【摘要】：Coxeter群在代數(shù),幾何,組合學和其它數(shù)學領域都有重要應用。在很多地方都用了代數(shù)和幾何觀點來闡述Coxeter群理論。這篇文章的主要目的是研究XYXsY,XYZ XsYtZ成立的條件和Bruhat區(qū)間的基數(shù)。在第一章中,我們簡要介紹了一些關于反射群和Coxeter群的相關概念,定義和命題,如長度函數(shù)、Bruhat序、子表達等。在第二章中,我們得首先到了XYXsY充要條件是要么s(?)R(X)∪￡(Y),要么s∈R(X)∩￡(Y)(見推論2.1.8);其次,在一定的條件下通過Coxeter變換我們可以得到XYZXsYtZ(見命題2.2.7)。在第三章中,設W(Nn和Hn的一個半直積)是Bn型Coxeter群。設w1,w2 ∈ ,令w1=w'1u1,w2= w'2u1,其中w'1,w'2∈Nn,u1,u2∈An-1。若w1 ≤ w2在Wn 中成立,則 w'1 ≤ w2'在Nn中成立(見第三章命題3.2.5)。在第四章中,我們主要討論了Bruhat區(qū)間的基數(shù)(C[w1,w2])。通過討論s是否在R(w1)中,w1 ≤ w2s或w1(?)w2s,我們得到了C[w1, w2]和C[w1,w2s],C[w1s,w2s], 者之間的關系(主要見命題4.1.6到命題4.1.9)。
[Abstract]:Coxeter groups have important applications in algebra, geometry, combinatorial science and other mathematical fields. In many places, algebra and geometry are used to explain Coxeter group theory. The main purpose of this paper is to study the conditions under which XYXsY,XYZ XsYtZ holds and the cardinality of Bruhat intervals. In the first chapter, we briefly introduce some related concepts, definitions and propositions about reflection group and Coxeter group, such as length function, Bruhat order, subexpression and so on. In the second chapter, we first get to the XYXsY if and only if either s (?) R (X) / (Y), or s / R (X) / (Y) (see corollary 2.1.8); Secondly, we can get XYZXsYtZ by Coxeter transformation under certain conditions (see Proposition 2.2.7). In Chapter 3, let W (Nn and Hn be a semi-direct product of type B n Coxeter group. Let W _ 1, W _ 2 鈭，

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