Combinatorial Hopf algebras of supercharacters of Lie type D

Abstract ii
Acknowledgements in
Table of Contents v
1 Introduction 1
1.1 Preliminaires ............................................................................................. 2
1.1.1 Character theory ........................................................................... 3
1.1.2 Combinatorial Hopf algebras......................................................... 12
1.1.3 Supercharacters .............................................................................. 18
2 Type A 23
2.1 TypeA ......................................................................................................... 24
2.1.1 Superclasses and supercharacters of Un(q ) ................................... 24
2.1.2 The space of superclass functions S C ......................................... 39
2.1.3 Representation theory operations on Un(q) ................................ 40
2.1.4 Product and Coproduct .................................................................. 44
3 Type D 49
3.1 Hopf algebra of type D .............................................................................. 50
3.1.1 Superclasses and supercharacters ............................................... 53
3.1.2 Product and coproduct .................................................................. 56
3.1.3 Sub Hopf algebras of type D ......................................................... 69
4 A coarser supercharacter theory 73
4.1 A coarser supercharacter theory ............................................................... 74
4.2 Species ....................................................................................................... 78
4.2.1 Hopf monoid of set partitions of type D ...................................... 83
4.2.2 Antipode in the P-basis .................................................................. 87
4.2.3 From Hopf monoid to Hopf algebra ............................................ 91
4.2.4 Primitives ....................................................................................... 94
5 Types B and C 98
5.1 types B and C ............................................................................................. 99
5.1.1 ty p e fl .......................................................................................... 99
5.1.2 ty p e C .......................................................................................... 100
6 Future work 103
6.1 Future w ork ................................................................................................ 104