Mixed Bruhat Operators and Yang-Baxter Equations for Weyl Groups

TLDR: Verma et al. as mentioned in this paper introduced and studied a family of operators which act in the group algebra of a Weyl group W and provided a multiparameter solution to the quantum Yang-Baxter equations of the corresponding type.

Abstract: 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 ∈W,we define a graded partial order onW called the tilted Bruhat order; this partial order has 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 whose every interval is Eulerian. This generalizes the well-known results of D.-N. Verma, A. Bjorner, M. Wachs, and M. Dyer.