1998/05/18 by Francesco Brenti, Sergey Fomin, Brenti, Francesco +3 · 1 citation
Mathematics · #05E15 #06A07 #14M15 #20F55 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.AG #math.CO #math.QA #math.RT #msc:05E15 #msc:06A07 #msc:14M15 #msc:20F55
paper · pdf · doi:10.48550/arxiv.math/9805079
19 pages
arxiv created 1998/05/18 · openalex publication_date 1998/05/18 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.