Abstract
Characterizing sets of permutations whose associated quasisymmetric function is symmetric and Schur-positive is a long-standing problem in algebraic combinatorics. In this paper, we present a general method to construct Schur-positive sets and multisets, based on geometric grid classes and the product operation. Our approach produces many new instances of Schur-positive sets and provides a broad framework that explains the existence of known such sets that until now were sporadic cases.
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Notes
Gessel was motivated by a well-known conjecture of Stanley [30, III, Ch. 21], which he reformulates as follows: If A is the set of linear extensions of a labeled poset P, then \(\mathcal {Q}(A)\) is symmetric if and only if P is isomorphic to the poset determined by a skew semistandard Young tableau.
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Acknowledgments
The authors thank Ron Adin, Michael Albert, Christos Athanasiadis, Mike Atkinson, Zach Hamaker and Bruce Sagan for useful discussions, comments and references. The authors also thank two anonymous referees for their thorough comments that have improved the presentation. The first author was partially supported by Grant #280575 from the Simons Foundation and by Grant H98230-14-1-0125 from the NSA. The second author was partially supported by Dartmouth’s Shapiro visitors fund.
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Elizalde, S., Roichman, Y. Schur-positive sets of permutations via products and grid classes. J Algebr Comb 45, 363–405 (2017). https://doi.org/10.1007/s10801-016-0710-x
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DOI: https://doi.org/10.1007/s10801-016-0710-x