The plethysm of two Schur functions at hook, near-hook, and two-row shapes and a class of (q,t)-symmetric functions arising for plethysm.

摘要

In this work we study applications arising from the plethysm operation on symmetric functions. One of the fundamental problems in the theory of symmetric functions is to expand the plethysm of two Schur functions, sl&sqbl0;sm&sqbr0; , as a sum of Schur functions. That is, we want to find the coefficients al,m,n where slsm =nal,m ,nsn. The problem of computing the al,m,n has proven to be difficult and explicit formulas are known for only a few special cases. In Chapter 1 we study, the coefficients al,m,n when n is a partition with one or two nonzero parts (a two-row shape), n = (1a, b) (a hook shape) or n = (1a, b, c) (a near-hook shape). We make extensive use of plethystic substitution of alphabets into a symmetric function. For example, the formula slX-Y =m⊂lsm X-1 l/ms l/m' Y shows that sl [1 - x] = 0 unless l is a hook. This gives a simple proof of an elegant result previously derived by Remmel that completely characterizes the hook shapes in sl&sqbl0;sm&sqbr0; . Similarly, to study two-row and near-hook shapes we examine sl [1 + x] and sl [l + x - y], respectively. These prove more difficult than the hook case and we are only able to derive explicit formulas for special cases.; We also study of a class of symmetric functions with a parameter q introduced by Brenti [4]. These are defined based on a plethysm with the power sum symmetric functions. For example, if we denote Brenti's q-symmetric function associated with a symmetric function f as fq, then pql=qll pl . Brenti gives combinatorial interpretations for the entries in the transition matrices that express the bases eql ,hql ,mql , and sql in terms of each of the standard bases {lcub} em {rcub}, {lcub} hm {rcub}, {lcub} mm {rcub} and {lcub} sm {rcub}. We give alternate expressions for many of these that involve counting significantly fewer objects and are more recognizable as q-analogues of the transition matrices between the standard bases.; Next, we generalize Brenti's results to the hyperoctahedral group and our expressions for the transition matrices generalize naturally to this new setting. We briefly discuss generalizing Brenti's results to the wreath product of an arbitrary cyclic group and the symmetric group.; Finally, we derive several new generating functions for permutation statistics for Sn, Bn, and C 3 &m22; Sn which follow from the classical identity n≥0unhn =1n≥0 -unen