Some KL Coefficients For An Affine Weyl Group Of Type ???₃ And Cells Of The Symmetric Groups

In Kazhdan-Lusztig?abbreviated as"KL"?theory,the KL polynomial is a core research object.For Weyl groups or affine Weyl groups,the leading coefficient of the KL polynomial?abbreviated as"the KL coefficient"?plays a very important role,which is not only help to understand all the KL polynomials,but also has a wide range of applications in lie theory and representation theory.However,what we know about the KL coefficients is very limited.In order to study the non-local finiteness of W-graphs for affine Weyl groups,Lusztig G introduced some semi-linear equations related to the KL coefficients,which provides an important method for solving the KL coefficients.Cell theory is another core theory in KL theory,which plays an important role in studying the algebraic structure of Coxeter groups.For the cells decomposition of symmetric groupsS_n and affine Weyl groups of type ???_n,professor Shi Jianyi gave a general theory.In view of existing researches,this dissertation mainly uses the semi-linear equation to calculate some KL coefficients of affine Weyl group of type ???₃.In addition,by using computer programming,the standard Young tableau is built by Matlab,the number of cells of some symmetric groups is obtained.The main results of this paper are as follows:First part,assume that W₀ is a Weyl group of type A₃,W is the affine Weyl group of type ???₃,???is the corresponding extended affine Weyl group.There are 5 two-sided cells in ???,andC₁,C ₂,C₃are three two-sided cells of them.We use the method of solving semi-linear equations to get some KL coefficients ????x,y?.Here x???C₁UC₂UC₃,y???C₂.Second part,for the ???₃ affine Weyl group,in order to get the KL coefficients on other two-sided cells,we give some results on the lowest two-sided cell C₀,and calculate the set d_w.In addition,in order to describe the relationship between group elements more clearly,we map group elements to corresponding coordinates,which lays a foundation for the subsequent calculation of KL coefficients on C₀.Third part,in the symmetric group S_n,for any given permutation element,with the help of Matlab software,the corresponding standard Young tableau is given.Secondly,using Matlab software,we study the two-sided cells and right?left?cells of S_n,and get the number of them corresponding to S_n under certain conditions.