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Mixed Bruhat operators and Yang-Baxter equations for Weyl groups. (English) Zbl 0978.22008

Introduction: We introduce and study a family of operators which act in the group algebra of a Weyl group \(W\) and provide a multiparameter solution to the quantum Yang-Baxter equations of the corresponding type. These operators are then used to derive new combinatorial properties of \(W\) and to obtain new proofs of known results concerning the Bruhat order of \(W\). The paper is organized as follows. Section 2 is devoted to preliminaries on Coxeter groups and associated Yang-Baxter equations. In Theorem 3.1 of Section 3, we describe our solution of these equations. In Section 4, we consider a certain limiting case of our solution, which leads to the quantum Bruhat operators. These operators play an important role in the explicit description of the (small) quantum cohomology ring of \(G/B\). Section 5 contains the proof of Theorem 3.1. Section 6 is devoted to combinatorial applications of our operators. For an arbitrary element \(u\in W\), we define a graded partial order on \(W\) called the tilted Bruhat order; this partial order has a unique minimal element \(u\). (The usual Bruhat order corresponds to the special case where \(u=e\), the identity element.) We then prove that tilted Bruhat orders are lexicographically shellable graded posets every interval of which is Eulerian. This generalizes the well-known results of D.-N. Verma, A. Björner, M. Wachs, and M. Dyer.

MSC:

22E30 Analysis on real and complex Lie groups
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
20F55 Reflection and Coxeter groups (group-theoretic aspects)
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