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Combinatorics of the Diagonal Harmonics

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Recent Trends in Algebraic Combinatorics

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 16))

Abstract

The Shuffle Theorem, recently proven by Carlsson and Mellit, states that the bigraded Frobenius characteristic of the diagonal harmonics is equal to a weighted sum of parking functions. In this introduction to the topic, we discuss the theorem and connections between it and the well-known Macdonald polynomials. Furthermore, we describe important combinatorial bijections which imply various restatements of the theorem and play an important role in its proof. Finally, we briefly discuss the proof and describe various generalizations of the theorem.

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Acknowledgements

The author would like to express her gratitude for the many helpful remarks of the anonymous reviewer.

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Authors and Affiliations

  1. Lehigh University, Bethlehem, PA, USA

    Angela Hicks

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  1. Angela Hicks
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Correspondence to Angela Hicks .

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  1. Mathematical Sciences Research Institute , Berkeley, CA, USA

    Hélène Barcelo

  2. Department of Mathematics, Pomona College, Claremont, CA, USA

    Gizem Karaali

  3. Department of Mathematics, Dartmouth College, Hanover, NH, USA

    Rosa Orellana

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Hicks, A. (2019). Combinatorics of the Diagonal Harmonics. In: Barcelo, H., Karaali, G., Orellana, R. (eds) Recent Trends in Algebraic Combinatorics. Association for Women in Mathematics Series, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-030-05141-9_5

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