Journal of Algebraic Combinatorics

, Volume 38, Issue 2, pp 285–327

Properties of the nonsymmetric Robinson–Schensted–Knuth algorithm

Article

DOI: 10.1007/s10801-012-0404-y

Cite this article as:
Haglund, J., Mason, S. & Remmel, J. J Algebr Comb (2013) 38: 285. doi:10.1007/s10801-012-0404-y
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Abstract

We introduce a generalization of the Robinson–Schensted–Knuth insertion algorithm for semi-standard augmented fillings whose basement is an arbitrary permutation σSn. If σ is the identity, then our insertion algorithm reduces to the insertion algorithm introduced by the second author (Sémin. Lothar. Comb. 57:B57e, 2006) for semi-standard augmented fillings and if σ is the reverse of the identity, then our insertion algorithm reduces to the original Robinson–Schensted–Knuth row insertion algorithm. We use our generalized insertion algorithm to obtain new decompositions of the Schur functions into nonsymmetric elements called generalized Demazure atoms (which become Demazure atoms when σ is the identity). Other applications include Pieri rules for multiplying a generalized Demazure atom by a complete homogeneous symmetric function or an elementary symmetric function, a generalization of Knuth’s correspondence between matrices of non-negative integers and pairs of tableaux, and a version of evacuation for composition tableaux whose basement is an arbitrary permutation σ.

Keywords

Symmetric functionsPermutationsNonsymmetric Macdonald polynomialsDemazure atomsPermuted basement fillings

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of PennsylvaniaPhiladelphiaUSA
  2. 2.Department of MathematicsWake Forest UniversityWinston-SalemUSA
  3. 3.Department of MathematicsUniversity of California, San DiegoLa JollaUSA