Journal of Combinatorics
Volume 9 (2018)
Number 2
Shifted dual equivalence and Schur -positivity
Pages: 279 – 308
DOI: https://dx.doi.org/10.4310/JOC.2018.v9.n2.a4
Author
Sami Assaf (Department of Mathematics, University of Southern California, Los Angeles, Calif., U.S.A.)
Abstract
By considering type analogs of permutations and tableaux, we extend abstract dual equivalence to type in two directions. In one direction, we define involutions on shifted tableaux that give a dual equivalence, thereby giving a new combinatorial proof of the Schur positivity of Schur - and -functions. In another direction, we define an abstract shifted dual equivalence parallel to dual equivalence and prove that it can be used to establish Schur -positivity of a function expressed as a sum of shifted fundamental quasisymmetric functions. As a first application, we give a new combinatorial proof that the product of Schur -functions is Schur -positive.
The author’s work was supported in part by NSF grant DMS-1265728.
Received 12 August 2015
Published 22 January 2018