The Bruhat Ordering In Coxeter Group And The Cardinality Of Bruhat Interval

Coxeter groups play a big role in algebra, geometry, combinatorics and certain other fields of mathe-matics. The theory of Coxeter groups has been exposited from algebraic and geometric points of view in several places. The purpose of this work is to present Some results for XY < XsY and XYZ < XsytZ and the cardinality of the Bruhat interval.In chapter one, we briefly introduce some notations, definitions and propositions about reflection groups and Coxeter groups, such as length function, Bruhat ordering, subexpression etc.In chapter two, we got XY < XsY if and only if either s (?) R(X) ??(Y) or s?R(X)??(Y) (Corollary 2.1.8); XYZ < XsYtZ in certain condition by Coxeter transformation (proposition 2.2.7).In chapter three, Assume that Wn(a semi-direct product of Nn and An-1) is a Coxeter group of type Bn. Let w1,W2? Wn,w1=w'1u1,w2= w'2u1, for w'1,w'2:?Nn,u1,u2?An-1. If w1? w2 in Wn,then w'1 ? w2' in Nn(see proposition 3.2.5).In chapter four, we primarily consider the cardinality of Bruhat interval [w1, w2] = {w ? W| w1?w ? W2}(C[w11I,w2]). We got relations between C[w1,w2] and C[w1,w2s], C[w1s,w2s] by considering whether s in R(w1) or not, w1 ? w2s or w1 (?)w2s for s in R(w2)(R(w)={s ? S | ws < w})(from proposition 4.1.6 to proposition 4.1.9).