The symmetric functions catalog

2019-07-22

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, [Ale19a], but the definitions differ. The families studied in [Ale19a] and [AS19a] 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 Demzure–Luzstig operators and representation theory.

References

• [Ale19a] Per Alexandersson. Non-symmetric Macdonald polynomials and Demazure–Lusztig operators. Séminaire Lotharingien de Combinatoire, 76, 2019.
• [AS19a] 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.
• [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.
• [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.
• [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. Discrete Mathematics and Theoretical Computer Science, 2013.