Motivated by a question in Schubert calculus, we study the interplay of quasisymmetric polynomials with 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 \(n-1\). We first show that divided symmetrization applied to a quasisymmetric polynomial in m indeterminates can be easily determined. Several examples with a strong combinatorial flavor are given. Then, we prove that the 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.

受舒伯特微积分中的一个问题的启发，我们研究了拟对称多项式与除对称化算子的相互作用，这是由 Postnikov 在置换面体的体积多项式的背景下引入的。除法对称化是一种线性形式，它作用于n 次\(n-1\)不确定项中的多项式空间。我们首先展示了应用于m 中的拟对称多项式的除对称化不确定性可以很容易地确定。给出了几个具有强烈组合风味的例子。然后，我们证明了由于 Aval-Bergeron-Bergeron 的直接和分解，可以自然地计算任何多项式的除对称化，涉及由n 个不确定项中的正度准对称多项式生成的理想。