Combinatorial solutions to the reflection equation

集合论Yang-Baxter方程与反射方程中的代数结构

Algebraic structures in set-theoretic Yang-Baxter & reflection equations

该研究围绕集合论Yang-Baxter方程与反射方程的解展开，梳理brace代数性质，分析对合解的参数化解与量子代数、Drinfeld扭与Yangian的关联，推导反射相关的代数关系并证明特殊反射映射下存在反射有限子代数。

On multiplicities of points on Schubert varieties in Graszmannians II

We prove a conjecture by Kreiman and Lakshmibai on a combinatorial description of multiplicities of points on Schubert varieties in Graszmannians in terms of certain sets of reflections in the corresponding Weyl group. The proof is accomplished by setting up a bijection between these sets of reflections and the author's previous combinatorial interpretation of these multiplicities in terms of nonintersecting lattice paths (Séminaire Lotharingien Combin. 45 (2001), Article B45c; see http://www.mat.univie.ac.at/~slc/wpapers/s45kratt.html or http://www.arxiv.org/abs/math.AG/0011129).

New Solutions to the Reflection Equation with Braces

The reflection equation of Cherednik is a counterpart to the celebrated Yang-Baxter equation, with importance in the theory of integrable systems. We obtain several new solutions of the reflection equation using braces building on the work of Smoktunowicz, Vendramin and Weston. In particular, we show that every brace yields several simple solutions and that a natural class of solutions is a near-ring. We find more solutions for factorizable rings. Some of our solutions apply to the original parameter-dependent equation.