Noncommutative Symmetric Functions, Lie Series and Descent Algebras

Abstract

We obtain the Schur positivity of spider graphs of the forms S(a, 2, 1) and S(a, 4, 1), which are considered to have the simpliest structures for which the Schur positivity was unknown. The proof outline has four steps. First, we find noncommutative analogs for the chromatic symmetric functions of the spider graphs S(a, b, 1). Secondly, we expand the analogs under the Λ-and R-bases, whose commutative images are the elementary and skew Schur symmetric functions, respectively. Thirdly, we recognize the Schur coefficients via the Littlewood-Richardson rule in terms of norms of multisets of Yamanouchi words. At last we establish the Schur positivity combinatorially together with the aid of computer assistance. Contents 1. Introduction 2. Preliminaries 2.1. Compositions, partitions and tableaux 2.2. Ribbons and ribbon tableaux 2.3. Commutative symmetric functions 2.4. Noncommutative symmetric functions 2.5. Maps between multisets 3. Noncommutative analogs of X Pn and X S(a,b,1) 4. The spiders S(a, 2, 1) and S(a, 4, 1) are Schur positive 4.1. The spiders S(a, 2, 1) are Schur positive 4.2. The spiders S(a, 4, 1) are Schur positive References Appendix A. A proof of Proposition 4.5