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Báo cáo toán học: "Superization and (q, t)-specialization in combinatorial Hopf algebras"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Superization and (q, t)-specialization in combinatorial Hopf algebras. | Superization and q -specialization in combinatorial Hopf algebras Jean-Christophe Novelli and Jean-Yves Thibon Universite Paris-Est Institut Gaspard Monge 5 Boulevard Descartes Champs-sur-Marne 77454 Marne-la-Vallee cedex 2 FRANCE novelli@univ-mlv.fr jyt@univ-mlv.fr Submitted Jul 6 2008 Accepted Aug 27 2009 Published Sep 4 2009 Mathematics Subject Classifications 05C05 16W30 18D50 Dedicated to Anders Bjorner on the occasion of his sixtieth birthday Abstract We extend a classical construction on symmetric functions the superization process to several combinatorial Hopf algebras and obtain analogs of the hook-content formula for the q -specializations of various bases. Exploiting the dendriform structures yields in particular q t -analogs of the Bjorner-Wachs q-hook-length formulas for binary trees and similar formulas for plane trees. Contents 1 Introduction 2 2 Background 4 2.1 Some conventions. 4 2.2 Noncommutative symmetric functions. 5 2.2.1 The algebra Sym. 5 2.2.2 Linear bases. 6 2.2.3 Hopf algebra structure. 6 2.2.4 The 1 q -transform. 6 2.2.5 Internal product. 7 2.3 Free quasi-symmetric functions. 8 2.4 Word quasi-symmetric functions . 8 3 Free super-quasi-symmetric functions 9 3.1 Supersymmetric functions. 9 3.2 Noncommutative supersymmetric functions. 10 3.2.1 A basis of Sym 2 . 10 3.2.2 Signed ribbons. 10 THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2 2009 R21 1 3.2.3 Internal product of Sym 2 . 11 3.3 Super-quasi-symmetric functions. 11 3.4 Free super-quasi-symmetric functions. 11 3.4.1 The superization map. 11 3.4.2 Conventions for signed words. 11 3.4.3 A combinatorial expression of the superization map . 12 4 An application the 1 t -transform 12 4.1 The canonical pro jection . 13 4.2 The dual transformation. 15 4.3 The q -specialization . 17 4.4 Hook-content formulas in FQSym . 17 4.5 Other approaches. 19 4.6 Graphical representations. 20 5 Dendriform operations and q -specialization 21 5.1 Dendriform algebras . 21 5.2 The half-products . 22 5.2.1 .