Báo cáo toán học: "The plethysm sλ[sµ] at hook and near-hook shapes"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: The plethysm sλ[sµ] at hook and near-hook shapes. | The plethysm a sJ at hook and near-hook shapes . Langley Department of Mathematics Rose-Hulman Institute of Technology Terre Haute IN 47803 . Remmel Department of Mathematics University of California San Diego La Jolla CA 92093 remmel@ Submitted Oct 18 2002 Accepted Mar 9 2003 Published Jan 23 2004 MR Subject Classifications 05E05 05E10 Abstract We completely characterize the appearance of Schur functions corresponding to partitions of the form V 1 b hook shapes in the Schur function expansion of the plethysm of two Schur functions sA M 12 ax v sv V Specifically we show that no Schur functions corresponding to hook shapes occur unless A and 1 are both hook shapes and give a new proof of a result of Carbonara Remmel and Yang that a single hook shape occurs in the expansion of the plethysm s ic d s 1 We also consider the problem of adding a row or column so that V is of the form 1a b c or 1a 2b c . This proves considerably more difficult than the hook case and we discuss these difficulties while deriving explicit formulas for a special case. 1 Introduction One of the fundamental problems in the theory of symmetric functions is to expand the plethysm of two Schur functions 3a sm as a sum of Schur functions. That is we want to find the coefficients ax ạ V where sA sM 52 ax v sv. V THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R11 1 X a I In general the problem of expanding different products of Schur functions as a sum of Schur functions arises in the representation theory of the symmetric group Sn. Specifically let CA be the conjugacy class of Sn associated with a partition A. Define a function 1a Sn C by setting D a x ơ E Ca for all ơ E Sn where for a statement A 0 if A is true 1 if A is false Let A n denote that A is a partition of the positive integer n. Then the set 1A AHn forms a basis for C Sn the center of the group algebra of Sn. There is a fundamental isometry between C Sn and An the vector space of homogeneous .