix: Sage

Appendix: Sage

Sage [151] is a completely open source general purpose mathematical software system, which appeared under the leadership of William Stein (University of Washington) and has developed explosively within the last 5 years. It is similar to Maple, MuPAD, Mathematica, Magma, and up to some point Matlab, and is based on the popular Python programming language. Sage has gained strong momentum in the mathematics community far beyond its initial focus in number theory, in particular in the field of combinatorics, see [140].

Tutorials and instructions on how to install Sage can be found at the main Sage website http://www.sagemath.org/. For example, for the basic Sage syntax and programming tricks see http://www.sagemath.org/doc/tutorial/programming.html.

Many aspects related to k-Schur functions and symmetric functions in general have been implemented in Sage and in fact are still being developed as an on-going project. Throughout the text we provide many examples on how to use Sage to do calculations related to k-Schur functions. Further information about the latest code and developments can be obtained from the Sage-Combinat website [140]. We suggest that the interested reader uses Sage version 5.13 or later to ensure that all features used in this book have been incorporated.

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Lam, T., Lapointe, L., Morse, J., Schilling, A., Shimozono, M., Zabrocki, M. (2014). Introduction. In: k-Schur Functions and Affine Schubert Calculus. Fields Institute Monographs, vol 33. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-0682-6_1

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