Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A note on exponents vs root heights for complex simple Lie algebras. | A note on exponents vs root heights for complex simple Lie algebras Sankaran Viswanath Department of Mathematics University of California Davis CA 95616 USA svis@ Submitted Sep 8 2006 Accepted Nov 26 2006 Published Dec 7 2006 Mathematics Subject Classification 05E15 Abstract We give an elementary combinatorial proof of a special case of a result due to Bazlov and Ion concerning the Fourier coefficients of the Cherednik kernel. This can be used to give yet another proof of the classical fact that for a complex simple Lie algebra g the partition formed by the exponents of g is dual to that formed by the numbers of positive roots at each height. Let g be a finite dimensional complex simple Lie algebra of rank n with associated root system A simple roots Oi i 1 n and set of positive roots A . Let Q be the root lattice of g and Q denote the set comprising Z-0 linear combinations of the Oi. For each O 2 Q let e denote the corresponding formal exponential these satisfy the usual rules e0 1 and e ạ e e. We define A Q t e_ n . Thus a typical element of A is a power series of the form 32Q Cp t e ạ where each Cạ t 2 Q t . Consider the element Ệ 2 A defined by 1 e-a Ệ n 1 Ĩ A 1 - te- oeA n 1 t - 1 e t t - 1 e-2 t2 t - 1 e-3 1 oeA Given 2 n 1 bịữị 2 Q define its height to be n ht 2 bi i 1 The main objective of this short note is to give an elementary combinatorial proof of the following proposition THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 N22 1 Proposition 1 For ft 2 A the coefficient of e 3 in Ệ is tht 3 h í 1 This proposition is the q 0 case of a more general q t theorem obtained by Bazlov 1 and Ion 2 . They consider K q t Y Y qie 2 1 qi 1 fl u 1 - tqie- 1 - tqi 1ea eA i 0 v y M y If K q t denotes the constant term coefficient of e0 of K q t one defines C q t K q t K q t upto a minor difference in convention this is called the Cherednik kernel in 2 . Bazlov and Ion compute the coefficient of e 3 in C q t for ft a positive root of g. Their approaches use