Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A Note on the Symmetric Powers of the Standard Representation of Sn. | A Note on the Symmetric Powers of the Standard Representation of Sn David Savitt1 Department of Mathematics Harvard University Cambridge MA 02138 USA dsavitt@ Richard P. Stanley2 Department of Mathematics Massachusetts Institute of Technology Cambridge MA 02139 USA rstan@ Submitted January 7 2000 Accepted February 12 2000 Abstract In this paper we prove that the dimension of the space spanned by the characters of the symmetric powers of the standard n-dimensional representation of Sn is asymptotic to n2 2. This is proved by using generating functions to obtain formulas for upper and lower bounds both asymptotic to n2 2 for this dimension. In particular for n 7 these characters do not span the full space of class functions on Sn. Primary AMS subject classification 05E10. Secondary 05A15 05A16 05E05. Notation Let P n denote the number of unordered partitions of n into positive integers and let fi denote the Euler totient function. Let V be the standard n-dimensional representation of Sn so that V Ce1 Cen with ơ ei eơi for Ơ 2 Sn. Let SNV denote the Nth symmetric power of V and let XN Sn Z denote its character. Finally let D n denote the dimension of the space of class functions on Sn spanned by all the XN N 0. Supported by an NSERC PGS-B fellowship 2Partially supported by NSF grant DMS-9500714 1 THE ELECTRONIC .JOURNAL OF COMBINATORICS 7 2000 R6 2 1 Preliminaries Our aim in this paper is to investigate the numbers D n . It is a fundamental problem of invariant theory to decompose the character of the symmetric powers of an irreducible representation of a finite group or more generally a reductive group . A special case with a nice theory is the reflection representation of a finite Coxeter group. This is essentially what we are looking at. The defining representation of Sn consists of the direct sum of the reflection representation and the trivial representation. This trivial summand has no significant effect on the theory. In this context
