Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Some new methods in the Theory of m-Quasi-Invariants. | Some new methods in the Theory of m-Quasi-Invariants J. Bell . Garsia and N. Wallach Department of Mathematics University of California San Diego USA agarsia@ Submitted Jan 29 2005 Accepted Jul 15 2005 Published Aug 30 2005 Abstract We introduce here a new approach to the study of m-quasi-invariants. This approach consists in representing m-quasi-invariants as Ntuples of invariants. Then conditions are sought which characterize such Ntuples. We study here the case of S3 m-quasi-invariants. This leads to an interesting free module of triplets of polynomials in the elementary symmetric functions e1 e2 e3 which explains certain observed properties of S3 m-quasi-invariants. We also use basic results on finitely generated graded algebras to derive some general facts about regular sequences of Sn m-quasi-invariants 1 Introduction The ring of polynomials in x1 x2 . xn with rational coefficients will be denoted Q Xn . For P 2 QI Xn we will write P x for P xi x2 . xn . Let us denote by Sj the transposition which interchanges xi with xj. Note that for any pair i j and exponents a b we have the identities x xj xi xj x - xj r a PrrT b a 1 r if n h xi x j w7 vr 0 xj xi 11 a u xỉx a b 1 xxa b 1 r if a b. Ẳy i Ay j J r_0 i j tA c This shows that the ratio in is always a polynomial that is symmetric in xi xj. It immediately follows from that the so-called divided differencd operator ỏij x. - x 1 - Sij sends polynomials into polynomials symmetric in xi xj. THE ELECTRONIC JOURNAL OF COMBINATORICS 12 2005 R20 1 It follows from this that for any P 2 Q Xn the highest power of xi Xj that divides the difference 1 Sjj P must necessarily be odd. This given a polynomial P 2 Q Xn is said to be m-quasi-invarianf if and only if for all pairs 1 i j n the difference 1 sij P x is divisible by xi Xj 2m 1. The space of m-quasi-invariant polynomials in x1 x2 . Xn will here and after be denoted QIm Xn or briefly QIm . Clearly QIm is a vector space over Q moreover since the
