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

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