Mixed Bruhat operators and Yang-Baxter equations for Weyl groups

Details Mixed Bruhat operators and Yang-Baxter equations for Weyl groups Mixed Bruhat operators and Yang-Baxter equations for Weyl groups

1998-05-18

arxiv.org/abs/math/9805079v1

Mathematics

Quantum Algebra

19 pages

Scientific paper

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 $w\in W$. (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.

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