Séminaire Lotharingien de Combinatoire, 84B.78 (2020), 12 pp.

Philippe Nadeau and Vasu Tewari

Divided Symmetrization and Quasisymmetric Functions

Abstract. We study various aspects of the divided symmetrization operator, which was introduced by Postnikov in the context of volume polynomials of permutahedra. Divided symmetrization is a linear form which acts on the space of polynomials in n indeterminates of degree n-1. Our main results are related to quasisymmetric polynomials. We show that divided symmetrization applied to a quasisymmetric polynomial in m <= n indeterminates has a natural interpretation. We further show that divided symmetrization of any polynomial can be naturally computed with respect to a direct sum decomposition due to Aval-Bergeron-Bergeron, involving the ideal generated by positive degree quasisymmetric polynomials in n indeterminates. Our main motivation for studying divided symmetrization comes from studying the cohomology class of the Peterson variety. o

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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