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There is a striking similarity between Macdonald's reduced word formula andthe image of the Schubert class in the cohomology ring of the permutahedralvariety $\mathrm{Perm}_n$ as computed by Klyachko. Toward understanding thisbetter, we undertake an in-depth study of a $q$-deformation of the$\mathbb{S}_n$-invariant part of the rational cohomology ring of$\mathrm{Perm}_n$, which we call the $q$-Klyachko algebra. We uncover intimatelinks between expansions in the basis of squarefree monomials in this algebraand various notions in algebraic combinatorics, thereby connecting seeminglyunrelated results by finding a common ground to study them.  Our main results are as follows. 1) A $q$-analog of divided symmetrization($q$-DS) using Yang-Baxter elements in the Hecke algebra. It is a linear formthat picks up coefficients in the squarefree basis. 2) A relation between$q$-DS and the ideal of quasisymmetric polynomials involving work ofAval--Bergeron--Bergeron. 3) A family of polynomials in $q$ with nonnegativeintegral coefficients that specialize to Postnikov's mixed Eulerian numberswhen $q=1$. We refer to these new polynomials as remixed Eulerian numbers. For$q>0$, their normalized versions occur as probabilities in the internaldiffusion limited aggregation (IDLA) stochastic process. 4) A lift ofMacdonald's reduced word identity in the $q$-Klyachko algebra. 5) The Schubertexpansion of the Chow class of the standard split Deligne--Lusztig variety intype $A$, when $q$ is a prime power.

A $q$-deformation of an algebra of Klyachko and Macdonald's reduced word formula