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Coxeter Groups and Hopf Algebras I Marcelo Aguiar and Swapneel

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Coxeter Groups and Hopf Algebras I Marcelo Aguiar and Swapneel Coxeter groups and Hopf algebras I Marcelo Aguiar and Swapneel Mahajan Department of Mathematics, Texas A&M University, College Station, Texas 77843, USA E-mail address: [email protected] URL: http://www.math.tamu.edu/∼maguiar Department of Mathematics, Indian Institute of Technology, Powai, Mumbai 400 076, India E-mail address: [email protected] URL: http://www.math.iitb.ac.in/∼swapneel 2000 Mathematics Subject Classification. 05E05, 06A11, 06A15, 16W30, 51E24. Key words and phrases. Coxeter group; descent; global descent; hyperplane arrangement; left regular band (LRB); projection map; projection poset; lunes; shuffles; bilinear forms; descent algebra; semisimple; Hopf algebra; (quasi) symmetric functions; (co)algebra axioms; (nested) set partitions; (nested) set compositions; (co)free (co)algebra. Foreword In the study of a mathematical system, algebraic structures allow for the discovery of more information. This is the motor behind the success of many areas of mathematics such as algebraic geometry, algebraic combinatorics, algebraic topology and others. This was certainly the motivation behind the observation of G.-C. Rota stating that various combinatorial objects possess natural product and coproduct structures. These struc- tures give rise to a graded Hopf algebra, which is usually referred to as a combinatorial Hopf algebra. Typically, it is a graded vector space where the homogeneous components are spanned by finite sets of combinatorial objects of a given type and the algebraic structures are given by some constructions on those objects. Recent foundational work has constructed many interesting combinatorial Hopf al- gebras and uncovered new connections between diverse subjects such as combinatorics, algebra, geometry, and theoretical physics. This has expanded the new and vibrant subject of combinatorial Hopf algebras. To give a few instances: • Connes and Kreimer showed that a certain renormalization problem in quantum field theory can be encoded and solved using a Hopf algebra indexed by rooted trees. • Loday and Ronco showed that a Hopf algebra indexed by planar binary trees is the free dendriform algebra on one generator. This is true for many types of algebras; the free algebra on one generator is a combinatorial Hopf algebra. • In the context of polytope theory, some interesting enumerative combinatorial in- variants induce a Hopf morphism from a Hopf algebra of posets to the Hopf algebra of quasi-symmetric functions. • Krob and Thibon showed that the representation theory of the Hecke algebras at q = 0 is intimately related to the Hopf algebra structure of quasi-symmetric functions and non-commutative symmetric functions. Some of the latest research in these areas has been the subject of a series of recent meetings, including an AMS/CMS meeting in Montr´eal in May 2002, a BIRS workshop in Banff in August 2004, and a CIRM workshop in Luminy in April 2005. It was suggested at the BIRS meeting that the draft text of M. Aguiar and S. Mahajan be expanded into the first monograph on the subject. Both are outstanding communicators. Their unified geometric approach using Coxeter complexes and projection maps allows us to construct many of the combinatorial Hopf algebras currently under study and further to understand their properties (freeness, cofreeness, etc.) and to describe morphisms among them. The current monograph is the result of this great effort and it is for me a great pleasure to introduce it. Nantel Bergeron Canada Research Chair York University 3 4 Contents Preface i 0.1 Thefirstpart:Chapters1-3 . i 0.2 Thesecondpart:Chapters4-8 . i 0.3 Futurework................................... i 0.4 Acknowledgements ............................... ii 0.5 Notation..................................... ii 1 Coxeter groups 1 1.1 Regular cell complexes and simplicial complexes . ........ 1 1.1.1 Gateproperty.............................. 1 1.1.2 Linkandjoin.............................. 2 1.2 Hyperplanearrangements . 2 1.2.1 Faces .................................. 2 1.2.2 Flats................................... 3 1.2.3 Sphericalpicture ............................ 3 1.2.4 Gatepropertyandotherfacts. 4 1.3 Reflectionarrangements . 4 1.3.1 Finitereflectiongroups . .. .. .. .. .. .. .. .. .. 4 1.3.2 Typesoffaces.............................. 5 1.3.3 TheCoxeterdiagram ......................... 5 1.3.4 Thedistancemap ........................... 6 1.3.5 TheBruhatorder............................ 6 1.3.6 The descent algebra: A geometric approach . ... 7 1.3.7 Linkandjoin.............................. 7 1.4 The Coxeter group of type An−1 ....................... 8 1.4.1 Thebraidarrangement . .. .. .. .. .. .. .. .. .. 8 1.4.2 Typesoffaces.............................. 9 1.4.3 Setcompositionsandpartitions. .. 9 1.4.4 TheBruhatorder............................ 10 2 Left regular bands 11 2.1 WhyLRBs?................................... 11 2.2 Facesandflats ................................. 12 2.2.1 Faces .................................. 12 2.2.2 Flats................................... 12 2.2.3 Chambers................................ 12 2.2.4 Examples ................................ 13 2.3 Pointedfacesandlunes ............................ 13 2.3.1 Pointedfaces .............................. 13 2.3.2 Lunes .................................. 13 2.3.3 TherelationofQandZwithΣandL . 14 2.3.4 Lunarregions.............................. 14 5 6 CONTENTS 2.3.5 Examples ................................ 15 2.4 LinkandjoinofLRBs ............................. 16 2.4.1 SubLRBandquotientLRB . 17 2.4.2 ProductofLRBs ............................ 17 2.5 BilinearformsrelatedtoaLRB. .. 17 2.5.1 The bilinear form on KQ ....................... 17 2.5.2 The pairing between KQ and KΣ................... 18 2.5.3 The bilinear form on KΣ ....................... 19 2.5.4 The bilinear form on KL........................ 19 2.5.5 The nondegeneracy of the form on KL................ 20 2.6 BilinearformsrelatedtoaCoxetergroup . ..... 21 2.6.1 The bilinear form on (KΣ)W ..................... 22 2.6.2 The bilinear form on (KL)W anditsnondegeneracy . 23 2.7 Projectionposets ................................ 24 2.7.1 Definitionandexamples . 24 2.7.2 Elementaryfacts ............................ 25 3 Hopf algebras 27 3.1 Hopfalgebras.................................. 27 3.1.1 Cofreegradedcoalgebras . 27 3.1.2 Thecoradicalfiltration . 28 3.1.3 Antipode ................................ 28 3.2 Hopfalgebras:Examples. 29 3.2.1 TheHopfalgebraΛ .......................... 29 3.2.2 TheHopfalgebraQΛ ......................... 31 3.2.3 TheHopfalgebraNΛ ......................... 32 3.2.4 ThedualitybetweenQΛandNΛ . 32 4 A brief overview 33 4.1 Abstract:Chapter5 .............................. 33 4.2 Abstract:Chapter6 .............................. 34 4.3 Abstract:Chapters7and8 . 35 5 The descent theory for Coxeter groups 37 5.1 Introduction................................... 37 5.1.1 Thefirstpart: Sections5.2-5.5 . 37 5.1.2 Thesecondpart: Sections5.6-5.7. .. 38 5.2 ThedescenttheoryforCoxetergroups . .... 39 5.2.1 Preliminaries .............................. 39 5.2.2 Summary ................................ 39 5.2.3 The posets Z and L .......................... 40 5.2.4 The partial orders on C×C andQ .................. 40 5.2.5 ThemapRoad ............................. 42 5.2.6 ThemapGRoad ............................ 43 5.2.7 ThemapΘ ............................... 44 5.2.8 Connectionamongthethreemaps . 45 5.3 The coinvariant descent theory for Coxeter groups . ........ 46 5.3.1 Themapdes .............................. 46 5.3.2 Themapgdes.............................. 47 5.3.3 The map θ ............................... 47 5.3.4 Connectionamongthethreemaps . 48 5.3.5 Shuffles ................................. 49 5.3.6 Sets related to the product in the M basisofSΛ .......... 51 5.4 The example of type An−1 ........................... 52 CONTENTS 7 5.4.1 The posets Σn and Ln ......................... 53 5.4.2 The posets Qn and Zn ......................... 53 n n 5.4.3 The quotient posets Q and L .................... 54 5.4.4 ThemapsRoad,GRoadandΘ . 54 5.4.5 The maps des, gdes and θ ....................... 55 5.4.6 Shuffles ................................. 56 ×(n−1) 5.5 The toy example of type A1 ....................... 56 5.5.1 The posets Σn and Ln ......................... 56 5.5.2 The posets Qn and Zn ......................... 57 n n 5.5.3 The quotient posets Q and L .................... 57 5.5.4 ThemapsDes,GDesandΘ. 58 5.5.5 The maps des, gdes and θ ....................... 58 5.6 Thecommutativediagram(5.8). .. 58 5.6.1 Theobjectsindiagram(5.8) . 59 5.6.2 The maps s,ΘandRoad ....................... 60 5.6.3 The bilinear form on KQ ....................... 61 5.6.4 Thetophalfofdiagram(5.8) . 62 5.6.5 The maps supp, lune and base∗ .................... 62 5.6.6 The dual maps supp∗, lune∗ andbase ................ 63 5.6.7 ThemapsΦandΥ........................... 63 5.6.8 Thebottomhalfofdiagram(5.8). 63 5.6.9 The algebra KL............................. 64 5.7 The coinvariant commutative diagram (5.17) . ...... 65 5.7.1 Theobjectsindiagram(5.17). 66 5.7.2 Themapsfrominvariants . 67 5.7.3 Themapstocoinvariants . 69 5.7.4 Themapsindiagram(5.17). 70 5.7.5 The algebra KL............................. 71 5.7.6 A different viewpoint relating diagrams (5.8) and (5.17) ...... 72 6 The construction of Hopf algebras 75 6.1 Introduction................................... 75 6.1.1 AdiagramofvectorspacesforaLRB . 75 6.1.2 A diagram of coalgebras and algebras for a family of LRBs .... 76 6.1.3 The example
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