A 𝑞-deformation of an algebra of Klyachko, and Macdonald’s reduced word formula

There is a striking similarity between Macdonald’s reduced word formula and the image of the Schubert class in the cohomology ring of the permutahedral variety P e r m n \mathrm {Perm}_n as computed by Klyachko. Toward understanding this better, we undertake an in-depth study of a q q -deformation of the S n \mathbb {S}_n -invariant part of the rational cohomology ring of P e r m n \mathrm {Perm}_n , which we call the q q -Klyachko algebra. We uncover intimate links between expansions in the basis of squarefree monomials in this algebra and various notions in algebraic combinatorics, thereby connecting seemingly unrelated results by finding a common ground to study them. Our main results are as follows. A q q -analogue of divided symmetrization ( q q -DS) using Yang–Baxter elements in the Hecke algebra. It is a linear form that picks up coefficients in the squarefree basis. A relation between q q -DS and the ideal of quasisymmetric polynomials involving work of Aval–Bergeron–Bergeron. A family of polynomials in q q with nonnegative integral coefficients that specialize to Postnikov’s mixed Eulerian numbers when q = 1 q=1 . We refer to these new polynomials as remixed Eulerian numbers. For q > 0 q>0 , their normalized versions occur as probabilities in the internal diffusion limited aggregation stochastic process. A lift of Macdonald’s reduced word identity in the q q -Klyachko algebra. The Schubert expansion of the Chow class of the standard split Deligne–Lusztig variety in type A A , when q q is a prime power.

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