Mixed Bruhat operators and Yang-Baxter equations for Weyl groups

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We introduce and study a family of operators which act in the span of a Weyl group W and provide a multi-parameter solution to the quantum Yang-Baxter equations of the corresponding type. Our operators generalize the "quantum Bruhat operators" that appear in the explicit description of the multiplicative structure of the (small) quantum cohomology ring of G/B. The main combinatorial applications concern the "tilted Bruhat order," a graded poset whose unique minimal element is an arbitrarily chosen element w∈ W. (The ordinary Bruhat order corresponds to the case w=1.) Using the mixed Bruhat operators, we prove that these posets are lexicographically shellable, and every interval in a tilted Bruhat order is Eulerian. This generalizes well known results of Verma, Bjorner, Wachs, and Dyer.