标题： 星和弱星不可约完全交换元在仿射类型 Coxeter 群中的研究 $\widetilde{B}$ 和 $\widetilde{D}$ 显示英文标题

标题： Star and weak star irreducible fully commutative elements in Coxeter groups of affine types $\widetilde{B}$ and $\widetilde{D}$

摘要： 星运算，最初由Kazhdan和Lusztig引入，后来由Ernst专门化为在Coxeter群的完全交换元素集合上的所谓弱星约简。 在本文中，我们对仿射类型$\widetilde{B}_{n+1}$和$\widetilde{D}_{n+2}$的Coxeter群中的星不可约和弱星不可约完全交换元素进行分类。 然后聚焦于类型$\widetilde{D}_{n+2}$的情况，我们利用星不可约元素的分类，提供对应广义Temperley-Lieb代数的图示表示的忠实性的新证明，并给出Lusztig的$\mathbf{a}$-函数的显式描述。 显示英文摘要

摘要： The star operation, originally introduced by Kazhdan and Lusztig, was later specialized by Ernst to the so-called weak star reduction on the set of fully commutative elements of a Coxeter group. In this paper, we classify the star and weak star irreducible fully commutative elements in Coxeter groups of affine types $\widetilde{B}_{n+1}$ and $\widetilde{D}_{n+2}$. Focusing then on the case of type $\widetilde{D}_{n+2}$, we use the classification of star irreducible elements to provide a new proof of the faithfulness of a diagrammatic representation of the corresponding generalized Temperley-Lieb algebra, along with an explicit description of Lusztig's $\mathbf{a}$-function.