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Báo cáo toán học: "Colored trees and noncommutative symmetric functions"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Colored trees and noncommutative symmetric functions. | Colored trees and noncommutative symmetric functions Matt Szczesny Department of Mathematics Boston University Boston MA USA szczesny@math.bu.edu Submitted Oct 16 2009 Accepted Mar 28 2010 Published Apr 5 2010 Abstract Let CRFs denote the category of S-colored rooted forests and HCRFs denote its Ringel-Hall algebra as introduced in 6 . We construct a homomorphism from a K CRFs -graded version of the Hopf algebra of noncommutative symmetric functions to Hcrfs . Dualizing we obtain a homomorphism from the Connes-Kreimer Hopf algebra to a K CRFs -graded version of the algebra of quasisymmetric functions. This homomorphism is a refinement of one considered by W. Zhao in 9 . 1 Introduction In 6 categories LRF LFG of labeled rooted forests and labeled Feynman graphs where constructed and were shown to possess many features in common with those of finitary abelian categories. In particular one can define their Ringel-Hall algebras HLRF Hlfg. If C is one of these categories HC is the algebra of functions on isomorphism classes of C equipped with the convolution product f j M f A g M A 1.1 ACM and the coproduct A f M N f M N 1.2 where M N denotes disjoint union of forests graphs. Together the structures 1.1 and 1.2 assemble to form a co-commutative Hopf algebra which was in 6 shown to be dual to the corresponding Connes-Kreimer Hopf algebra 5 2 . In 6 we also defined the Grothendieck groups K0 C for C L R.F C FQ and showed that HC is naturally graded by K C - the effective cone inside Ko C . From the point of view of Ringel-Hall algebras of finitary abelian categories the characteristic functions of classes in K are interesting. If A is such a category and a G K A THE ELECTRONIC JOURNAL OF COMBINATORICS 17 2010 N19 1 we may consider Ka - the characteristic function of the locus of objects of class a inside Iso A for a precise definition see 4 . It is shown there that the Ka satisfy A Ka Kai 0 Ka2 1.3 ai a2 a ai 2eK A In this note we show that these identities hold also when