Hopf Algebras in Combinatorics (version Containing Solutions)

Introduction 5 1. What is a Hopf algebra? 7 1.1. Algebras 7 1.2. Coalgebras 8 1.3. Morphisms, tensor products, and bialgebras 9 1.4. Antipodes and Hopf algebras 13 1.5. Commutativity, cocommutativity 20 1.6. Duals 22 2. Review of symmetric functions Λ as Hopf algebra 28 2.1. Definition of Λ 28 2.2. Other Bases 30 2.3. Comultiplications 34 2.4. The antipode, the involution ω, and algebra generators 37 2.5. Cauchy product, Hall inner product, self-duality 39 2.6. Bialternants, Littlewood-Richardson: Stembridge’s concise proof 47 2.7. The Pieri and Assaf-McNamara skew Pieri rule 51 2.8. Skewing and Lam’s proof of the skew Pieri rule 54 2.9. Assorted exercises on symmetric functions 57 3. Zelevinsky’s structure theory of positive self-dual Hopf algebras 71 3.1. Self-duality implies polynomiality 71 3.2. The decomposition theorem 74 3.3. Λ is the unique indecomposable PSH 77 4. Complex representations for Sn, wreath products, GLn(Fq) 83 4.1. Review of complex character theory 83 4.2. Three towers of groups 91 4.3. Bialgebra and double cosets 93 4.4. Symmetric groups 100 4.5. Wreath products 103 4.6. General linear groups 105 4.7. Steinberg’s unipotent characters 106 4.8. Examples: GL2(F2) and GL3(F2) 107 4.9. The Hall algebra 109 5. Quasisymmetric functions and P -partitions 115 5.1. Definitions, and Hopf structure 115 5.2. The fundamental basis and P -partitions 120 5.3. The Hopf algebra NSym dual to QSym 127 6. Polynomial generators for QSym and Lyndon words 134 6.1. Lyndon words 134 6.2. Shuffles and Lyndon words 148 6.3. Radford’s theorem on the shuffle algebra 160 6.4. Polynomial freeness of QSym: statement and easy parts 163