The symmetric functions catalog

#Generalized Demazure atoms

The family of polynomials referred to as generalized Demazure atoms were introduced in [HMR12] and further studied in [TR13].

The generalized Demazure atoms are related to the fillings in the combinatorial model for non-symmetric Macdonald polynomials, [HHL08] and permuted-basement Macdonald polynomials, [Ale19], but the definitions differ. The families studied in [Ale19] and [AS19] are referred to as permuted-basement atoms and are not the same as generalized Demazure atoms.

The generalized Demazure atoms have nice properties with respect to insertion algorithms and RSK, while the permuted-basement atoms are compatible with Demazure–Lusztig operators and representation theory. For a fixed basement \(\sigma,\) Haglund–Mason–Remmel’s generalized Demazure atoms \(\widehat{E}^{\sigma}_\gamma\) refine Schur functions: \[\schurS_\beta(\xvec)=\sum_{\lambda(\gamma)=\beta} \widehat{E}^{\sigma}_\gamma(\xvec).\] Here, \(\lambda(\gamma)\) is the decreasing rearrangement of \(\gamma.\) For the reverse-identity basement this sum has a single nonzero term, so Schur functions occur as a subfamily. For the identity basement one recovers Mason’s Demazure atoms. The \(t\)-deformed basement families are recorded here as permuted-basement \(t\)-atoms.

J. L. Tiefenbruck and J. B. Remmel prove a Murnaghan–Nakayama rule for generalized Demazure atoms [TR13]. Their rule expands the product of a power-sum symmetric function with a generalized Demazure atom as a signed sum of generalized Demazure atoms.

Bibliography

1. [Ale19]Per Alexandersson. Non-symmetric Macdonald polynomials and Demazure–Lusztig operators. Séminaire Lotharingien de Combinatoire, 76, 2019.
   .bib

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   @article{Alexandersson2015gbMacdonald,
     author = {Per Alexandersson},
     title = {Non-symmetric {M}acdonald polynomials and {D}emazure--{L}usztig operators},
     journal = {Séminaire Lotharingien de Combinatoire},
     volume = {76},
     year = {2019},
     url = {https://www.mat.univie.ac.at/~slc/wpapers/s76alexand.html}
   }
   

2. [AS19]Per Alexandersson and Mehtaab Sawhney. Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$. Annals of Combinatorics, 23(2):219–239, May 2019.
   .bib

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   @article{AlexanderssonSawhney2019,
     doi = {10.1007/s00026-019-00432-z},
     url2 = {https://doi.org/10.1007/s00026-019-00432-z},
     volume = {23},
     number = {2},
     pages = {219--239},
     year = {2019},
     month = may,
     publisher = {Springer Science and Business Media {LLC}},
     author = {Per Alexandersson and Mehtaab Sawhney},
     title = {Properties of Non-symmetric {M}acdonald Polynomials at $q=1$ and $q=0$},
     journal = {Annals of Combinatorics}
   }
   

3. [HHL08]James Haglund, Mark D. Haiman and Nicholas A. Loehr. A combinatorial formula for nonsymmetric Macdonald polynomials. American Journal of Mathematics, 130(2):359–383, 2008.
   .bib

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   @article{HaglundHaimanLoehr2008,
    ISSN = {00029327, 10806377},
    URL2 = {http://www.jstor.org/stable/40068131},
    doi = {10.1353/ajm.2008.0015},
    author = {James Haglund and Mark D. Haiman and Nicholas A. Loehr},
    journal = {American Journal of Mathematics},
    number = {2},
    pages = {359--383},
    publisher = {The Johns Hopkins University Press},
    title = {A combinatorial formula for nonsymmetric {M}acdonald polynomials},
    volume = {130},
    year = {2008}
   }
   

4. [HMR12]James Haglund, Sarah K. Mason and Jeffrey B. Remmel. Properties of the nonsymmetric Robinson–Schensted–Knuth algorithm. J. Algebr. Comb, 38(2):285–327, October 2012.
   .bib

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   @article{HaglundMasonRemmel2012,
     doi = {10.1007/s10801-012-0404-y},
     year  = {2012},
     month = oct,
     publisher = {Springer Science $\mathplus$ Business Media},
     volume = {38},
     number = {2},
     pages = {285--327},
     author = {James Haglund and Sarah K. Mason and Jeffrey B. Remmel},
     title = {Properties of the nonsymmetric {R}obinson--{S}chensted--{K}nuth algorithm},
     journal = {J. Algebr. Comb}
   }
   

5. [TR13]Janine LoBue Tiefenbruck and Jeffrey B. Remmel. A Murnaghan–Nakayama rule for generalized Demazure atoms. 25th International conference on formal power series and algebraic combinatorics:969–980, 2013.
   .bib

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   @inproceedings{RemmelTiefenbruck2013,
     title = {A {M}urnaghan--{N}akayama Rule for Generalized {D}emazure Atoms},
     author = {Janine LoBue Tiefenbruck and Jeffrey B. Remmel},
     url = {https://hal.inria.fr/hal-01229701/file/dmAS0182.pdf},
     booktitle = {25th {I}nternational Conference on Formal Power Series and Algebraic Combinatorics},
     venue = {Paris},
     publisher = {Discrete Mathematics and Theoretical Computer Science},
     series = {DMTCS Proceedings},
     pages = {969--980},
     year = {2013}
   }