tailieunhanh - Báo cáo toán học: "A note on exponents vs root heights for complex simple Lie algebras"

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