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Báo cáo toán học: "A Plethysm Formula for pµ (x) ◦ hλ (x) William F. Doran IV Department of Mathematics California Institute of Technology Pasadena, CA 91125"

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 .