Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Depth reduction of a class of Witten zeta functions. | Depth reduction of a class of Witten zeta functions Xia Zhou Department of mathematics Zhejiang University Hangzhou 310027 David M. Bradley Department of Mathematics Statistics University of Maine 5752 Neville Hall Orono Maine 04469-5752 xiazhou0821@ bradley@ dbradley@ Tianxin Cai Department of mathematics Zhejiang University Hangzhou 310027 caitianxin@ Submitted Apr 6 2008 Accepted Jul 21 2009 Published Jul 31 2009 Mathematics Subject Classifications 11A07 11A63 Abstract We show that if a b c d f are positive integers such that a b c d f is even then the Witten zeta value Cst 4 a b c d 0 f is expressible in terms of Witten zeta functions with fewer arguments. 1 Introduction Let N be the set of positive integers Q the field of rational numbers C the field of complex numbers. For any semisimple Lie algebra g the Witten zeta function cf. 5 is defined by C s dim p -s p where s G C and p runs over all finite dimensional irreducible representations of g. In order the calculate the volumes of certain moduli space Witten 7 introduced the values The first and third authors are supported by the National Natural Science Foundation of China Project 10871169. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2009 N27 1 Zg 2k for k G N and showed that n 2klZg 2k G Q where l is the number of positive roots of g. For positive integer r Matsumoto and Tsumura 5 defined a multi-variate extension called the Witten multiple zeta-function associated with sl r 1 by Cst r 1 s ro r r-j 1 j k-1 X Ẽ nn Ẹm. . mr 1 j 1 k 1 v k 1 . . . G r . . . . In particular 5 section 2 Prop if m G N we denote s r 1 2m k - j Gl r 1 2m . . 2m . 1 j k r 1 r r 1 2 As in 1 given the Witten multiple zeta-function 1 we define the depth to be r. Further if the zeta functions y1 . yk have depth r1 . rk respectively then for a1 . ak G C we define the depth of a1y1 akyk to be max ri 1 i k . We would like to know which sums can be expressed in .
