Quasisymmetric Schur functions
type: Item
instance of: scholarly article
title: Quasisymmetric Schur functions
zbMATH Open document ID: 1229.05270
author: J. Haglund
author: Sarah K. Mason
author: Stephanie van Willigenburg
author: Kurt W. Luoto
published in: Journal of Combinatorial Theory. Series A
publication date: 2011-01-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/0810.2489
review text: The authors define the quasisymmetric Schur functions as a basis for quasisymmetric functions. The functions arise from a specialization of nonsymmetric Macdonald polynomials developed by \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [``Keys \& standard bases,'' Invariant theory and tableaux, Proc. Workshop, Minneapolis/MN (USA) 1988, IMA Vol.\ Math.\ Appl.\ 19, 125--144 (1990; Zbl 0815.20013)]; the authors call the functions Demazure atoms, because they decompose Demazure characters into their smallest parts. The construction of quasisymmetric Schur functions is fully combinatorial; semistandard (reverse) tableaux indexed by partition diagrams are generalized to tableaux indexed by composition diagrams, and the quasisymmetric Schur function is defined as a generating function indexed by an enumeration of these tableaux. These quasisymmetric Schur functions have many of the properties of the classical Schur functions. There are analogues of the Kostka numbers for expressing a quasisymmetric Schur function as a sum of monomial quasisymmetric functions and a similar rule for fundamental quasisymmetric functions, and of the Pieri rule for multiplying a quasisymmetric Schur function by a row or column quasisymmetic Schur function. The parameter \(t\) can be incorporated into the construction, providing a decomposition of the Hall-Littlewood polynomial into quasisymmetric functions; this is compared to a decomposition of the Macdonald polynomials into the fundamental quasisymmetric functions by \textit{J. Haglund} [The \(q,t\)-Catalan numbers and the space of diagonal harmonics. With an appendix on the combinatorics of Macdonald polynomials. University Lecture Series 41. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1142.05074)].
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 5836882
zbMATH Keywords: compositions
zbMATH Keywords: Hall-Littlewood polynomials
zbMATH Keywords: Kostka coefficients
zbMATH Keywords: nonsymmetric Macdonald polynomials
zbMATH Keywords: Pieri rule
zbMATH Keywords: quasisymmetric functions
zbMATH Keywords: Schur function
zbMATH Keywords: tableaux
reviewed by: David Grabiner
MaRDI profile type: Publication
OpenAlex ID: W1974746744
Wikidata QID: Q56625059
arXiv ID: 0810.2489
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DOI: 10.1016/J.JCTA.2009.11.002
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description: scientific article