Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: A Plethysm Formula for pµ (x) ◦ hλ (x) William F. Doran IV Department of Mathematics California Institute of Technology Pasadena, CA 91125. | A Plethysm Formula for p x o h x William F. Doran IV Department of Mathematics California Institute of Technology Pasadena CA 91125 . Submitted September 10 1996 Accepted May 2 1997 Abstract This paper gives a new formula for the plethysm of power-sum symmetric functions and complete symmetric functions. The form of the main result is that for 1 p b and A p a with length t then p x o h x majc T ssh T x where the sum is over semistandard tableaux of weight A A Ab and m lj is a root of unity which depends on fl t and T. 1. Introduction This paper gives a formula for the plethysm of power-sum symmetric functions and complete symmetric functions. In Section 1 some tableaux and symmetric function notation is given. In Section 2 the work of D is reviewed. The key result in this section is that for 1 p b p x o ha x majM T Ssh T x where the sum is over all semistandard tableaux T of weight ab and majM T is a root of unity which depends on T and fl. In Section 3 a technical result which is needed later is proven. In Section 4 the main result of the paper is proven. The form of this result is that for l p b and A p a with length t p x o h x T Ssh T x where the sum is over semistandard tableaux T of weight A A . Ab and m lj is a root of unity which depends on l t and T. Keywords Symmetric Functions Plethysm Mathematics Subject Classification Primary 05E05 Secondary 05E10 the electronic journal of combinatorics 4 1997 R14 2 . A. of n is a weakly decreasing sequence A Al Az of positive integers which sum to n. Either A n or A h n is used to denote that A is a partition of n. The value l is called the - of A and is denoted l A . Let A i j 1 i l A and 1 j Ai c Z2. The set A is the. of A and is thought of as a collection of boxes arranged using matrix coordinates. A. of n is a sequence a 1 . ai of positive integers which sum to n. For notational purposes let al1 a b . ab denote the composition a1y_-- ai 0 2 . a2 . ai . 0 . b1 times b2 times bl times Also given a composition a 1 .
